Research ArticleThe Third Five-Parametric HypergeometricQuantum-Mechanical Potential

T A Ishkhanyan123 and A M Ishkhanyan 12

1Russian-Armenian University Yerevan 0051 Armenia2Institute for Physical Research Ashtarak 0203 Armenia3Moscow Institute of Physics and Technology Dolgoprudny 141700 Russia

Correspondence should be addressed to A M Ishkhanyan aishkhanyangmailcom

Received 12 August 2018 Revised 25 September 2018 Accepted 27 September 2018 Published 17 October 2018

Guest Editor Andrzej Okninski

Copyright copy 2018 T A Ishkhanyan and A M Ishkhanyan This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We introduce the third five-parametric ordinary hypergeometric energy-independent quantum-mechanical potential after theEckart and Poschl-Teller potentials which is proportional to an arbitrary variable parameter and has a shape that is independentof that parameter Depending on an involved parameter the potential presents either a short-range singular well (which behaves asinverse square root at the origin and vanishes exponentially at infinity) or a smooth asymmetric step-barrier (with variable heightand steepness) The general solution of the Schrodinger equation for this potential which is a member of a general Heun familyof potentials is written through fundamental solutions each of which presents an irreducible linear combination of two Gaussordinary hypergeometric functions

1 Introduction

The solutions of the Schrodinger equation in terms of specialmathematical functions for energy-independent potentialswhich are proportional to an arbitrary variable parameter andhave a shape independent of that parameter are very rare [1ndash10] (see the discussion in [11]) It is a common conventionto refer to such potentials as exactly solvable in order todistinguish them from the conditionally integrable ones forwhich a condition is imposed on the potential parameterssuch that the shape of the potential is not independent of thepotential strength (eg a parameter is fixed to a constant ordifferent term-strengths are not varied independently)Whilethere is a relatively large set of potentials of the latter type (seeeg [12ndash20] for some examples discussed in the past and [21ndash25] for some recent examples) the list of the known exactlyintegrable potentials is rather limited even for the potentialsof themost flexible hypergeometric classThe list of the exactlysolvable hypergeometric potentials currently involves onlyten items [1ndash10] Six of these potentials are solved in termsof the confluent hypergeometric functions [1ndash6] These arethe classical Coulomb [1] harmonic oscillator [2] andMorse

[3] potentials and the three recently derived potentials whichare the inverse square root [4] the Lambert-W step [5]and Lambert-W singular [6] potentials The remaining fourexactly integrable potentials which are solved in terms of theGauss ordinary hypergeometric functions are the classicalEckart [7] and Poschl-Teller [8] potentials and the two newpotentials that we have introduced recently [9 10]

An observation worth mentioning here is that all fiveclassical hypergeometric potentials both confluent and ordi-nary involve five arbitrary variable parameters while all newpotentials are four-parametric In this communication weshow that the two four-parametric ordinary hypergeometricpotentials [9 10] are in fact particular cases of a more generalfive-parametric potential which is solved in terms of thehypergeometric functions This generalization thus suggeststhe third five-parametric ordinary hypergeometric quantum-mechanical potential after the ones by Eckart [7] and Poschl-Teller [8]

The potential we introduce belongs to one of the elevenindependent eight-parametric general Heun families [25](see also [26]) From the mathematical point of view apeculiarity of the potential is that this is the only known

HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 2769597 8 pageshttpsdoiorg10115520182769597

2 Advances in High Energy Physics

case when the location of a singularity of the equation towhich the Schrodinger equation is reduced is not fixed to aparticular point but stands for a variable potential-parameterPrecisely in our case the third finite singularity of the Heunequation located at a point 119911 = 119886 of the complex 119911-plane(that is the singularity which is additional if compared withthe ordinary hypergeometric equation) is not fixed but isvariable it stands for the fifth free parameter of the potential

The potential is in general defined parametrically as apair of functions 119881(119911) 119909(119911) However in several cases thecoordinate transformation 119909(119911) is inverted thus producingexplicitly written potentials given as 119881 = 119881(119911(119909)) throughan elementary function 119911 = 119911(119909) All these cases are achievedby fixing the parameter 119886 to a particular value hence all theseparticular potentials are four-parametricThementioned tworecently presented four-parametric ordinary hypergeometricpotentials [9 10] are just such cases

The potential we present is either a singular well (whichbehaves as the inverse square root in the vicinity of theorigin and exponentially vanishes at infinity) or a smoothasymmetric step-barrier (with variable height steepnessand asymmetry) The general solution of the Schrodingerequation for this potential is written through fundamen-tal solutions each of which presents an irreducible linearcombination of two ordinary hypergeometric functions 21198651The singular version of the potential describes a short-rangeinteraction and for this reason supports only a finite numberof bound states We derive the exact equation for energyspectrum and estimate the number of bound states

2 The Potential

The potential is given parametrically as

119881(119911) = 1198810 + 1198811119911 (1)

119909 (119911) = 1199090 + 120590 (119886 ln (119911 minus 119886) minus ln (119911 minus 1)) (2)

where 119886 = 0 1 and 1199090 120590 1198810 1198811 are arbitrary (real or complex)constants Rewriting the coordinate transformation as

(119911 minus 119886)119886119911 minus 1 = 119890(119909minus1199090)120590 (3)

it is seen that for real rational 119886 the transformation is rewrittenas a polynomial equation for 119911 hence in several cases it canbe inverted

Since 119886 = 0 1 the possible simplest case is when thepolynomial equation is quadratic This is achieved for 119886 =minus1 12 2 It is checked however that these three cases leadto four-parametric subpotentials which are equivalent in thesense that each is derived from another by specifications ofthe involved parameters For 119886 = minus1 the potential reads [9]

119881 (119909) = 1198810 + 1198811radic1 + 119890(119909minus1199090)120590 (4)

where we have changed 120590 997888rarr minus120590The next are the cubic polynomial reductions which are

achieved in six cases 119886 = minus2 minus12 13 23 32 3 It is

0 1 2 3 4 5 6 7x

0204060810z(x)

1 2 3 4x

minus20

minus15

minus10

minus5

0V(x)

Figure 1 Potential (1) (2) for 119886 = minus2 and (120590 1199090 1198810 1198811) =(2 0 5 minus5)The inset presents the coordinate transformation 119911(119909) isin(0 1) for 119909 isin (0infin)

again checked however that these choices produce only oneindependent potential This is the four-parametric potentialpresented in [10]

119881 = 1198810 + 1198811119911 119911 = minus1 + 1

(119890119909(2120590) + radic1 + 119890119909120590)23+ (119890119909(2120590) + radic1 + 119890119909120590)23

(5)

where one should replace 119909 by 119909 minus 1199090 Similar potentials interms of elementary functions through quartic and quinticreductions of (3) are rather cumbersome we omit those

For arbitrary real 119886 = 0 1 assuming 119911 isin (0 1) andshifting

1199090 997888rarr 1199090 minus 120590119886 ln (minus119886) + 119894120587120590 (6)

the potential (1) (2) presents a singular well In the vicinity ofthe origin it behaves as 119909minus12

119881|119909997888rarr0 sim radic (119886 minus 1) 1205902119886 1198811radic119909 (7)

and exponentially approaches a constant 1198810 + 1198811 at infinity119881|119909997888rarr+infin sim (119886 minus 1119886 )119886 1198811119890minus119909120590 (8)

The potential and the two asymptotes are shown in Figure 1A potential of a different type is constructed if one allows

the parameterization variable 119911 to vary within the interval 119911 isin(1infin) for 119886 lt 1 or within the interval 119911 isin (1 119886) for 119886 gt 1This time shifting (compare with (6))

1199090 997888rarr 1199090 minus 120590119886 ln (1 minus 119886) (9)

we derive an asymmetric step-barrier the height of whichdepends on 1198810 and 1198811 while the asymmetry and steepness

Advances in High Energy Physics 3

minus2 0 2 4x

246810

z

-12-2

-1

minus10 minus5 5 10x

02

04

06

08

10

minus8 minus6 minus4 minus2 0 2 4x

105110115120125

z

-2-1 -12

minus15 minus10 minus5 5 10 15x

02

04

06

08

10V(x) V(x)

Figure 2 Potential (1) (2) for 119886 = minus2 and (1199090 1198810 1198811) = (0 1 minus1) (left figure) and for 119886 = 125 and (1199090 1198810 1198811) = (0 5 minus5) (right figure)120590 = minus2 minus1 minus12 The fixed points are marked by filled circles The insets present the coordinate transformation 119911(119909) for 120590 = minus1

are controlled by the parameters 119886 and 120590 The shape of thepotential is shown in Figure 2 for 119886 = minus2 and 119886 = 125We note that in the limit 120590 997888rarr 0 the potential turns intothe abrupt-step potential and that the subfamily of barriersgenerated by variation of 120590 at constant 1198810 and 1198811 has a 120590-independent fixed point located at119909 = 1199090 (marked in Figure 2by filled circles)

3 Reduction to the General Heun Equation

The solution of the one-dimensional Schrodinger equationfor potential (1) (2)

11988921205951198891199092 + 2119898ℏ2 (119864 minus 119881 (119909)) 120595 = 0 (10)

is constructed via reduction to the general Heun equation[27ndash29]

11988921199061198891199112 + ( 120574119911 minus 1198861 +120575119911 minus 1198862 +

120576119911 minus 1198863)119889119906119889119911

+ 120572120573119911 minus 119902(119911 minus 1198861) (119911 minus 1198862) (119911 minus 1198863)119906 = 0(11)

The details of the technique are presented in [11 25] Ithas been shown that the energy-independent general-Heunpotentials which are proportional to an arbitrary variableparameter and have shapes which are independent of thatparameter are constructed by the coordinate transformation119911 = 119911(119909) of the Manning form [30] given as

119889119911119889119909 = (119911 minus 1198861)1198981 (119911 minus 1198862)1198982 (119911 minus 1198863)1198983120590 (12)

where119898123 are integers or half-integers and 120590 is an arbitraryscaling constant As it is seen the coordinate transformationis solely defined by the singularities 119886123 of the general Heunequation The canonical form of the Heun equation assumestwo of the three finite singularities at 0 and 1 and the thirdone at a point 119886 so that 119886123 = (0 1 119886) [27ndash29] However itmay be convenient for practical purposes to apply a different

specification of the singularities so for a moment we keep theparameters 119886123 unspecified

The coordinate transformation is followed by the changeof the dependent variable

120595 = (119911 minus 1198861)1205721 (119911 minus 1198862)1205722 (119911 minus 1198863)1205723 119906 (119911) (13)

and application of the ansatz

119881 (119911) = V0 + V1119911 + V21199112 + V31199113 + V41199114(119911 minus 1198861)2 (119911 minus 1198862)2 (119911 minus 1198863)2 (119889119911119889119909)2

V01234 = const(14)

The form of this ansatz and the permissible sets of the param-eters 119898123 are revealed through the analysis of the behaviorof the solution in the vicinity of the finite singularities ofthe general Heun equation [11] This is a crucial point whichwarrants that all the parameters involved in the resultingpotentials can be varied independently

It has been shown that there exist in total thirty-fivepermissible choices for the coordinate transformation eachbeing defined by a triad (1198981 1198982 1198983) satisfying the inequali-ties minus1 le 119898123 le 1 and 1 le 1198981 +1198982 +1198983 le 3 [25] Howeverbecause of the symmetry of the general Heun equation withrespect to the transpositions of its singularities only elevenof the resultant potentials are independent [25]The potential(1) (2) belongs to the fifth independent family with 119898123 =(1 1 minus1) for which from (14) we have

119881 (119911) = 1198814 + 1198813119911 + 11988121199112 + 11988111199113 + 11988101199114(119911 minus 1198863)4 (15)

with arbitrary 11988101234 = const and from (12)

119909 minus 1199090120590 = 1198861 minus 11988631198861 minus 1198862 ln (119911 minus 1198861) + 1198863 minus 11988621198861 minus 1198862 ln (119911 minus 1198862) (16)

4 Advances in High Energy Physics

It is now convenient to have a potential which does notexplicitly involve the singularities Hence we put 1198863 = 0 andapply the specification 119886123 = (119886 1 0) to derive the potential

119881(119911) = 1198810 + 1198811119911 + 11988121199112 + 11988131199113 + 11988141199114 (17)

with(119909 minus 1199090)120590 (119886 minus 1) = 119886 ln (119911 minus 119886) minus ln (119911 minus 1) (18)

The solution of the Schrodinger equation (10) for thispotential is written in terms of the general Heun function119867119866as

120595 = (119911 minus 119886)1205721 1199111205722 (119911 minus 1)1205723sdot 119867119866 (1198861 1198862 1198863 119902 120572 120573 120574 120575 120576 119911) (19)

where the involved parameters 120572 120573 120574 120575 120576 and 119902 are giventhrough the parameters 11988101234 of potential (17) and theexponents 120572123 of the prefactor by the equations [25]

(120574 120575 120576) = (1 + 21205721 1 + 21205722 minus1 + 21205723) (20)

1 + 120572 + 120573 = 120574 + 120575 + 120576120572120573 = (1205721 + 1205722 + 1205723)2 + 21198981205902 (119864 minus 1198810)ℏ2 (21)

119902 = 21198981205902ℏ2 (1198811 minus (1 + 119886) (119864 minus 1198810))+ (minus12057222 + (minus1 + 1205721 + 1205723) (1205721 + 1205723))+ 119886 (minus12057221 + (minus1 + 1205722 + 1205723) (1205722 + 1205723))

(22)

the exponents 120572123 of the prefactor are defined by theequations

12057221 = 211989812059021198862 (119886 minus 1)2 ℏ2 (1198814 + 1198861198813 + 11988621198812 + 11988631198811+ 1198864 (1198810 minus 119864))

(23)

12057222 = minus 21198981205902(119886 minus 1)2 ℏ2 (119864 minus 1198810 minus 1198811 minus 1198812 minus 1198813 minus 1198814) (24)

1205723 (1205723 minus 2) = 2119898120590211988141198862ℏ2 (25)

