Solution & visualization in The Sturm – Liouville problem

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JASS 2009Alexandra Zykova

superviser: PhD Vadim Monakhov

Department of Computational PhysicsSPSU

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Outline

1. Introduction. The regular Sturm - Liouville problem.

2. The Two – Center Problem in quantum mechanics.

3. SLEIGN23.1 Manual for program package SLEIGN23.2 SLEIGN2 with BARSIC

),()()()()()')'()((

baxxyxwxyxqxyxp

∈=+− λ

(a, b) – open interval

{p, q, w} - coefficients defined on the open interval ( a, b)

λ - spectral parameter

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1. Introduction. The regular Sturm - Liouville problem.

Boundary conditions:

• Regular (R)• Singular (S)• Separated• Coupled

Regular boundary conditions:

• separated boundary conditions

0)()('0)()('

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=+=+

byBbyBayAayA

• coupled boundary conditions

))('())('()()(

bpyapybyay=

=

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Eigen solution of the regular Sturm - Liouville problem -

},{ yλ

eigenvalue, for which differential equation hasnontrivial solution

−λ

−y eigenfunction, corresponds to eigenvalue, satisfies boundary conditions

ψ

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The regular Sturm - Liouville problem & Schrödinger problem

),(),()()()()')'()(( baxxyxwxyxqxyxp ∈=+− λ

(a, b) – is the integration interval and the boundaryconditions

),(),()()(')'( baxxyxyxqxy ∈=+− λ

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(a, b) – is the integration interval and the boundaryconditions

2. The Two – Center Problem in quantum mechanics.

0);()(2);(2

2

1

1 =−−+Δ RrrZ

rZERr ψψ

1r 2r 1Z2Z

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eme ==h

and - nuclear charges

and – distance between electron and ,

1Z

2Z

)(REE= - energy term

R - internuclear distance

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Prolate spheroidal coordinate system

)/arctan(,, 2121 xyR

rrR

rr=

−=

+= ϕηζ

- coordinates of charge

- coordinates of charge

1Z

1,1 −== ηζ 2Z1,1 +== ηζ

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ϕηζϕηζψψ immqmkkqmkqmj eRYRXRNR );();()();,,( ==

- set of quantum numbers

- principal quantum number

- orbital quantum number

- magnetic quantum number

},,{ mqkj =

k

q

m

∫ =V

mmqqkkmqkkqm dVRR '''''' );,,();,,(* δδδϕηξψϕηξψ

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0);(]1

)1([);()1(

0);(]1

)1([);()1(

2

222

~2

2

2222

=−

−+−−−+−

=−

−+−−−+−

RYmbpRYdd

dd

RXmapRXdd

dd

mqmq

mkmk

ηη

ηηληη

ηη

ζζ

ζζλζζ

ζζ

RZZbRZZa

pRE

p jj

)(,)(

)0(2

1221

22

−=+=

>−= - energy parameter

- charge parameter

),()( apmkζλλ = ),()(

~bpmq

ηλλ =

),(),( )()( bpap mqmkηζ λλ =

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, - separation constants

Quasiradial equation

),1[,0);(,);1(

0);(]1

)1([);()1( 2

2222

∞∈∞<

=−

−+−−−+−

→∞→

ζζ

ζζ

ζζλζζ

ζζ

ζ

RXRX

RXmapRXdd

dd

mkmk

mkmk

Jaffé expansion

- transformation of variable

- transformation of equilibrium points

- transformation of interval

)1/()1( +−= ζζt

1,01,1, +→∞→+∞+→−→tζ

)1;0[);1[ ∈→∞∈ tζ

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ss

s

mpmk geRX )

11()1()1();(

1

2/2

+−

+−= Σ∞

=

−

ζζζζζ σζ

)1(2/ +−= mpaδ

- three - termed relation between coefficients sg

011 =+− −+ ssssss ggg γβα

)1)(1(2)1)(()2(2

)1)(1(

δδγλδδδβ

α

−−−−−=+−++−−+=

+++=

msspmmpss

mss

s

s

s

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Eigenvalues of the problem are found from the condition of nullifying of the continued fraction.

0...),,(

2

211

100 =

−−

−=

βγαβ

γαβλapF

)()41(41

2

2

1

1

spO

sp

sss

ss +−→∞→−

−

ββγα

Coefficients of three – termed relation converge for all It follows from the equation above.