4 The Solution of the Schroumldinger Equation inTerms of the Gauss Functions

Having determined the parameters of the Heun equationthe next step is to examine the cases when the general Heunfunction119867119866 is written in terms of the Gauss hypergeometricfunctions 21198651 An observation here is that the direct one-termHeun-to-hypergeometric reductions discussed bymanyauthors (see eg [27 28 31ndash34]) are achieved by suchrestrictions and imposed on the involved parameters (threeor more conditions) which are either not satisfied by the

Heun potentials or produce very restrictive potentials It ischecked that the less restrictive reductions reproduce theclassical Eckart and Poschl-Teller potentials while the otherreductions result in conditionally integrable potentials

More advanced are the finite-sum solutions achieved bytermination of the series expansions of the general Heunfunction in terms of the hypergeometric functions [35ndash39]For such reductions only two restrictions are imposed on theinvolved parameters and notably these restrictions are suchthat in many cases they are satisfied The solutions for theabove-mentioned four-parametric subpotentials [9 10] havebeen constructed right in this way Other examples achievedby termination of the hypergeometric series expansions ofthe functions of the Heun class include the recently reportedinverse square root [4] Lambert-W step [5] and Lambert-Wsingular [6] potentials

The series expansions of the general Heun functionin terms of the Gauss ordinary hypergeometric functionsare governed by three-term recurrence relations for thecoefficients of the successive terms of the expansion A usefulparticular expansion in terms of the functions of the form21198651(120572 120573 1205740 minus 119899 119911) which leads to simpler coefficients of therecurrence relation is presented in [25 39] If the expansionfunctions are assumed irreducible to simpler functions thetermination of this series occurs if 120576 = minus119873 119899 = 0 1 2 and a (119873+1)th degree polynomial equation for the accessoryparameter 119902 is satisfied For 120576 = 0 the latter equation is119902 = 119886120572120573 which corresponds to the trivial direct reductionof the general Heun equation to the Gauss hypergeometricequation This case reproduces the classical Eckart andPoschl-Teller potentials [25] For the first nontrivial case 120576 =minus1 the termination condition for singularities 119886123 = (119886 1 0)takes a particularly simple form

1199022 + 119902 (120574 minus 1 + 119886 (120575 minus 1)) + 119886120572120573 = 0 (26)

The solution of the Heun equation for a root of this equationis written as [39]

119906 = 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 120574 minus 1119902 + 119886 (120575 minus 1)sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1)

(27)

This solution has a representation through Clausenrsquos general-ized hypergeometric function 31198652 [40 41]

Consider if the termination condition (26) for 120576 = minus1 issatisfied for the parameters given by (20)-(25) From (20) wefind that for 120576 = minus1 holds 1205723 = 0 It then follows from (25)that 1198814 = 0 With this (26) is reduced to

1198812 + 1198813 (1 + 119886119886 minus 211989812059021198862ℏ2 1198813) = 0 (28)

This equation generally defines a conditionally integrablepotential in that the potential parameters 1198812 and 1198813 are notvaried independently Alternatively if the potential parame-ters are assumed independent the equation is satisfied onlyif 1198812 = 1198813 = 0 Thus we put 119881234 = 0 and potential

Advances in High Energy Physics 5

(17) is reduced to that given by (1) Furthermore since 120590 isarbitrary in order for (18) to exactly reproduce the coordinatetransformation (2) we replace 120590(1 minus 119886) 997888rarr 120590

With this the solution of the Schrodinger equation (10)for potential (1) is written as

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 ( 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1)+ 212057211198861205722 minus 1205721 sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1))

(29)

with (120572 120573 120574) = (1205721 + 1205722 + 1205720 1205721 + 1205722 minus 1205720 1 + 21205721) (30)

120572012 = (plusmnradic21198981205902 (119886 minus 1)2ℏ2 (1198810 minus 119864)

plusmn radic211989812059021198862ℏ2 (1198810 minus 119864 + 1198811119886 )plusmn radic21198981205902ℏ2 (1198810 minus 119864 + 1198811))

(31)

This solution applies for any real or complex set of theinvolved parameters Furthermore we note that any combi-nation for the signs of 12057212 is applicable Hence by choosingdifferent combinations one can construct different indepen-dent fundamental solutions Thus this solution supports thegeneral solution of the Schrodinger equation

A final remark is that using the contiguous functionsrelations for the hypergeometric functions one can replacethe second hypergeometric function in (29) by a linearcombination of the first hypergeometric function and itsderivative In this way we arrive at the following representa-tion of the general solution of the Schrodinger equation

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 (119865 + 119911 minus 1198861205721 + 1198861205722119889119865119889119911 ) (32)

where 119865 = 1198881 sdot 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 1198882sdot 21198651 (120572 120573 1 + 120572 + 120573 minus 120574 119911 minus 1119886 minus 1)

(33)

5 Bound States

Consider the bound states supported by the singular versionof potential (1) (2) achieved by shifting 1199090 997888rarr 1199090 minus120590119886 ln(minus119886) + 119894120587120590 in (2) Since the potential vanishes atinfinity exponentially it is understood that this is a short-range potential The integral of the function 119909119881(119909) over thesemiaxis 119909 isin (0 +infin) is finite hence according to the generalcriterion [42ndash46] the potential supports only a finite numberof bound states These states are derived by demanding thewave function to vanish both at infinity and in the origin (seethe discussion in [47]) We recall that for this potential thecoordinate transformation maps the interval 119909 isin (0 +infin)onto the interval 119911 isin (0 1) Thus we demand 120595(119911 = 0) =120595(119911 = 1) = 0

minus4 minus3 minus2 minus1E

minus10

minus5

5

10S(E)

Figure 3 Graphical representation of the spectrum equation (35)for119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2

The condition119881(+infin) = 0 assumes1198810 +1198811 = 0 hence 1205722is real for negative energy Choosing for definiteness the plussigns in (31) we have 1205722 gt 0 Then examining the equation120595(119911 = 1) = 0 we find that

1205951003816100381610038161003816119911997888rarr1 sim 11988811198601 (1 minus 119911)minus1205722 + 11988821198602 (1 minus 119911)1205722 (34)

with some constants 11986012 Since for positive 1205722 the first termdiverges we conclude 1198881 = 0 The condition 120595(119911 = 0) = 0then gives the following exact equation for the spectrum

119878 (119864) equiv 1 + 1205721 + 11988612057222 (1 minus 119886) 1205722sdot 21198651 (120572 + 1 120573 + 1 1 + 21205722 1 (1 minus 119886))

21198651 (120572 120573 21205722 1 (1 minus 119886)) = 0 (35)

The graphical representation of this equation is shown inFigure 3 The function 119878(119864) has a finite number of zeros Forthe parameters 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 applied in thefigure there are just three bound states

According to the general theory the number of boundstates is equal to the number of zeros (not counting 119909 = 0) ofthe zero-energy solution which vanishes at the origin [42ndash46] We note that for 119864 = 0 the lower parameter of thesecond hypergeometric function in (33) vanishes 1 + 120572 +120573 minus 120574 = 0 Hence a different second independent solutionshould be applied This solution is constructed by using thefirst hypergeometric functionwith1205721 everywhere replaced byminus1205721The result is rather cumbersome It is more convenientlywritten in terms of the Clausen functions as

120595119864=0 = 1198881 (119911 minus 119886)1205721 31198652 (minusradic119886 minus 1119886 1205721 + 1205721 radic 119886 minus 1119886 1205721+ 1205721 1 + 1205721 1205721 1 + 21205721 119886 minus 119911119886 minus 1) + 1198882 (119911 minus 119886)minus1205721sdot 31198652 (minusradic119886 minus 1119886 1205721 minus 1205721 radic 119886 minus 1119886 1205721 minus 1205721 1 minus 1205721119886 minus 1205721119886 1 119911 minus 1119886 minus 1)

(36)

6 Advances in High Energy Physics

5 10 15x

minus2000

0

2000

4000

6000

0(x)

Figure 4 The zero-energy solution for 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2The dashed line shows the logarithmic asymptote at infinity1205950|119909997888rarrinfin sim 119860 + 119861 ln(1 minus 119911)

where 1205721 = radic2119886(119886 minus 1)11989812059021198810ℏ2 and the relation between 1198881and 1198882 is readily derived from the condition 120595119864=0(0) = 0 Thissolution is shown in Figure 4 It is seen that for parameters119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 used in Figure 3 the number ofzeros (excluded the origin) is indeed 3

For practical purposes it is useful to have an estimate forthe number of bound states The absolute upper limit for thisnumber is given by the integral [42 43]

119868119861 = intinfin0

119903 10038161003816100381610038161003816100381610038161003816119881(119909 997888rarr 119903ℏradic2119898)10038161003816100381610038161003816100381610038161003816 119889119903 = (1 minus 119886)sdot (1198711198942 ( 11 minus 119886) + 2119886 cothminus1 (1 minus 2119886)2) 211989812059021198810ℏ2

(37)

where 1198711198942 is Jonquierersquos polylogarithm function of order 2[48 49] Though of general importance however in manycases this is a rather overestimating limit Indeed for theparameters applied in Figure 3 it gives 119899 le 119868119861 asymp 24

More stringent are the estimates by Calogero [44] andChadan [45] which are specialized for everywhere monoton-ically nondecreasing attractive central potentials Calogerorsquosestimate reads 119899 le 119868119862 with [44]

119868119862 = 2120587ℏradic2119898 intinfin0

radicminus119881 (119909)119889119909= (1 + (radic1 minus 119886 minus radicminus119886)2)radic211989812059021198810ℏ2

(38)

We note that 119868119862 asymp radic2119868119861 The result by Chadan further tunesthe upper limit for the number of bound states to the half ofthat by Calogero that is 119899 le 1198681198622 [45] For the parametersapplied in Figure 3 this gives 119899 le 348 which is indeed anaccurate estimate The dependence of the function 119899119888 = 1198681198622on the parameter 119886 for 119886 isin (minusinfin 0) cup (1 +infin) is shown inFigure 5 It is seen that more bound states are available for119886 close to zero The maximum number achieved in the limit119886 997888rarr 0 is radic211989812059021198810ℏ2 hence for sufficiently small 1198810 or 120590such that 211989812059021198810 lt ℏ2 bound states are not possible at all

minus3 minus2 minus1 0 1 2 3a

1

2

3

4

5

6

7nc

Figure 5 The dependence of Chadanrsquos estimate 119899119888 = 1198681198622 for thenumber of bound states on the parameter 119886 (119898 ℏ1198810 120590 = 1 1 5 2)

6 Discussion

Thus we have presented the third five-parametric quantum-mechanical potential for which the solution of theSchrodinger equation is written in terms of the Gaussordinary hypergeometric functions The potential involvesfive (generally complex) parameters which are variedindependently Depending on the particular specificationsof these parameters the potential suggests two differentappearances In one version we have a smooth step-barrierwith variable height steepness and asymmetry while in theother version this is a singular potential-well which behavesas the inverse square root in the vicinity of the origin andexponentially vanishes at infinity

The potential is in general given parametrically how-ever in several cases the involved coordinate transformationallows inversion thus leading to particular potentials whichare explicitly written in terms of elementary functions Thesereductions are achieved by particular specifications of aparameter standing for the third finite singularity of thegeneral Heun equation The resultant subpotentials all arefour-parametric (see eg [9 10]) These particular cases aredefined by coordinate transformations which are roots ofpolynomial equations It turns out that different polynomialequations of the same degree produce the same potential(with altered parameters) The reason for this is well under-stood in the case of quadratic equations In that case thethird singularity of the general Heun equation to which theSchrodinger equation is reduced is specified as 119886 = minus1 12or 2 We then note that the form-preserving transformationsof the independent variable map the four singularities of theHeun equation 119911 = 0 1 119886infin onto the points 119911 = 0 1 1198861infinwith 1198861 adopting one of the six possible values 119886 1119886 1 minus119886 1(1minus119886) 119886(1minus119886) (119886minus1)119886 [27ndash29] It is seen that the triad(minus1 12 2) is a specific set which remains invariant at form-preserving transformations of the independent variable

The potential belongs to the general Heun family119898123 =(1 1 minus1) This family allows several conditionally integrablereductions too [25] A peculiarity of the exactly integrablepotential that we have presented here is that the location of

Advances in High Energy Physics 7

a finite singularity of the general Heun equation is not fixedto a particular point of the complex 119911-plane but serves as avariable potential-parameter In the step-barrier version ofthe potential this parameter stands for the asymmetry of thepotential

The solution of the Schrodinger equation for the poten-tial we have presented is constructed via termination of aseries expansion of the general Heun equation in terms ofthe Gauss ordinary hypergeometric functions The generalsolution of the problem is composed of fundamental solu-tions each of which is an irreducible combination of twohypergeometric functions Several other potentials allowingsolutions of this type have been reported recently [4ndash6 910 23ndash25] Further cases involve the solutions for super-symmetric partner potentials much discussed in the past[15 50 51] and for several nonanalytic potentials discussedrecently [52ndash54] One should distinguish these solutionsfrom the case of reducible hypergeometric functions [55ndash59]when the solutions eventually reduce to quasi-polynomialseg discussed in the context of quasi-exactly solvability[57ndash59] We note that owing to the contiguous functionsrelations [60] the two-term structure of the solution is ageneral property of all finite-sum hypergeometric reductionsof the general Heun functions achieved via termination ofseries solutions It is checked that in our case the linearcombination of the involved Gauss functions is expressedthrough a single generalized hypergeometric function 31198652[40 41]

We have presented the explicit solution of the problemand discussed the bound states supported by the singularversion of the potential We have derived the exact equationfor the energy spectrum and estimated the number of boundstates The exact number of bound states is given by thenumber of zeros of the zero-energy solution which we havealso presented

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the Russian-Armenian(Slavonic) University at the expense of the Ministry of Edu-cation and Science of the Russian Federation the ArmenianScience Committee (SC Grants no 18RF-139 and no 18T-1C276) and the Armenian National Science and Educa-tion Fund (ANSEF Grant no PS-4986) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy inArmenia for a doctoral grant as well as theAgenceuniversitaire de la Francophonie with Armenian ScienceCommittee for a Scientific Mobility grant

References

[1] E Schrodinger ldquoQuantisierung als eigenwertproblemrdquoAnnalender Physik vol 384 no 4 pp 361ndash376 1926

[2] E Schrodinger ldquoQuantisierung als Eigenwertproblem ZweiteMitteilungrdquo Annalen der Physik vol 384 no 6 pp 489ndash5271926

[3] P M Morse ldquoDiatomic molecules according to the wavemechanics II Vibrational levelsrdquo Physical Review A AtomicMolecular and Optical Physics vol 34 no 1 pp 57ndash64 1929

[4] A M Ishkhanyan ldquoExact solution of the Schrodinger equationfor the inverse square root potential V0radicxrdquo EPL (EurophysicsLetters) vol 112 no 1 2015

[5] AM Ishkhanyan ldquoThe Lambert-W step-potential ndash an exactlysolvable confluent hypergeometric potentialrdquo Physics Letters Avol 380 no 5-6 pp 640ndash644 2016

[6] A M Ishkhanyan ldquoA singular Lambert-W Schrodinger poten-tial exactly solvable in terms of the confluent hypergeometricfunctionsrdquo Modern Physics Letters A vol 31 no 33 1650177 11pages 2016

[7] C Eckart ldquoThe penetration of a potential barrier by electronsrdquoPhysical Review A Atomic Molecular and Optical Physics vol35 no 11 article 1303 1930

[8] G Poschl and E Teller ldquoBemerkungen zur Quantenmechanikdes anharmonischen Oszillatorsrdquo Zeitschrift fur Physik vol 83no 3-4 pp 143ndash151 1933

[9] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[10] T A Ishkhanyan V A Manukyan A H Harutyunyan andA M Ishkhanyan ldquoA new exactly integrable hypergeometricpotential for the Schrodinger equationrdquo AIP Advances vol 8no 3 2018

[11] A Ishkhanyan and V Krainov ldquoDiscretization of Natanzonpotentialsrdquo The European Physical Journal Plus vol 131 no 92016

[12] F H Stillinger ldquoSolution of a quantum mechanical eigenvalueproblem with long range potentialsrdquo Journal of MathematicalPhysics vol 20 no 9 pp 1891ndash1895 1979

[13] G P Flessas and A Watt ldquoAn exact solution of the Schrodingerequation for a multiterm potentialrdquo Journal of Physics AMathematical and General vol 14 no 9 pp L315ndashL318 1981

[14] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[15] FCooper J N Ginocchio andAKhare ldquoRelationship betweensupersymmetry and solvable potentialsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 36 no 8 pp2458ndash2473 1987

[16] A De Souza Dutra ldquoConditionally exactly soluble class ofquantum potentialsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 4 pp R2435ndashR2437 1993

[17] R Dutt A Khare and Y P Varshni ldquoNew class of conditionallyexactly solvable potentials in quantum mechanicsrdquo Journal ofPhysics A Mathematical and General vol 28 no 3 pp L107ndashL113 1995

[18] C Grosche ldquoConditionally solvable path integral problemsrdquoJournal of Physics A Mathematical and General vol 28 no 20pp 5889ndash5902 1995

[19] H Exton ldquoThe exact solution of two new types of Schrodingerequationrdquo Journal of Physics A Mathematical and General vol28 no 23 pp 6739ndash6741 1995

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

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Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 3

minus2 0 2 4x

246810

z

-12-2

-1

minus10 minus5 5 10x

02

04

06

08

10

minus8 minus6 minus4 minus2 0 2 4x

105110115120125

z

-2-1 -12

minus15 minus10 minus5 5 10 15x

02

04

06

08

10V(x) V(x)

Figure 2 Potential (1) (2) for 119886 = minus2 and (1199090 1198810 1198811) = (0 1 minus1) (left figure) and for 119886 = 125 and (1199090 1198810 1198811) = (0 5 minus5) (right figure)120590 = minus2 minus1 minus12 The fixed points are marked by filled circles The insets present the coordinate transformation 119911(119909) for 120590 = minus1

are controlled by the parameters 119886 and 120590 The shape of thepotential is shown in Figure 2 for 119886 = minus2 and 119886 = 125We note that in the limit 120590 997888rarr 0 the potential turns intothe abrupt-step potential and that the subfamily of barriersgenerated by variation of 120590 at constant 1198810 and 1198811 has a 120590-independent fixed point located at119909 = 1199090 (marked in Figure 2by filled circles)

3 Reduction to the General Heun Equation

The solution of the one-dimensional Schrodinger equationfor potential (1) (2)

11988921205951198891199092 + 2119898ℏ2 (119864 minus 119881 (119909)) 120595 = 0 (10)

is constructed via reduction to the general Heun equation[27ndash29]

11988921199061198891199112 + ( 120574119911 minus 1198861 +120575119911 minus 1198862 +

120576119911 minus 1198863)119889119906119889119911

+ 120572120573119911 minus 119902(119911 minus 1198861) (119911 minus 1198862) (119911 minus 1198863)119906 = 0(11)

The details of the technique are presented in [11 25] Ithas been shown that the energy-independent general-Heunpotentials which are proportional to an arbitrary variableparameter and have shapes which are independent of thatparameter are constructed by the coordinate transformation119911 = 119911(119909) of the Manning form [30] given as

119889119911119889119909 = (119911 minus 1198861)1198981 (119911 minus 1198862)1198982 (119911 minus 1198863)1198983120590 (12)

where119898123 are integers or half-integers and 120590 is an arbitraryscaling constant As it is seen the coordinate transformationis solely defined by the singularities 119886123 of the general Heunequation The canonical form of the Heun equation assumestwo of the three finite singularities at 0 and 1 and the thirdone at a point 119886 so that 119886123 = (0 1 119886) [27ndash29] However itmay be convenient for practical purposes to apply a different

specification of the singularities so for a moment we keep theparameters 119886123 unspecified

The coordinate transformation is followed by the changeof the dependent variable

120595 = (119911 minus 1198861)1205721 (119911 minus 1198862)1205722 (119911 minus 1198863)1205723 119906 (119911) (13)

and application of the ansatz

119881 (119911) = V0 + V1119911 + V21199112 + V31199113 + V41199114(119911 minus 1198861)2 (119911 minus 1198862)2 (119911 minus 1198863)2 (119889119911119889119909)2

V01234 = const(14)

The form of this ansatz and the permissible sets of the param-eters 119898123 are revealed through the analysis of the behaviorof the solution in the vicinity of the finite singularities ofthe general Heun equation [11] This is a crucial point whichwarrants that all the parameters involved in the resultingpotentials can be varied independently

It has been shown that there exist in total thirty-fivepermissible choices for the coordinate transformation eachbeing defined by a triad (1198981 1198982 1198983) satisfying the inequali-ties minus1 le 119898123 le 1 and 1 le 1198981 +1198982 +1198983 le 3 [25] Howeverbecause of the symmetry of the general Heun equation withrespect to the transpositions of its singularities only elevenof the resultant potentials are independent [25]The potential(1) (2) belongs to the fifth independent family with 119898123 =(1 1 minus1) for which from (14) we have

119881 (119911) = 1198814 + 1198813119911 + 11988121199112 + 11988111199113 + 11988101199114(119911 minus 1198863)4 (15)

with arbitrary 11988101234 = const and from (12)

119909 minus 1199090120590 = 1198861 minus 11988631198861 minus 1198862 ln (119911 minus 1198861) + 1198863 minus 11988621198861 minus 1198862 ln (119911 minus 1198862) (16)

4 Advances in High Energy Physics

It is now convenient to have a potential which does notexplicitly involve the singularities Hence we put 1198863 = 0 andapply the specification 119886123 = (119886 1 0) to derive the potential

119881(119911) = 1198810 + 1198811119911 + 11988121199112 + 11988131199113 + 11988141199114 (17)

with(119909 minus 1199090)120590 (119886 minus 1) = 119886 ln (119911 minus 119886) minus ln (119911 minus 1) (18)

The solution of the Schrodinger equation (10) for thispotential is written in terms of the general Heun function119867119866as

120595 = (119911 minus 119886)1205721 1199111205722 (119911 minus 1)1205723sdot 119867119866 (1198861 1198862 1198863 119902 120572 120573 120574 120575 120576 119911) (19)

where the involved parameters 120572 120573 120574 120575 120576 and 119902 are giventhrough the parameters 11988101234 of potential (17) and theexponents 120572123 of the prefactor by the equations [25]

(120574 120575 120576) = (1 + 21205721 1 + 21205722 minus1 + 21205723) (20)

1 + 120572 + 120573 = 120574 + 120575 + 120576120572120573 = (1205721 + 1205722 + 1205723)2 + 21198981205902 (119864 minus 1198810)ℏ2 (21)

119902 = 21198981205902ℏ2 (1198811 minus (1 + 119886) (119864 minus 1198810))+ (minus12057222 + (minus1 + 1205721 + 1205723) (1205721 + 1205723))+ 119886 (minus12057221 + (minus1 + 1205722 + 1205723) (1205722 + 1205723))

(22)

the exponents 120572123 of the prefactor are defined by theequations

12057221 = 211989812059021198862 (119886 minus 1)2 ℏ2 (1198814 + 1198861198813 + 11988621198812 + 11988631198811+ 1198864 (1198810 minus 119864))

(23)

12057222 = minus 21198981205902(119886 minus 1)2 ℏ2 (119864 minus 1198810 minus 1198811 minus 1198812 minus 1198813 minus 1198814) (24)

1205723 (1205723 minus 2) = 2119898120590211988141198862ℏ2 (25)

4 The Solution of the Schroumldinger Equation inTerms of the Gauss Functions

Having determined the parameters of the Heun equationthe next step is to examine the cases when the general Heunfunction119867119866 is written in terms of the Gauss hypergeometricfunctions 21198651 An observation here is that the direct one-termHeun-to-hypergeometric reductions discussed bymanyauthors (see eg [27 28 31ndash34]) are achieved by suchrestrictions and imposed on the involved parameters (threeor more conditions) which are either not satisfied by the

Heun potentials or produce very restrictive potentials It ischecked that the less restrictive reductions reproduce theclassical Eckart and Poschl-Teller potentials while the otherreductions result in conditionally integrable potentials

More advanced are the finite-sum solutions achieved bytermination of the series expansions of the general Heunfunction in terms of the hypergeometric functions [35ndash39]For such reductions only two restrictions are imposed on theinvolved parameters and notably these restrictions are suchthat in many cases they are satisfied The solutions for theabove-mentioned four-parametric subpotentials [9 10] havebeen constructed right in this way Other examples achievedby termination of the hypergeometric series expansions ofthe functions of the Heun class include the recently reportedinverse square root [4] Lambert-W step [5] and Lambert-Wsingular [6] potentials

The series expansions of the general Heun functionin terms of the Gauss ordinary hypergeometric functionsare governed by three-term recurrence relations for thecoefficients of the successive terms of the expansion A usefulparticular expansion in terms of the functions of the form21198651(120572 120573 1205740 minus 119899 119911) which leads to simpler coefficients of therecurrence relation is presented in [25 39] If the expansionfunctions are assumed irreducible to simpler functions thetermination of this series occurs if 120576 = minus119873 119899 = 0 1 2 and a (119873+1)th degree polynomial equation for the accessoryparameter 119902 is satisfied For 120576 = 0 the latter equation is119902 = 119886120572120573 which corresponds to the trivial direct reductionof the general Heun equation to the Gauss hypergeometricequation This case reproduces the classical Eckart andPoschl-Teller potentials [25] For the first nontrivial case 120576 =minus1 the termination condition for singularities 119886123 = (119886 1 0)takes a particularly simple form

1199022 + 119902 (120574 minus 1 + 119886 (120575 minus 1)) + 119886120572120573 = 0 (26)

The solution of the Heun equation for a root of this equationis written as [39]

119906 = 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 120574 minus 1119902 + 119886 (120575 minus 1)sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1)

(27)

This solution has a representation through Clausenrsquos general-ized hypergeometric function 31198652 [40 41]

Consider if the termination condition (26) for 120576 = minus1 issatisfied for the parameters given by (20)-(25) From (20) wefind that for 120576 = minus1 holds 1205723 = 0 It then follows from (25)that 1198814 = 0 With this (26) is reduced to

1198812 + 1198813 (1 + 119886119886 minus 211989812059021198862ℏ2 1198813) = 0 (28)

This equation generally defines a conditionally integrablepotential in that the potential parameters 1198812 and 1198813 are notvaried independently Alternatively if the potential parame-ters are assumed independent the equation is satisfied onlyif 1198812 = 1198813 = 0 Thus we put 119881234 = 0 and potential

Advances in High Energy Physics 5

(17) is reduced to that given by (1) Furthermore since 120590 isarbitrary in order for (18) to exactly reproduce the coordinatetransformation (2) we replace 120590(1 minus 119886) 997888rarr 120590

With this the solution of the Schrodinger equation (10)for potential (1) is written as

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 ( 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1)+ 212057211198861205722 minus 1205721 sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1))

(29)

with (120572 120573 120574) = (1205721 + 1205722 + 1205720 1205721 + 1205722 minus 1205720 1 + 21205721) (30)

120572012 = (plusmnradic21198981205902 (119886 minus 1)2ℏ2 (1198810 minus 119864)

plusmn radic211989812059021198862ℏ2 (1198810 minus 119864 + 1198811119886 )plusmn radic21198981205902ℏ2 (1198810 minus 119864 + 1198811))

(31)

This solution applies for any real or complex set of theinvolved parameters Furthermore we note that any combi-nation for the signs of 12057212 is applicable Hence by choosingdifferent combinations one can construct different indepen-dent fundamental solutions Thus this solution supports thegeneral solution of the Schrodinger equation

A final remark is that using the contiguous functionsrelations for the hypergeometric functions one can replacethe second hypergeometric function in (29) by a linearcombination of the first hypergeometric function and itsderivative In this way we arrive at the following representa-tion of the general solution of the Schrodinger equation

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 (119865 + 119911 minus 1198861205721 + 1198861205722119889119865119889119911 ) (32)

where 119865 = 1198881 sdot 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 1198882sdot 21198651 (120572 120573 1 + 120572 + 120573 minus 120574 119911 minus 1119886 minus 1)

(33)

5 Bound States

Consider the bound states supported by the singular versionof potential (1) (2) achieved by shifting 1199090 997888rarr 1199090 minus120590119886 ln(minus119886) + 119894120587120590 in (2) Since the potential vanishes atinfinity exponentially it is understood that this is a short-range potential The integral of the function 119909119881(119909) over thesemiaxis 119909 isin (0 +infin) is finite hence according to the generalcriterion [42ndash46] the potential supports only a finite numberof bound states These states are derived by demanding thewave function to vanish both at infinity and in the origin (seethe discussion in [47]) We recall that for this potential thecoordinate transformation maps the interval 119909 isin (0 +infin)onto the interval 119911 isin (0 1) Thus we demand 120595(119911 = 0) =120595(119911 = 1) = 0

minus4 minus3 minus2 minus1E

minus10

minus5

5

10S(E)

Figure 3 Graphical representation of the spectrum equation (35)for119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2

The condition119881(+infin) = 0 assumes1198810 +1198811 = 0 hence 1205722is real for negative energy Choosing for definiteness the plussigns in (31) we have 1205722 gt 0 Then examining the equation120595(119911 = 1) = 0 we find that

1205951003816100381610038161003816119911997888rarr1 sim 11988811198601 (1 minus 119911)minus1205722 + 11988821198602 (1 minus 119911)1205722 (34)

with some constants 11986012 Since for positive 1205722 the first termdiverges we conclude 1198881 = 0 The condition 120595(119911 = 0) = 0then gives the following exact equation for the spectrum

119878 (119864) equiv 1 + 1205721 + 11988612057222 (1 minus 119886) 1205722sdot 21198651 (120572 + 1 120573 + 1 1 + 21205722 1 (1 minus 119886))

21198651 (120572 120573 21205722 1 (1 minus 119886)) = 0 (35)

The graphical representation of this equation is shown inFigure 3 The function 119878(119864) has a finite number of zeros Forthe parameters 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 applied in thefigure there are just three bound states

According to the general theory the number of boundstates is equal to the number of zeros (not counting 119909 = 0) ofthe zero-energy solution which vanishes at the origin [42ndash46] We note that for 119864 = 0 the lower parameter of thesecond hypergeometric function in (33) vanishes 1 + 120572 +120573 minus 120574 = 0 Hence a different second independent solutionshould be applied This solution is constructed by using thefirst hypergeometric functionwith1205721 everywhere replaced byminus1205721The result is rather cumbersome It is more convenientlywritten in terms of the Clausen functions as

120595119864=0 = 1198881 (119911 minus 119886)1205721 31198652 (minusradic119886 minus 1119886 1205721 + 1205721 radic 119886 minus 1119886 1205721+ 1205721 1 + 1205721 1205721 1 + 21205721 119886 minus 119911119886 minus 1) + 1198882 (119911 minus 119886)minus1205721sdot 31198652 (minusradic119886 minus 1119886 1205721 minus 1205721 radic 119886 minus 1119886 1205721 minus 1205721 1 minus 1205721119886 minus 1205721119886 1 119911 minus 1119886 minus 1)

(36)

6 Advances in High Energy Physics

5 10 15x

minus2000

0

2000

4000

6000

0(x)

Figure 4 The zero-energy solution for 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2The dashed line shows the logarithmic asymptote at infinity1205950|119909997888rarrinfin sim 119860 + 119861 ln(1 minus 119911)

where 1205721 = radic2119886(119886 minus 1)11989812059021198810ℏ2 and the relation between 1198881and 1198882 is readily derived from the condition 120595119864=0(0) = 0 Thissolution is shown in Figure 4 It is seen that for parameters119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 used in Figure 3 the number ofzeros (excluded the origin) is indeed 3

For practical purposes it is useful to have an estimate forthe number of bound states The absolute upper limit for thisnumber is given by the integral [42 43]

119868119861 = intinfin0

119903 10038161003816100381610038161003816100381610038161003816119881(119909 997888rarr 119903ℏradic2119898)10038161003816100381610038161003816100381610038161003816 119889119903 = (1 minus 119886)sdot (1198711198942 ( 11 minus 119886) + 2119886 cothminus1 (1 minus 2119886)2) 211989812059021198810ℏ2

(37)

where 1198711198942 is Jonquierersquos polylogarithm function of order 2[48 49] Though of general importance however in manycases this is a rather overestimating limit Indeed for theparameters applied in Figure 3 it gives 119899 le 119868119861 asymp 24

More stringent are the estimates by Calogero [44] andChadan [45] which are specialized for everywhere monoton-ically nondecreasing attractive central potentials Calogerorsquosestimate reads 119899 le 119868119862 with [44]

119868119862 = 2120587ℏradic2119898 intinfin0

radicminus119881 (119909)119889119909= (1 + (radic1 minus 119886 minus radicminus119886)2)radic211989812059021198810ℏ2

(38)

We note that 119868119862 asymp radic2119868119861 The result by Chadan further tunesthe upper limit for the number of bound states to the half ofthat by Calogero that is 119899 le 1198681198622 [45] For the parametersapplied in Figure 3 this gives 119899 le 348 which is indeed anaccurate estimate The dependence of the function 119899119888 = 1198681198622on the parameter 119886 for 119886 isin (minusinfin 0) cup (1 +infin) is shown inFigure 5 It is seen that more bound states are available for119886 close to zero The maximum number achieved in the limit119886 997888rarr 0 is radic211989812059021198810ℏ2 hence for sufficiently small 1198810 or 120590such that 211989812059021198810 lt ℏ2 bound states are not possible at all

minus3 minus2 minus1 0 1 2 3a

1

2

3

4

5

6

7nc

Figure 5 The dependence of Chadanrsquos estimate 119899119888 = 1198681198622 for thenumber of bound states on the parameter 119886 (119898 ℏ1198810 120590 = 1 1 5 2)

6 Discussion

Thus we have presented the third five-parametric quantum-mechanical potential for which the solution of theSchrodinger equation is written in terms of the Gaussordinary hypergeometric functions The potential involvesfive (generally complex) parameters which are variedindependently Depending on the particular specificationsof these parameters the potential suggests two differentappearances In one version we have a smooth step-barrierwith variable height steepness and asymmetry while in theother version this is a singular potential-well which behavesas the inverse square root in the vicinity of the origin andexponentially vanishes at infinity

The potential is in general given parametrically how-ever in several cases the involved coordinate transformationallows inversion thus leading to particular potentials whichare explicitly written in terms of elementary functions Thesereductions are achieved by particular specifications of aparameter standing for the third finite singularity of thegeneral Heun equation The resultant subpotentials all arefour-parametric (see eg [9 10]) These particular cases aredefined by coordinate transformations which are roots ofpolynomial equations It turns out that different polynomialequations of the same degree produce the same potential(with altered parameters) The reason for this is well under-stood in the case of quadratic equations In that case thethird singularity of the general Heun equation to which theSchrodinger equation is reduced is specified as 119886 = minus1 12or 2 We then note that the form-preserving transformationsof the independent variable map the four singularities of theHeun equation 119911 = 0 1 119886infin onto the points 119911 = 0 1 1198861infinwith 1198861 adopting one of the six possible values 119886 1119886 1 minus119886 1(1minus119886) 119886(1minus119886) (119886minus1)119886 [27ndash29] It is seen that the triad(minus1 12 2) is a specific set which remains invariant at form-preserving transformations of the independent variable

The potential belongs to the general Heun family119898123 =(1 1 minus1) This family allows several conditionally integrablereductions too [25] A peculiarity of the exactly integrablepotential that we have presented here is that the location of

Advances in High Energy Physics 7

a finite singularity of the general Heun equation is not fixedto a particular point of the complex 119911-plane but serves as avariable potential-parameter In the step-barrier version ofthe potential this parameter stands for the asymmetry of thepotential

The solution of the Schrodinger equation for the poten-tial we have presented is constructed via termination of aseries expansion of the general Heun equation in terms ofthe Gauss ordinary hypergeometric functions The generalsolution of the problem is composed of fundamental solu-tions each of which is an irreducible combination of twohypergeometric functions Several other potentials allowingsolutions of this type have been reported recently [4ndash6 910 23ndash25] Further cases involve the solutions for super-symmetric partner potentials much discussed in the past[15 50 51] and for several nonanalytic potentials discussedrecently [52ndash54] One should distinguish these solutionsfrom the case of reducible hypergeometric functions [55ndash59]when the solutions eventually reduce to quasi-polynomialseg discussed in the context of quasi-exactly solvability[57ndash59] We note that owing to the contiguous functionsrelations [60] the two-term structure of the solution is ageneral property of all finite-sum hypergeometric reductionsof the general Heun functions achieved via termination ofseries solutions It is checked that in our case the linearcombination of the involved Gauss functions is expressedthrough a single generalized hypergeometric function 31198652[40 41]

We have presented the explicit solution of the problemand discussed the bound states supported by the singularversion of the potential We have derived the exact equationfor the energy spectrum and estimated the number of boundstates The exact number of bound states is given by thenumber of zeros of the zero-energy solution which we havealso presented

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the Russian-Armenian(Slavonic) University at the expense of the Ministry of Edu-cation and Science of the Russian Federation the ArmenianScience Committee (SC Grants no 18RF-139 and no 18T-1C276) and the Armenian National Science and Educa-tion Fund (ANSEF Grant no PS-4986) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy inArmenia for a doctoral grant as well as theAgenceuniversitaire de la Francophonie with Armenian ScienceCommittee for a Scientific Mobility grant

References

[1] E Schrodinger ldquoQuantisierung als eigenwertproblemrdquoAnnalender Physik vol 384 no 4 pp 361ndash376 1926

[2] E Schrodinger ldquoQuantisierung als Eigenwertproblem ZweiteMitteilungrdquo Annalen der Physik vol 384 no 6 pp 489ndash5271926

[3] P M Morse ldquoDiatomic molecules according to the wavemechanics II Vibrational levelsrdquo Physical Review A AtomicMolecular and Optical Physics vol 34 no 1 pp 57ndash64 1929

[4] A M Ishkhanyan ldquoExact solution of the Schrodinger equationfor the inverse square root potential V0radicxrdquo EPL (EurophysicsLetters) vol 112 no 1 2015

[5] AM Ishkhanyan ldquoThe Lambert-W step-potential ndash an exactlysolvable confluent hypergeometric potentialrdquo Physics Letters Avol 380 no 5-6 pp 640ndash644 2016

[6] A M Ishkhanyan ldquoA singular Lambert-W Schrodinger poten-tial exactly solvable in terms of the confluent hypergeometricfunctionsrdquo Modern Physics Letters A vol 31 no 33 1650177 11pages 2016

[7] C Eckart ldquoThe penetration of a potential barrier by electronsrdquoPhysical Review A Atomic Molecular and Optical Physics vol35 no 11 article 1303 1930

[8] G Poschl and E Teller ldquoBemerkungen zur Quantenmechanikdes anharmonischen Oszillatorsrdquo Zeitschrift fur Physik vol 83no 3-4 pp 143ndash151 1933

[9] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[10] T A Ishkhanyan V A Manukyan A H Harutyunyan andA M Ishkhanyan ldquoA new exactly integrable hypergeometricpotential for the Schrodinger equationrdquo AIP Advances vol 8no 3 2018

[11] A Ishkhanyan and V Krainov ldquoDiscretization of Natanzonpotentialsrdquo The European Physical Journal Plus vol 131 no 92016

[12] F H Stillinger ldquoSolution of a quantum mechanical eigenvalueproblem with long range potentialsrdquo Journal of MathematicalPhysics vol 20 no 9 pp 1891ndash1895 1979

[13] G P Flessas and A Watt ldquoAn exact solution of the Schrodingerequation for a multiterm potentialrdquo Journal of Physics AMathematical and General vol 14 no 9 pp L315ndashL318 1981

[14] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[15] FCooper J N Ginocchio andAKhare ldquoRelationship betweensupersymmetry and solvable potentialsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 36 no 8 pp2458ndash2473 1987

[16] A De Souza Dutra ldquoConditionally exactly soluble class ofquantum potentialsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 4 pp R2435ndashR2437 1993

[17] R Dutt A Khare and Y P Varshni ldquoNew class of conditionallyexactly solvable potentials in quantum mechanicsrdquo Journal ofPhysics A Mathematical and General vol 28 no 3 pp L107ndashL113 1995

[18] C Grosche ldquoConditionally solvable path integral problemsrdquoJournal of Physics A Mathematical and General vol 28 no 20pp 5889ndash5902 1995

[19] H Exton ldquoThe exact solution of two new types of Schrodingerequationrdquo Journal of Physics A Mathematical and General vol28 no 23 pp 6739ndash6741 1995

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

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4 Advances in High Energy Physics

It is now convenient to have a potential which does notexplicitly involve the singularities Hence we put 1198863 = 0 andapply the specification 119886123 = (119886 1 0) to derive the potential

119881(119911) = 1198810 + 1198811119911 + 11988121199112 + 11988131199113 + 11988141199114 (17)

with(119909 minus 1199090)120590 (119886 minus 1) = 119886 ln (119911 minus 119886) minus ln (119911 minus 1) (18)

The solution of the Schrodinger equation (10) for thispotential is written in terms of the general Heun function119867119866as

120595 = (119911 minus 119886)1205721 1199111205722 (119911 minus 1)1205723sdot 119867119866 (1198861 1198862 1198863 119902 120572 120573 120574 120575 120576 119911) (19)

where the involved parameters 120572 120573 120574 120575 120576 and 119902 are giventhrough the parameters 11988101234 of potential (17) and theexponents 120572123 of the prefactor by the equations [25]

(120574 120575 120576) = (1 + 21205721 1 + 21205722 minus1 + 21205723) (20)

1 + 120572 + 120573 = 120574 + 120575 + 120576120572120573 = (1205721 + 1205722 + 1205723)2 + 21198981205902 (119864 minus 1198810)ℏ2 (21)

119902 = 21198981205902ℏ2 (1198811 minus (1 + 119886) (119864 minus 1198810))+ (minus12057222 + (minus1 + 1205721 + 1205723) (1205721 + 1205723))+ 119886 (minus12057221 + (minus1 + 1205722 + 1205723) (1205722 + 1205723))

(22)

the exponents 120572123 of the prefactor are defined by theequations

12057221 = 211989812059021198862 (119886 minus 1)2 ℏ2 (1198814 + 1198861198813 + 11988621198812 + 11988631198811+ 1198864 (1198810 minus 119864))

(23)

12057222 = minus 21198981205902(119886 minus 1)2 ℏ2 (119864 minus 1198810 minus 1198811 minus 1198812 minus 1198813 minus 1198814) (24)

1205723 (1205723 minus 2) = 2119898120590211988141198862ℏ2 (25)

4 The Solution of the Schroumldinger Equation inTerms of the Gauss Functions

Having determined the parameters of the Heun equationthe next step is to examine the cases when the general Heunfunction119867119866 is written in terms of the Gauss hypergeometricfunctions 21198651 An observation here is that the direct one-termHeun-to-hypergeometric reductions discussed bymanyauthors (see eg [27 28 31ndash34]) are achieved by suchrestrictions and imposed on the involved parameters (threeor more conditions) which are either not satisfied by the

Heun potentials or produce very restrictive potentials It ischecked that the less restrictive reductions reproduce theclassical Eckart and Poschl-Teller potentials while the otherreductions result in conditionally integrable potentials

More advanced are the finite-sum solutions achieved bytermination of the series expansions of the general Heunfunction in terms of the hypergeometric functions [35ndash39]For such reductions only two restrictions are imposed on theinvolved parameters and notably these restrictions are suchthat in many cases they are satisfied The solutions for theabove-mentioned four-parametric subpotentials [9 10] havebeen constructed right in this way Other examples achievedby termination of the hypergeometric series expansions ofthe functions of the Heun class include the recently reportedinverse square root [4] Lambert-W step [5] and Lambert-Wsingular [6] potentials

The series expansions of the general Heun functionin terms of the Gauss ordinary hypergeometric functionsare governed by three-term recurrence relations for thecoefficients of the successive terms of the expansion A usefulparticular expansion in terms of the functions of the form21198651(120572 120573 1205740 minus 119899 119911) which leads to simpler coefficients of therecurrence relation is presented in [25 39] If the expansionfunctions are assumed irreducible to simpler functions thetermination of this series occurs if 120576 = minus119873 119899 = 0 1 2 and a (119873+1)th degree polynomial equation for the accessoryparameter 119902 is satisfied For 120576 = 0 the latter equation is119902 = 119886120572120573 which corresponds to the trivial direct reductionof the general Heun equation to the Gauss hypergeometricequation This case reproduces the classical Eckart andPoschl-Teller potentials [25] For the first nontrivial case 120576 =minus1 the termination condition for singularities 119886123 = (119886 1 0)takes a particularly simple form

1199022 + 119902 (120574 minus 1 + 119886 (120575 minus 1)) + 119886120572120573 = 0 (26)

The solution of the Heun equation for a root of this equationis written as [39]

119906 = 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 120574 minus 1119902 + 119886 (120575 minus 1)sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1)

(27)

This solution has a representation through Clausenrsquos general-ized hypergeometric function 31198652 [40 41]

Consider if the termination condition (26) for 120576 = minus1 issatisfied for the parameters given by (20)-(25) From (20) wefind that for 120576 = minus1 holds 1205723 = 0 It then follows from (25)that 1198814 = 0 With this (26) is reduced to

1198812 + 1198813 (1 + 119886119886 minus 211989812059021198862ℏ2 1198813) = 0 (28)

This equation generally defines a conditionally integrablepotential in that the potential parameters 1198812 and 1198813 are notvaried independently Alternatively if the potential parame-ters are assumed independent the equation is satisfied onlyif 1198812 = 1198813 = 0 Thus we put 119881234 = 0 and potential

Advances in High Energy Physics 5

(17) is reduced to that given by (1) Furthermore since 120590 isarbitrary in order for (18) to exactly reproduce the coordinatetransformation (2) we replace 120590(1 minus 119886) 997888rarr 120590

With this the solution of the Schrodinger equation (10)for potential (1) is written as

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 ( 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1)+ 212057211198861205722 minus 1205721 sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1))

(29)

with (120572 120573 120574) = (1205721 + 1205722 + 1205720 1205721 + 1205722 minus 1205720 1 + 21205721) (30)

120572012 = (plusmnradic21198981205902 (119886 minus 1)2ℏ2 (1198810 minus 119864)

plusmn radic211989812059021198862ℏ2 (1198810 minus 119864 + 1198811119886 )plusmn radic21198981205902ℏ2 (1198810 minus 119864 + 1198811))

(31)

This solution applies for any real or complex set of theinvolved parameters Furthermore we note that any combi-nation for the signs of 12057212 is applicable Hence by choosingdifferent combinations one can construct different indepen-dent fundamental solutions Thus this solution supports thegeneral solution of the Schrodinger equation

A final remark is that using the contiguous functionsrelations for the hypergeometric functions one can replacethe second hypergeometric function in (29) by a linearcombination of the first hypergeometric function and itsderivative In this way we arrive at the following representa-tion of the general solution of the Schrodinger equation

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 (119865 + 119911 minus 1198861205721 + 1198861205722119889119865119889119911 ) (32)

where 119865 = 1198881 sdot 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 1198882sdot 21198651 (120572 120573 1 + 120572 + 120573 minus 120574 119911 minus 1119886 minus 1)

(33)

5 Bound States

Consider the bound states supported by the singular versionof potential (1) (2) achieved by shifting 1199090 997888rarr 1199090 minus120590119886 ln(minus119886) + 119894120587120590 in (2) Since the potential vanishes atinfinity exponentially it is understood that this is a short-range potential The integral of the function 119909119881(119909) over thesemiaxis 119909 isin (0 +infin) is finite hence according to the generalcriterion [42ndash46] the potential supports only a finite numberof bound states These states are derived by demanding thewave function to vanish both at infinity and in the origin (seethe discussion in [47]) We recall that for this potential thecoordinate transformation maps the interval 119909 isin (0 +infin)onto the interval 119911 isin (0 1) Thus we demand 120595(119911 = 0) =120595(119911 = 1) = 0

minus4 minus3 minus2 minus1E

minus10

minus5

5

10S(E)

Figure 3 Graphical representation of the spectrum equation (35)for119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2

The condition119881(+infin) = 0 assumes1198810 +1198811 = 0 hence 1205722is real for negative energy Choosing for definiteness the plussigns in (31) we have 1205722 gt 0 Then examining the equation120595(119911 = 1) = 0 we find that

1205951003816100381610038161003816119911997888rarr1 sim 11988811198601 (1 minus 119911)minus1205722 + 11988821198602 (1 minus 119911)1205722 (34)

with some constants 11986012 Since for positive 1205722 the first termdiverges we conclude 1198881 = 0 The condition 120595(119911 = 0) = 0then gives the following exact equation for the spectrum

119878 (119864) equiv 1 + 1205721 + 11988612057222 (1 minus 119886) 1205722sdot 21198651 (120572 + 1 120573 + 1 1 + 21205722 1 (1 minus 119886))

21198651 (120572 120573 21205722 1 (1 minus 119886)) = 0 (35)

The graphical representation of this equation is shown inFigure 3 The function 119878(119864) has a finite number of zeros Forthe parameters 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 applied in thefigure there are just three bound states

According to the general theory the number of boundstates is equal to the number of zeros (not counting 119909 = 0) ofthe zero-energy solution which vanishes at the origin [42ndash46] We note that for 119864 = 0 the lower parameter of thesecond hypergeometric function in (33) vanishes 1 + 120572 +120573 minus 120574 = 0 Hence a different second independent solutionshould be applied This solution is constructed by using thefirst hypergeometric functionwith1205721 everywhere replaced byminus1205721The result is rather cumbersome It is more convenientlywritten in terms of the Clausen functions as

120595119864=0 = 1198881 (119911 minus 119886)1205721 31198652 (minusradic119886 minus 1119886 1205721 + 1205721 radic 119886 minus 1119886 1205721+ 1205721 1 + 1205721 1205721 1 + 21205721 119886 minus 119911119886 minus 1) + 1198882 (119911 minus 119886)minus1205721sdot 31198652 (minusradic119886 minus 1119886 1205721 minus 1205721 radic 119886 minus 1119886 1205721 minus 1205721 1 minus 1205721119886 minus 1205721119886 1 119911 minus 1119886 minus 1)

(36)

6 Advances in High Energy Physics

5 10 15x

minus2000

0

2000

4000

6000

0(x)

Figure 4 The zero-energy solution for 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2The dashed line shows the logarithmic asymptote at infinity1205950|119909997888rarrinfin sim 119860 + 119861 ln(1 minus 119911)

where 1205721 = radic2119886(119886 minus 1)11989812059021198810ℏ2 and the relation between 1198881and 1198882 is readily derived from the condition 120595119864=0(0) = 0 Thissolution is shown in Figure 4 It is seen that for parameters119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 used in Figure 3 the number ofzeros (excluded the origin) is indeed 3

For practical purposes it is useful to have an estimate forthe number of bound states The absolute upper limit for thisnumber is given by the integral [42 43]

119868119861 = intinfin0

119903 10038161003816100381610038161003816100381610038161003816119881(119909 997888rarr 119903ℏradic2119898)10038161003816100381610038161003816100381610038161003816 119889119903 = (1 minus 119886)sdot (1198711198942 ( 11 minus 119886) + 2119886 cothminus1 (1 minus 2119886)2) 211989812059021198810ℏ2

(37)

where 1198711198942 is Jonquierersquos polylogarithm function of order 2[48 49] Though of general importance however in manycases this is a rather overestimating limit Indeed for theparameters applied in Figure 3 it gives 119899 le 119868119861 asymp 24

More stringent are the estimates by Calogero [44] andChadan [45] which are specialized for everywhere monoton-ically nondecreasing attractive central potentials Calogerorsquosestimate reads 119899 le 119868119862 with [44]

119868119862 = 2120587ℏradic2119898 intinfin0

radicminus119881 (119909)119889119909= (1 + (radic1 minus 119886 minus radicminus119886)2)radic211989812059021198810ℏ2

(38)

We note that 119868119862 asymp radic2119868119861 The result by Chadan further tunesthe upper limit for the number of bound states to the half ofthat by Calogero that is 119899 le 1198681198622 [45] For the parametersapplied in Figure 3 this gives 119899 le 348 which is indeed anaccurate estimate The dependence of the function 119899119888 = 1198681198622on the parameter 119886 for 119886 isin (minusinfin 0) cup (1 +infin) is shown inFigure 5 It is seen that more bound states are available for119886 close to zero The maximum number achieved in the limit119886 997888rarr 0 is radic211989812059021198810ℏ2 hence for sufficiently small 1198810 or 120590such that 211989812059021198810 lt ℏ2 bound states are not possible at all

minus3 minus2 minus1 0 1 2 3a

1

2

3

4

5

6

7nc

Figure 5 The dependence of Chadanrsquos estimate 119899119888 = 1198681198622 for thenumber of bound states on the parameter 119886 (119898 ℏ1198810 120590 = 1 1 5 2)

6 Discussion

Thus we have presented the third five-parametric quantum-mechanical potential for which the solution of theSchrodinger equation is written in terms of the Gaussordinary hypergeometric functions The potential involvesfive (generally complex) parameters which are variedindependently Depending on the particular specificationsof these parameters the potential suggests two differentappearances In one version we have a smooth step-barrierwith variable height steepness and asymmetry while in theother version this is a singular potential-well which behavesas the inverse square root in the vicinity of the origin andexponentially vanishes at infinity

The potential is in general given parametrically how-ever in several cases the involved coordinate transformationallows inversion thus leading to particular potentials whichare explicitly written in terms of elementary functions Thesereductions are achieved by particular specifications of aparameter standing for the third finite singularity of thegeneral Heun equation The resultant subpotentials all arefour-parametric (see eg [9 10]) These particular cases aredefined by coordinate transformations which are roots ofpolynomial equations It turns out that different polynomialequations of the same degree produce the same potential(with altered parameters) The reason for this is well under-stood in the case of quadratic equations In that case thethird singularity of the general Heun equation to which theSchrodinger equation is reduced is specified as 119886 = minus1 12or 2 We then note that the form-preserving transformationsof the independent variable map the four singularities of theHeun equation 119911 = 0 1 119886infin onto the points 119911 = 0 1 1198861infinwith 1198861 adopting one of the six possible values 119886 1119886 1 minus119886 1(1minus119886) 119886(1minus119886) (119886minus1)119886 [27ndash29] It is seen that the triad(minus1 12 2) is a specific set which remains invariant at form-preserving transformations of the independent variable

The potential belongs to the general Heun family119898123 =(1 1 minus1) This family allows several conditionally integrablereductions too [25] A peculiarity of the exactly integrablepotential that we have presented here is that the location of

Advances in High Energy Physics 7

a finite singularity of the general Heun equation is not fixedto a particular point of the complex 119911-plane but serves as avariable potential-parameter In the step-barrier version ofthe potential this parameter stands for the asymmetry of thepotential

The solution of the Schrodinger equation for the poten-tial we have presented is constructed via termination of aseries expansion of the general Heun equation in terms ofthe Gauss ordinary hypergeometric functions The generalsolution of the problem is composed of fundamental solu-tions each of which is an irreducible combination of twohypergeometric functions Several other potentials allowingsolutions of this type have been reported recently [4ndash6 910 23ndash25] Further cases involve the solutions for super-symmetric partner potentials much discussed in the past[15 50 51] and for several nonanalytic potentials discussedrecently [52ndash54] One should distinguish these solutionsfrom the case of reducible hypergeometric functions [55ndash59]when the solutions eventually reduce to quasi-polynomialseg discussed in the context of quasi-exactly solvability[57ndash59] We note that owing to the contiguous functionsrelations [60] the two-term structure of the solution is ageneral property of all finite-sum hypergeometric reductionsof the general Heun functions achieved via termination ofseries solutions It is checked that in our case the linearcombination of the involved Gauss functions is expressedthrough a single generalized hypergeometric function 31198652[40 41]

We have presented the explicit solution of the problemand discussed the bound states supported by the singularversion of the potential We have derived the exact equationfor the energy spectrum and estimated the number of boundstates The exact number of bound states is given by thenumber of zeros of the zero-energy solution which we havealso presented

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the Russian-Armenian(Slavonic) University at the expense of the Ministry of Edu-cation and Science of the Russian Federation the ArmenianScience Committee (SC Grants no 18RF-139 and no 18T-1C276) and the Armenian National Science and Educa-tion Fund (ANSEF Grant no PS-4986) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy inArmenia for a doctoral grant as well as theAgenceuniversitaire de la Francophonie with Armenian ScienceCommittee for a Scientific Mobility grant

References

[1] E Schrodinger ldquoQuantisierung als eigenwertproblemrdquoAnnalender Physik vol 384 no 4 pp 361ndash376 1926

[2] E Schrodinger ldquoQuantisierung als Eigenwertproblem ZweiteMitteilungrdquo Annalen der Physik vol 384 no 6 pp 489ndash5271926

[3] P M Morse ldquoDiatomic molecules according to the wavemechanics II Vibrational levelsrdquo Physical Review A AtomicMolecular and Optical Physics vol 34 no 1 pp 57ndash64 1929

[4] A M Ishkhanyan ldquoExact solution of the Schrodinger equationfor the inverse square root potential V0radicxrdquo EPL (EurophysicsLetters) vol 112 no 1 2015

[5] AM Ishkhanyan ldquoThe Lambert-W step-potential ndash an exactlysolvable confluent hypergeometric potentialrdquo Physics Letters Avol 380 no 5-6 pp 640ndash644 2016

[6] A M Ishkhanyan ldquoA singular Lambert-W Schrodinger poten-tial exactly solvable in terms of the confluent hypergeometricfunctionsrdquo Modern Physics Letters A vol 31 no 33 1650177 11pages 2016

[7] C Eckart ldquoThe penetration of a potential barrier by electronsrdquoPhysical Review A Atomic Molecular and Optical Physics vol35 no 11 article 1303 1930

[8] G Poschl and E Teller ldquoBemerkungen zur Quantenmechanikdes anharmonischen Oszillatorsrdquo Zeitschrift fur Physik vol 83no 3-4 pp 143ndash151 1933

[9] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[10] T A Ishkhanyan V A Manukyan A H Harutyunyan andA M Ishkhanyan ldquoA new exactly integrable hypergeometricpotential for the Schrodinger equationrdquo AIP Advances vol 8no 3 2018

[11] A Ishkhanyan and V Krainov ldquoDiscretization of Natanzonpotentialsrdquo The European Physical Journal Plus vol 131 no 92016

[12] F H Stillinger ldquoSolution of a quantum mechanical eigenvalueproblem with long range potentialsrdquo Journal of MathematicalPhysics vol 20 no 9 pp 1891ndash1895 1979

[13] G P Flessas and A Watt ldquoAn exact solution of the Schrodingerequation for a multiterm potentialrdquo Journal of Physics AMathematical and General vol 14 no 9 pp L315ndashL318 1981

[14] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[15] FCooper J N Ginocchio andAKhare ldquoRelationship betweensupersymmetry and solvable potentialsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 36 no 8 pp2458ndash2473 1987

[16] A De Souza Dutra ldquoConditionally exactly soluble class ofquantum potentialsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 4 pp R2435ndashR2437 1993

[17] R Dutt A Khare and Y P Varshni ldquoNew class of conditionallyexactly solvable potentials in quantum mechanicsrdquo Journal ofPhysics A Mathematical and General vol 28 no 3 pp L107ndashL113 1995

[18] C Grosche ldquoConditionally solvable path integral problemsrdquoJournal of Physics A Mathematical and General vol 28 no 20pp 5889ndash5902 1995

[19] H Exton ldquoThe exact solution of two new types of Schrodingerequationrdquo Journal of Physics A Mathematical and General vol28 no 23 pp 6739ndash6741 1995

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

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Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 5

(17) is reduced to that given by (1) Furthermore since 120590 isarbitrary in order for (18) to exactly reproduce the coordinatetransformation (2) we replace 120590(1 minus 119886) 997888rarr 120590

With this the solution of the Schrodinger equation (10)for potential (1) is written as

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 ( 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1)+ 212057211198861205722 minus 1205721 sdot 21198651 (120572 120573 120574 minus 1 119886 minus 119911119886 minus 1))

(29)

with (120572 120573 120574) = (1205721 + 1205722 + 1205720 1205721 + 1205722 minus 1205720 1 + 21205721) (30)

120572012 = (plusmnradic21198981205902 (119886 minus 1)2ℏ2 (1198810 minus 119864)

plusmn radic211989812059021198862ℏ2 (1198810 minus 119864 + 1198811119886 )plusmn radic21198981205902ℏ2 (1198810 minus 119864 + 1198811))

(31)

This solution applies for any real or complex set of theinvolved parameters Furthermore we note that any combi-nation for the signs of 12057212 is applicable Hence by choosingdifferent combinations one can construct different indepen-dent fundamental solutions Thus this solution supports thegeneral solution of the Schrodinger equation

A final remark is that using the contiguous functionsrelations for the hypergeometric functions one can replacethe second hypergeometric function in (29) by a linearcombination of the first hypergeometric function and itsderivative In this way we arrive at the following representa-tion of the general solution of the Schrodinger equation

120595 = (119911 minus 119886)1205721 (119911 minus 1)1205722 (119865 + 119911 minus 1198861205721 + 1198861205722119889119865119889119911 ) (32)

where 119865 = 1198881 sdot 21198651 (120572 120573 120574 119886 minus 119911119886 minus 1) + 1198882sdot 21198651 (120572 120573 1 + 120572 + 120573 minus 120574 119911 minus 1119886 minus 1)

(33)

5 Bound States

Consider the bound states supported by the singular versionof potential (1) (2) achieved by shifting 1199090 997888rarr 1199090 minus120590119886 ln(minus119886) + 119894120587120590 in (2) Since the potential vanishes atinfinity exponentially it is understood that this is a short-range potential The integral of the function 119909119881(119909) over thesemiaxis 119909 isin (0 +infin) is finite hence according to the generalcriterion [42ndash46] the potential supports only a finite numberof bound states These states are derived by demanding thewave function to vanish both at infinity and in the origin (seethe discussion in [47]) We recall that for this potential thecoordinate transformation maps the interval 119909 isin (0 +infin)onto the interval 119911 isin (0 1) Thus we demand 120595(119911 = 0) =120595(119911 = 1) = 0

minus4 minus3 minus2 minus1E

minus10

minus5

5

10S(E)

Figure 3 Graphical representation of the spectrum equation (35)for119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2

The condition119881(+infin) = 0 assumes1198810 +1198811 = 0 hence 1205722is real for negative energy Choosing for definiteness the plussigns in (31) we have 1205722 gt 0 Then examining the equation120595(119911 = 1) = 0 we find that

1205951003816100381610038161003816119911997888rarr1 sim 11988811198601 (1 minus 119911)minus1205722 + 11988821198602 (1 minus 119911)1205722 (34)

with some constants 11986012 Since for positive 1205722 the first termdiverges we conclude 1198881 = 0 The condition 120595(119911 = 0) = 0then gives the following exact equation for the spectrum

119878 (119864) equiv 1 + 1205721 + 11988612057222 (1 minus 119886) 1205722sdot 21198651 (120572 + 1 120573 + 1 1 + 21205722 1 (1 minus 119886))

21198651 (120572 120573 21205722 1 (1 minus 119886)) = 0 (35)

The graphical representation of this equation is shown inFigure 3 The function 119878(119864) has a finite number of zeros Forthe parameters 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 applied in thefigure there are just three bound states

According to the general theory the number of boundstates is equal to the number of zeros (not counting 119909 = 0) ofthe zero-energy solution which vanishes at the origin [42ndash46] We note that for 119864 = 0 the lower parameter of thesecond hypergeometric function in (33) vanishes 1 + 120572 +120573 minus 120574 = 0 Hence a different second independent solutionshould be applied This solution is constructed by using thefirst hypergeometric functionwith1205721 everywhere replaced byminus1205721The result is rather cumbersome It is more convenientlywritten in terms of the Clausen functions as

120595119864=0 = 1198881 (119911 minus 119886)1205721 31198652 (minusradic119886 minus 1119886 1205721 + 1205721 radic 119886 minus 1119886 1205721+ 1205721 1 + 1205721 1205721 1 + 21205721 119886 minus 119911119886 minus 1) + 1198882 (119911 minus 119886)minus1205721sdot 31198652 (minusradic119886 minus 1119886 1205721 minus 1205721 radic 119886 minus 1119886 1205721 minus 1205721 1 minus 1205721119886 minus 1205721119886 1 119911 minus 1119886 minus 1)

(36)

6 Advances in High Energy Physics

5 10 15x

minus2000

0

2000

4000

6000

0(x)

Figure 4 The zero-energy solution for 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2The dashed line shows the logarithmic asymptote at infinity1205950|119909997888rarrinfin sim 119860 + 119861 ln(1 minus 119911)

where 1205721 = radic2119886(119886 minus 1)11989812059021198810ℏ2 and the relation between 1198881and 1198882 is readily derived from the condition 120595119864=0(0) = 0 Thissolution is shown in Figure 4 It is seen that for parameters119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 used in Figure 3 the number ofzeros (excluded the origin) is indeed 3

For practical purposes it is useful to have an estimate forthe number of bound states The absolute upper limit for thisnumber is given by the integral [42 43]

119868119861 = intinfin0

119903 10038161003816100381610038161003816100381610038161003816119881(119909 997888rarr 119903ℏradic2119898)10038161003816100381610038161003816100381610038161003816 119889119903 = (1 minus 119886)sdot (1198711198942 ( 11 minus 119886) + 2119886 cothminus1 (1 minus 2119886)2) 211989812059021198810ℏ2

(37)

where 1198711198942 is Jonquierersquos polylogarithm function of order 2[48 49] Though of general importance however in manycases this is a rather overestimating limit Indeed for theparameters applied in Figure 3 it gives 119899 le 119868119861 asymp 24

More stringent are the estimates by Calogero [44] andChadan [45] which are specialized for everywhere monoton-ically nondecreasing attractive central potentials Calogerorsquosestimate reads 119899 le 119868119862 with [44]

119868119862 = 2120587ℏradic2119898 intinfin0

radicminus119881 (119909)119889119909= (1 + (radic1 minus 119886 minus radicminus119886)2)radic211989812059021198810ℏ2

(38)

We note that 119868119862 asymp radic2119868119861 The result by Chadan further tunesthe upper limit for the number of bound states to the half ofthat by Calogero that is 119899 le 1198681198622 [45] For the parametersapplied in Figure 3 this gives 119899 le 348 which is indeed anaccurate estimate The dependence of the function 119899119888 = 1198681198622on the parameter 119886 for 119886 isin (minusinfin 0) cup (1 +infin) is shown inFigure 5 It is seen that more bound states are available for119886 close to zero The maximum number achieved in the limit119886 997888rarr 0 is radic211989812059021198810ℏ2 hence for sufficiently small 1198810 or 120590such that 211989812059021198810 lt ℏ2 bound states are not possible at all

minus3 minus2 minus1 0 1 2 3a

1

2

3

4

5

6

7nc

Figure 5 The dependence of Chadanrsquos estimate 119899119888 = 1198681198622 for thenumber of bound states on the parameter 119886 (119898 ℏ1198810 120590 = 1 1 5 2)

6 Discussion

Thus we have presented the third five-parametric quantum-mechanical potential for which the solution of theSchrodinger equation is written in terms of the Gaussordinary hypergeometric functions The potential involvesfive (generally complex) parameters which are variedindependently Depending on the particular specificationsof these parameters the potential suggests two differentappearances In one version we have a smooth step-barrierwith variable height steepness and asymmetry while in theother version this is a singular potential-well which behavesas the inverse square root in the vicinity of the origin andexponentially vanishes at infinity

The potential is in general given parametrically how-ever in several cases the involved coordinate transformationallows inversion thus leading to particular potentials whichare explicitly written in terms of elementary functions Thesereductions are achieved by particular specifications of aparameter standing for the third finite singularity of thegeneral Heun equation The resultant subpotentials all arefour-parametric (see eg [9 10]) These particular cases aredefined by coordinate transformations which are roots ofpolynomial equations It turns out that different polynomialequations of the same degree produce the same potential(with altered parameters) The reason for this is well under-stood in the case of quadratic equations In that case thethird singularity of the general Heun equation to which theSchrodinger equation is reduced is specified as 119886 = minus1 12or 2 We then note that the form-preserving transformationsof the independent variable map the four singularities of theHeun equation 119911 = 0 1 119886infin onto the points 119911 = 0 1 1198861infinwith 1198861 adopting one of the six possible values 119886 1119886 1 minus119886 1(1minus119886) 119886(1minus119886) (119886minus1)119886 [27ndash29] It is seen that the triad(minus1 12 2) is a specific set which remains invariant at form-preserving transformations of the independent variable

The potential belongs to the general Heun family119898123 =(1 1 minus1) This family allows several conditionally integrablereductions too [25] A peculiarity of the exactly integrablepotential that we have presented here is that the location of

Advances in High Energy Physics 7

a finite singularity of the general Heun equation is not fixedto a particular point of the complex 119911-plane but serves as avariable potential-parameter In the step-barrier version ofthe potential this parameter stands for the asymmetry of thepotential

The solution of the Schrodinger equation for the poten-tial we have presented is constructed via termination of aseries expansion of the general Heun equation in terms ofthe Gauss ordinary hypergeometric functions The generalsolution of the problem is composed of fundamental solu-tions each of which is an irreducible combination of twohypergeometric functions Several other potentials allowingsolutions of this type have been reported recently [4ndash6 910 23ndash25] Further cases involve the solutions for super-symmetric partner potentials much discussed in the past[15 50 51] and for several nonanalytic potentials discussedrecently [52ndash54] One should distinguish these solutionsfrom the case of reducible hypergeometric functions [55ndash59]when the solutions eventually reduce to quasi-polynomialseg discussed in the context of quasi-exactly solvability[57ndash59] We note that owing to the contiguous functionsrelations [60] the two-term structure of the solution is ageneral property of all finite-sum hypergeometric reductionsof the general Heun functions achieved via termination ofseries solutions It is checked that in our case the linearcombination of the involved Gauss functions is expressedthrough a single generalized hypergeometric function 31198652[40 41]

We have presented the explicit solution of the problemand discussed the bound states supported by the singularversion of the potential We have derived the exact equationfor the energy spectrum and estimated the number of boundstates The exact number of bound states is given by thenumber of zeros of the zero-energy solution which we havealso presented

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the Russian-Armenian(Slavonic) University at the expense of the Ministry of Edu-cation and Science of the Russian Federation the ArmenianScience Committee (SC Grants no 18RF-139 and no 18T-1C276) and the Armenian National Science and Educa-tion Fund (ANSEF Grant no PS-4986) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy inArmenia for a doctoral grant as well as theAgenceuniversitaire de la Francophonie with Armenian ScienceCommittee for a Scientific Mobility grant

References

[1] E Schrodinger ldquoQuantisierung als eigenwertproblemrdquoAnnalender Physik vol 384 no 4 pp 361ndash376 1926

[2] E Schrodinger ldquoQuantisierung als Eigenwertproblem ZweiteMitteilungrdquo Annalen der Physik vol 384 no 6 pp 489ndash5271926

[3] P M Morse ldquoDiatomic molecules according to the wavemechanics II Vibrational levelsrdquo Physical Review A AtomicMolecular and Optical Physics vol 34 no 1 pp 57ndash64 1929

[4] A M Ishkhanyan ldquoExact solution of the Schrodinger equationfor the inverse square root potential V0radicxrdquo EPL (EurophysicsLetters) vol 112 no 1 2015

[5] AM Ishkhanyan ldquoThe Lambert-W step-potential ndash an exactlysolvable confluent hypergeometric potentialrdquo Physics Letters Avol 380 no 5-6 pp 640ndash644 2016

[6] A M Ishkhanyan ldquoA singular Lambert-W Schrodinger poten-tial exactly solvable in terms of the confluent hypergeometricfunctionsrdquo Modern Physics Letters A vol 31 no 33 1650177 11pages 2016

[7] C Eckart ldquoThe penetration of a potential barrier by electronsrdquoPhysical Review A Atomic Molecular and Optical Physics vol35 no 11 article 1303 1930

[8] G Poschl and E Teller ldquoBemerkungen zur Quantenmechanikdes anharmonischen Oszillatorsrdquo Zeitschrift fur Physik vol 83no 3-4 pp 143ndash151 1933

[9] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[10] T A Ishkhanyan V A Manukyan A H Harutyunyan andA M Ishkhanyan ldquoA new exactly integrable hypergeometricpotential for the Schrodinger equationrdquo AIP Advances vol 8no 3 2018

[11] A Ishkhanyan and V Krainov ldquoDiscretization of Natanzonpotentialsrdquo The European Physical Journal Plus vol 131 no 92016

[12] F H Stillinger ldquoSolution of a quantum mechanical eigenvalueproblem with long range potentialsrdquo Journal of MathematicalPhysics vol 20 no 9 pp 1891ndash1895 1979

[13] G P Flessas and A Watt ldquoAn exact solution of the Schrodingerequation for a multiterm potentialrdquo Journal of Physics AMathematical and General vol 14 no 9 pp L315ndashL318 1981

[14] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[15] FCooper J N Ginocchio andAKhare ldquoRelationship betweensupersymmetry and solvable potentialsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 36 no 8 pp2458ndash2473 1987

[16] A De Souza Dutra ldquoConditionally exactly soluble class ofquantum potentialsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 4 pp R2435ndashR2437 1993

[17] R Dutt A Khare and Y P Varshni ldquoNew class of conditionallyexactly solvable potentials in quantum mechanicsrdquo Journal ofPhysics A Mathematical and General vol 28 no 3 pp L107ndashL113 1995

[18] C Grosche ldquoConditionally solvable path integral problemsrdquoJournal of Physics A Mathematical and General vol 28 no 20pp 5889ndash5902 1995

[19] H Exton ldquoThe exact solution of two new types of Schrodingerequationrdquo Journal of Physics A Mathematical and General vol28 no 23 pp 6739ndash6741 1995

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

6 Advances in High Energy Physics

5 10 15x

minus2000

0

2000

4000

6000

0(x)

Figure 4 The zero-energy solution for 119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2The dashed line shows the logarithmic asymptote at infinity1205950|119909997888rarrinfin sim 119860 + 119861 ln(1 minus 119911)

where 1205721 = radic2119886(119886 minus 1)11989812059021198810ℏ2 and the relation between 1198881and 1198882 is readily derived from the condition 120595119864=0(0) = 0 Thissolution is shown in Figure 4 It is seen that for parameters119898 ℏ1198810 120590 119886 = 1 1 5 2 minus2 used in Figure 3 the number ofzeros (excluded the origin) is indeed 3

For practical purposes it is useful to have an estimate forthe number of bound states The absolute upper limit for thisnumber is given by the integral [42 43]

119868119861 = intinfin0

119903 10038161003816100381610038161003816100381610038161003816119881(119909 997888rarr 119903ℏradic2119898)10038161003816100381610038161003816100381610038161003816 119889119903 = (1 minus 119886)sdot (1198711198942 ( 11 minus 119886) + 2119886 cothminus1 (1 minus 2119886)2) 211989812059021198810ℏ2

(37)

where 1198711198942 is Jonquierersquos polylogarithm function of order 2[48 49] Though of general importance however in manycases this is a rather overestimating limit Indeed for theparameters applied in Figure 3 it gives 119899 le 119868119861 asymp 24

More stringent are the estimates by Calogero [44] andChadan [45] which are specialized for everywhere monoton-ically nondecreasing attractive central potentials Calogerorsquosestimate reads 119899 le 119868119862 with [44]

119868119862 = 2120587ℏradic2119898 intinfin0

radicminus119881 (119909)119889119909= (1 + (radic1 minus 119886 minus radicminus119886)2)radic211989812059021198810ℏ2

(38)

We note that 119868119862 asymp radic2119868119861 The result by Chadan further tunesthe upper limit for the number of bound states to the half ofthat by Calogero that is 119899 le 1198681198622 [45] For the parametersapplied in Figure 3 this gives 119899 le 348 which is indeed anaccurate estimate The dependence of the function 119899119888 = 1198681198622on the parameter 119886 for 119886 isin (minusinfin 0) cup (1 +infin) is shown inFigure 5 It is seen that more bound states are available for119886 close to zero The maximum number achieved in the limit119886 997888rarr 0 is radic211989812059021198810ℏ2 hence for sufficiently small 1198810 or 120590such that 211989812059021198810 lt ℏ2 bound states are not possible at all

minus3 minus2 minus1 0 1 2 3a

1

2

3

4

5

6

7nc

Figure 5 The dependence of Chadanrsquos estimate 119899119888 = 1198681198622 for thenumber of bound states on the parameter 119886 (119898 ℏ1198810 120590 = 1 1 5 2)

6 Discussion

Thus we have presented the third five-parametric quantum-mechanical potential for which the solution of theSchrodinger equation is written in terms of the Gaussordinary hypergeometric functions The potential involvesfive (generally complex) parameters which are variedindependently Depending on the particular specificationsof these parameters the potential suggests two differentappearances In one version we have a smooth step-barrierwith variable height steepness and asymmetry while in theother version this is a singular potential-well which behavesas the inverse square root in the vicinity of the origin andexponentially vanishes at infinity

The potential is in general given parametrically how-ever in several cases the involved coordinate transformationallows inversion thus leading to particular potentials whichare explicitly written in terms of elementary functions Thesereductions are achieved by particular specifications of aparameter standing for the third finite singularity of thegeneral Heun equation The resultant subpotentials all arefour-parametric (see eg [9 10]) These particular cases aredefined by coordinate transformations which are roots ofpolynomial equations It turns out that different polynomialequations of the same degree produce the same potential(with altered parameters) The reason for this is well under-stood in the case of quadratic equations In that case thethird singularity of the general Heun equation to which theSchrodinger equation is reduced is specified as 119886 = minus1 12or 2 We then note that the form-preserving transformationsof the independent variable map the four singularities of theHeun equation 119911 = 0 1 119886infin onto the points 119911 = 0 1 1198861infinwith 1198861 adopting one of the six possible values 119886 1119886 1 minus119886 1(1minus119886) 119886(1minus119886) (119886minus1)119886 [27ndash29] It is seen that the triad(minus1 12 2) is a specific set which remains invariant at form-preserving transformations of the independent variable

The potential belongs to the general Heun family119898123 =(1 1 minus1) This family allows several conditionally integrablereductions too [25] A peculiarity of the exactly integrablepotential that we have presented here is that the location of

Advances in High Energy Physics 7

a finite singularity of the general Heun equation is not fixedto a particular point of the complex 119911-plane but serves as avariable potential-parameter In the step-barrier version ofthe potential this parameter stands for the asymmetry of thepotential

The solution of the Schrodinger equation for the poten-tial we have presented is constructed via termination of aseries expansion of the general Heun equation in terms ofthe Gauss ordinary hypergeometric functions The generalsolution of the problem is composed of fundamental solu-tions each of which is an irreducible combination of twohypergeometric functions Several other potentials allowingsolutions of this type have been reported recently [4ndash6 910 23ndash25] Further cases involve the solutions for super-symmetric partner potentials much discussed in the past[15 50 51] and for several nonanalytic potentials discussedrecently [52ndash54] One should distinguish these solutionsfrom the case of reducible hypergeometric functions [55ndash59]when the solutions eventually reduce to quasi-polynomialseg discussed in the context of quasi-exactly solvability[57ndash59] We note that owing to the contiguous functionsrelations [60] the two-term structure of the solution is ageneral property of all finite-sum hypergeometric reductionsof the general Heun functions achieved via termination ofseries solutions It is checked that in our case the linearcombination of the involved Gauss functions is expressedthrough a single generalized hypergeometric function 31198652[40 41]

We have presented the explicit solution of the problemand discussed the bound states supported by the singularversion of the potential We have derived the exact equationfor the energy spectrum and estimated the number of boundstates The exact number of bound states is given by thenumber of zeros of the zero-energy solution which we havealso presented

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the Russian-Armenian(Slavonic) University at the expense of the Ministry of Edu-cation and Science of the Russian Federation the ArmenianScience Committee (SC Grants no 18RF-139 and no 18T-1C276) and the Armenian National Science and Educa-tion Fund (ANSEF Grant no PS-4986) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy inArmenia for a doctoral grant as well as theAgenceuniversitaire de la Francophonie with Armenian ScienceCommittee for a Scientific Mobility grant

References

[1] E Schrodinger ldquoQuantisierung als eigenwertproblemrdquoAnnalender Physik vol 384 no 4 pp 361ndash376 1926

[2] E Schrodinger ldquoQuantisierung als Eigenwertproblem ZweiteMitteilungrdquo Annalen der Physik vol 384 no 6 pp 489ndash5271926

[3] P M Morse ldquoDiatomic molecules according to the wavemechanics II Vibrational levelsrdquo Physical Review A AtomicMolecular and Optical Physics vol 34 no 1 pp 57ndash64 1929

[4] A M Ishkhanyan ldquoExact solution of the Schrodinger equationfor the inverse square root potential V0radicxrdquo EPL (EurophysicsLetters) vol 112 no 1 2015

[5] AM Ishkhanyan ldquoThe Lambert-W step-potential ndash an exactlysolvable confluent hypergeometric potentialrdquo Physics Letters Avol 380 no 5-6 pp 640ndash644 2016

[6] A M Ishkhanyan ldquoA singular Lambert-W Schrodinger poten-tial exactly solvable in terms of the confluent hypergeometricfunctionsrdquo Modern Physics Letters A vol 31 no 33 1650177 11pages 2016

[7] C Eckart ldquoThe penetration of a potential barrier by electronsrdquoPhysical Review A Atomic Molecular and Optical Physics vol35 no 11 article 1303 1930

[8] G Poschl and E Teller ldquoBemerkungen zur Quantenmechanikdes anharmonischen Oszillatorsrdquo Zeitschrift fur Physik vol 83no 3-4 pp 143ndash151 1933

[9] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[10] T A Ishkhanyan V A Manukyan A H Harutyunyan andA M Ishkhanyan ldquoA new exactly integrable hypergeometricpotential for the Schrodinger equationrdquo AIP Advances vol 8no 3 2018

[11] A Ishkhanyan and V Krainov ldquoDiscretization of Natanzonpotentialsrdquo The European Physical Journal Plus vol 131 no 92016

[12] F H Stillinger ldquoSolution of a quantum mechanical eigenvalueproblem with long range potentialsrdquo Journal of MathematicalPhysics vol 20 no 9 pp 1891ndash1895 1979

[13] G P Flessas and A Watt ldquoAn exact solution of the Schrodingerequation for a multiterm potentialrdquo Journal of Physics AMathematical and General vol 14 no 9 pp L315ndashL318 1981

[14] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[15] FCooper J N Ginocchio andAKhare ldquoRelationship betweensupersymmetry and solvable potentialsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 36 no 8 pp2458ndash2473 1987

[16] A De Souza Dutra ldquoConditionally exactly soluble class ofquantum potentialsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 4 pp R2435ndashR2437 1993

[17] R Dutt A Khare and Y P Varshni ldquoNew class of conditionallyexactly solvable potentials in quantum mechanicsrdquo Journal ofPhysics A Mathematical and General vol 28 no 3 pp L107ndashL113 1995

[18] C Grosche ldquoConditionally solvable path integral problemsrdquoJournal of Physics A Mathematical and General vol 28 no 20pp 5889ndash5902 1995

[19] H Exton ldquoThe exact solution of two new types of Schrodingerequationrdquo Journal of Physics A Mathematical and General vol28 no 23 pp 6739ndash6741 1995

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Advances in High Energy Physics 7

a finite singularity of the general Heun equation is not fixedto a particular point of the complex 119911-plane but serves as avariable potential-parameter In the step-barrier version ofthe potential this parameter stands for the asymmetry of thepotential

The solution of the Schrodinger equation for the poten-tial we have presented is constructed via termination of aseries expansion of the general Heun equation in terms ofthe Gauss ordinary hypergeometric functions The generalsolution of the problem is composed of fundamental solu-tions each of which is an irreducible combination of twohypergeometric functions Several other potentials allowingsolutions of this type have been reported recently [4ndash6 910 23ndash25] Further cases involve the solutions for super-symmetric partner potentials much discussed in the past[15 50 51] and for several nonanalytic potentials discussedrecently [52ndash54] One should distinguish these solutionsfrom the case of reducible hypergeometric functions [55ndash59]when the solutions eventually reduce to quasi-polynomialseg discussed in the context of quasi-exactly solvability[57ndash59] We note that owing to the contiguous functionsrelations [60] the two-term structure of the solution is ageneral property of all finite-sum hypergeometric reductionsof the general Heun functions achieved via termination ofseries solutions It is checked that in our case the linearcombination of the involved Gauss functions is expressedthrough a single generalized hypergeometric function 31198652[40 41]

We have presented the explicit solution of the problemand discussed the bound states supported by the singularversion of the potential We have derived the exact equationfor the energy spectrum and estimated the number of boundstates The exact number of bound states is given by thenumber of zeros of the zero-energy solution which we havealso presented

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The research was supported by the Russian-Armenian(Slavonic) University at the expense of the Ministry of Edu-cation and Science of the Russian Federation the ArmenianScience Committee (SC Grants no 18RF-139 and no 18T-1C276) and the Armenian National Science and Educa-tion Fund (ANSEF Grant no PS-4986) T A Ishkhanyanacknowledges the support from SPIE through a 2017 Opticsand Photonics Education Scholarship and thanks the FrenchEmbassy inArmenia for a doctoral grant as well as theAgenceuniversitaire de la Francophonie with Armenian ScienceCommittee for a Scientific Mobility grant

References

[1] E Schrodinger ldquoQuantisierung als eigenwertproblemrdquoAnnalender Physik vol 384 no 4 pp 361ndash376 1926

[2] E Schrodinger ldquoQuantisierung als Eigenwertproblem ZweiteMitteilungrdquo Annalen der Physik vol 384 no 6 pp 489ndash5271926

[3] P M Morse ldquoDiatomic molecules according to the wavemechanics II Vibrational levelsrdquo Physical Review A AtomicMolecular and Optical Physics vol 34 no 1 pp 57ndash64 1929

[4] A M Ishkhanyan ldquoExact solution of the Schrodinger equationfor the inverse square root potential V0radicxrdquo EPL (EurophysicsLetters) vol 112 no 1 2015

[5] AM Ishkhanyan ldquoThe Lambert-W step-potential ndash an exactlysolvable confluent hypergeometric potentialrdquo Physics Letters Avol 380 no 5-6 pp 640ndash644 2016

[6] A M Ishkhanyan ldquoA singular Lambert-W Schrodinger poten-tial exactly solvable in terms of the confluent hypergeometricfunctionsrdquo Modern Physics Letters A vol 31 no 33 1650177 11pages 2016

[7] C Eckart ldquoThe penetration of a potential barrier by electronsrdquoPhysical Review A Atomic Molecular and Optical Physics vol35 no 11 article 1303 1930

[8] G Poschl and E Teller ldquoBemerkungen zur Quantenmechanikdes anharmonischen Oszillatorsrdquo Zeitschrift fur Physik vol 83no 3-4 pp 143ndash151 1933

[9] A Ishkhanyan ldquoThe third exactly solvable hypergeometricquantum-mechanical potentialrdquo EPL (Europhysics Letters) vol115 no 2 2016

[10] T A Ishkhanyan V A Manukyan A H Harutyunyan andA M Ishkhanyan ldquoA new exactly integrable hypergeometricpotential for the Schrodinger equationrdquo AIP Advances vol 8no 3 2018

[11] A Ishkhanyan and V Krainov ldquoDiscretization of Natanzonpotentialsrdquo The European Physical Journal Plus vol 131 no 92016

[12] F H Stillinger ldquoSolution of a quantum mechanical eigenvalueproblem with long range potentialsrdquo Journal of MathematicalPhysics vol 20 no 9 pp 1891ndash1895 1979

[13] G P Flessas and A Watt ldquoAn exact solution of the Schrodingerequation for a multiterm potentialrdquo Journal of Physics AMathematical and General vol 14 no 9 pp L315ndashL318 1981

[14] J N Ginocchio ldquoA class of exactly solvable potentials I One-dimensional Schrodinger equationrdquo Annals of Physics vol 152no 1 pp 203ndash219 1984

[15] FCooper J N Ginocchio andAKhare ldquoRelationship betweensupersymmetry and solvable potentialsrdquo Physical Review DParticles Fields Gravitation and Cosmology vol 36 no 8 pp2458ndash2473 1987

[16] A De Souza Dutra ldquoConditionally exactly soluble class ofquantum potentialsrdquo Physical Review A Atomic Molecular andOptical Physics vol 47 no 4 pp R2435ndashR2437 1993

[17] R Dutt A Khare and Y P Varshni ldquoNew class of conditionallyexactly solvable potentials in quantum mechanicsrdquo Journal ofPhysics A Mathematical and General vol 28 no 3 pp L107ndashL113 1995

[18] C Grosche ldquoConditionally solvable path integral problemsrdquoJournal of Physics A Mathematical and General vol 28 no 20pp 5889ndash5902 1995

[19] H Exton ldquoThe exact solution of two new types of Schrodingerequationrdquo Journal of Physics A Mathematical and General vol28 no 23 pp 6739ndash6741 1995

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

8 Advances in High Energy Physics

[20] G Junker and P Roy ldquoConditionally exactly solvable problemsand non-linear algebrasrdquo Physics Letters A vol 232 no 3-4 pp155ndash161 1997

[21] B W Williams ldquoExact solutions of a Schrodinger equationbased on the Lambert functionrdquo Physics Letters A vol 334 no2-3 pp 117ndash122 2005

[22] A Lopez-Ortega ldquoNew conditionally exactly solvable inversepower law potentialsrdquo Physica Scripta vol 90 no 8 p 0852022015

[23] G Junker and P Roy ldquoConditionally exactly solvable potentialsa supersymmetric constructionmethodrdquoAnnals of Physics vol270 no 1 pp 155ndash177 1998

[24] A M Ishkhanyan ldquoA conditionally exactly solvable generaliza-tion of the inverse square root potentialrdquo Physics Letters A vol380 pp 3786ndash3790 2016

[25] A M Ishkhanyan ldquoSchrodinger potentials solvable in terms ofthe generalHeun functionsrdquoAnnals of Physics vol 388 pp 456ndash471 2018

[26] A Lemieux and A K Bose Construction de Potentiels PourLesquels Lrsquoequation De Schrodinger Est Soluble vol 10 Annalesde lrsquoInstitut Henri Poincare 1969

[27] A Ronveaux Heunrsquos Differential Equations Oxford UniversityPress Oxford UK 1995

[28] S Y Slavyanov and W Lay Special Functions A Unified The-ory Based on Singularities Oxford Mathematical MonographsOxford 2000

[29] FW J Olver DW Lozier R F Boisvert and CW ClarkNISTHandbook of Mathematical Functions Cambridge UniversityPress 2010

[30] M F Manning ldquoExact solutions of the schrodinger equationrdquoPhysical Review A Atomic Molecular and Optical Physics vol48 no 2 pp 161ndash164 1935

[31] R S Maier ldquoOn reducing the Heun equation to the hypergeo-metric equationrdquo Journal of Differential Equations vol 213 no1 pp 171ndash203 2005

[32] R Vidunas and G Filipuk ldquoParametric transformationsbetween the Heun and Gauss hypergeometric functionsrdquo Funk-cialaj Ekvacioj Serio Internacia vol 56 no 2 pp 271ndash321 2013

[33] R Vidunas and G Filipuk ldquoA classification of coveringsyielding Heun-to-hypergeometric reductionsrdquoOsaka Journal ofMathematics vol 51 no 4 pp 867ndash903 2014

[34] M van Hoeij and R Vidunas ldquoBelyi functions for hyperbolichypergeometric-to-Heun transformationsrdquo Journal of Algebravol 441 pp 609ndash659 2015

[35] N Svartholm ldquoDie Losung der Fuchsrsquoschen Differentialgle-ichung zweiter Ordnung durch Hypergeometrische PolynomerdquoMathematische Annalen vol 116 no 1 pp 413ndash421 1939

[36] A Erdelyi ldquoThe Fuchsian equation of second order with foursingularitiesrdquo Duke Mathematical Journal vol 9 pp 48ndash581942

[37] A Erdelyi ldquoCertain expansions of solutions of the Heun equa-tionrdquoQuarterly Journal of Mathematics vol 15 pp 62ndash69 1944

[38] E G Kalnins and J Miller ldquoHypergeometric expansions ofHeun polynomialsrdquo SIAM Journal on Mathematical Analysisvol 22 no 5 pp 1450ndash1459 1991

[39] T A Ishkhanyan T A Shahverdyan and A M IshkhanyanldquoExpansions of the Solutions of the General Heun EquationGoverned by Two-Term Recurrence Relations for CoefficientsrdquoAdvances in High Energy Physics vol 2018 Article ID 42636789 pages 2018

[40] J Letessier G Valent and JWimp ldquoSome differential equationssatisfied by hypergeometric functionsrdquo in Approximation andcomputation (West Lafayette IN 1993) vol 119 of Internat SerNumer Math pp 371ndash381 Birkhauser Boston Boston MassUSA 1994

[41] R S Maier ldquoP-symbols Heun identities and 3 F2 identitiesrdquoin Special Functions and Orthogonal Polynomials vol 471 ofContemp Math pp 139ndash159 Amer Math Soc Providence RI2008

[42] V Bargmann ldquoOn the number of bound states in a central fieldof forcerdquo Proceedings of the National Acadamy of Sciences of theUnited States of America vol 38 pp 961ndash966 1952

[43] J Schwinger ldquoOn the bound states of a given potentialrdquoProceedings of the National Acadamy of Sciences of the UnitedStates of America vol 47 pp 122ndash129 1961

[44] F Calogero ldquoUpper and lower limits for the number ofbound states in a given central potentialrdquo Communications inMathematical Physics vol 1 pp 80ndash88 1965

[45] K Chadan ldquoThe asymptotic behaviour of the number of boundstates of a given potential in the limit of large couplingrdquo Il NuovoCimento A vol 58 no 1 pp 191ndash204 1968

[46] F Brau ldquoLimits on the number of bound states and conditionsfor their existencerdquo in Studies in mathematical physics researchpp 1ndash54 Nova Sci Publ New York 2004

[47] M Znojil ldquoComment on ldquoConditionally exactly soluble class ofquantumpotentialsrdquordquoPhysical Review AAtomicMolecularandOptical Physics vol 61 no 6 2000

[48] L Euler ldquoInstitutiones Calculi IntegralisrdquoOpera Omnia vol 11pp 110ndash113 1768

[49] A Jonquiere ldquoNote sur la serierdquo Bulletin de la SocieteMathematique de France vol 17 pp 142ndash152 1889

[50] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995

[51] B K Bagchi Supersymmetry in Quantum And Classical Me-chanics Chapman amp HallCRC 2000

[52] R Sasaki and M Znojil ldquoOne-dimensional Schrodinger equa-tion with non-analytic potential and its exact Bessel-functionsolvabilityrdquo Journal of Physics AMathematical and General vol49 no 44 Article ID 445303 2016

[53] M Znojil ldquoSymmetrized exponential oscillatorrdquo ModernPhysics Letters A vol 31 no 34 1650195 11 pages 2016

[54] M Znojil ldquoMorse potential symmetric Morse potential andbracketed bound-state energiesrdquoModern Physics Letters A vol31 no 14 p 1650088 2016

[55] S K Bose ldquoExact bound states for the central fraction powersingular potentialrdquo Nuovo Cimento vol 109 no 11 pp 1217ndash1220 1994

[56] J Karwowski and H A Witek ldquoBiconfluent Heun equation inquantumchemistry Harmonium and related systemsrdquoTheoret-ical Chemistry Accounts vol 133 no 7 2014

[57] A G Ushveridze Quasi-Exactly Solvable Models in QuantumMechanics IOP Bristol UK 1994

[58] CMBender andMMonou ldquoNewquasi-exactly solvable sexticpolynomial potentialsrdquo Journal of Physics A Mathematical andGeneral vol 38 no 10 pp 2179ndash2187 2005

[59] A V Turbiner ldquoOne-dimensional quasi-exactly solvableSchrodinger equationsrdquo Physics Reports vol 642 pp 1ndash712016

[60] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

High Energy PhysicsAdvances in

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

AstronomyAdvances in

Antennas andPropagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal of

Geophysics

Advances inOpticalTechnologies

Hindawiwwwhindawicom

Volume 2018

Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Hindawiwwwhindawicom Volume 2018

ChemistryAdvances in

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

Submit your manuscripts atwwwhindawicom