0>p

The ratio of the series coefficients

)(211

spO

sp

s

s

s

gg

+−→∞→

+

provides the convergence of Jaffé expansion on the complete interval )1;0[∈t or );1[ ∞∈ζ 14

Quasiangular equation

0);(]1

)1([);()1( 2

222

~2 =

−−+−−−+− RYmbpRY

dd

dd

mqmq ζη

ηηληη

ηη

11;);1( +≤≤−∞<± ηRYmq

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Baber – Hasse expansion

∑∞

=+

−=0

)();(s

mmss

pmq PceRY ηη η

- relation between three – terms coefficients sc

011 =+− −+ ssssss ccc δχρ

1)(2)](2[

)1)((3)(2

))1(2)(12(

−+++

=

−+++=++

++−++=

msmspbs

msmsms

mspbms

s

s

s

δ

λχ

ρ

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The ratio of succeeding series coefficients at large indexes:

0~1 →∞→

+

ss

s

sp

cc

Solutions of can be found from the equation

0...),,(

2

211

100

)( =

−−

−=

kk

kbpFδρ

δρλη

),(),( )()( bpap qmkmηζ λλ =

Continuous fraction converges due to following limit:

2

1

1 2⎟⎠⎞

⎜⎝⎛→

∞→−

−

sp

sss

ss

χχδρ

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General-purpose programs for computing the eigenvalues and eigenfunctions of Sturm - Liouville problem

• Program SLEIGN has been developed by Bailey, Gordon and Shampine , programming language FORTRAN

• Code in the NAG Library has been developed by Pryce and Marletta , programming language FORTRAN

• Program SLEDGE has been developed by Fulton and Pruess , programming language FORTRAN

• Program SLEIGN2 has been developed by Bailey Everitt and Zettl , programming language FORTRAN

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3. SLEIGN2

To meet the needs of numerical computing techniques was made the following assumptions:

1. The interval ( a, b) of R may be bounded or unbounded2. p, q and w are real-valued functions on (a, b)3. p, q and w piecewise continuous on (a, b)4. p and w strictly positive on (a, b)

Conditions on the coefficients:Minimal conditions:

Smoothness conditions:

),(,, 11 baLwqp ∈−

),(,,', baCwqpp ∈

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0)(),( >xwxp

0)(),( >xwxp

Endpoint classification

To classify endpoints a and b, it is convenient to choose a point ),( bac ∈

• а is Regular (R), if

p, q, w - piecewise continuous on [ a, c] p ( a ) > 0,w ( a ) > 0

•А is Singular (S), ifa or

a R, but ∫ +∞=++−c

a

dxxwxqxp )}()())({( 1∈

±∞=

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∞+<<∞− a

• The singular endpoint a is Limit Point (LP) if for some real at least one solution of differential equation satisfies the condition

• The endpoint a is Weakly Regular (WR) if &

∫ +∞=c

a

dxyw 2

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∫ ∞+<++−c

a

dxwqp )( 1

λ

a<∞−

• The singular endpoint is Limit-Circle Non-Oscillatory(LCNO) if for some real value of spectral parameter ALL real-valued solutions satisfy the conditions

and has at most a finite number of zeros in (a, c ]

• The singular endpoint is Limit-Circle Oscillatory(LCO)if for some real value of spectral parameter ALL real-valued solutions satisfy the conditions

and has an infinite number of zeros in (a, c ]

),( λxy

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λ

∞+<∫2c

a

yw

∞+<∫2c

a

yw

),( λxy

λ),( λxy

),( λxy

SLP problems are classified into various classes based on the classification of the endpoints and on whether the boundary conditions are separated (S) or coupled (C). We have the following categories:

1. R/R, Separated2. R/R, Coupled3. R/LCNO LCNO/R, Separated4. R/LCNO LCNO/R, Coupled5. R/LCO LCO/R, Separated6. R/LCO LCO/R, Coupled7. LCNO /LCO LCO/ LCNO LCO/ LCO, Separated 8. LCNO /LCO LCO/ LCNO LCO/ LCO, Coupled9. LP/R LP/LCNO LP/LCO R/LP LCNO/LP LCO/LP10.LP/LP

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The algorithm in SLEIGN2

• Initial interval (a, b) is converted to interval (-1,+1) in the SLEIGN2 package

• The computation procedure is implemented by the use of Prüfer Transform.

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Prüfer Transform.

))(cos()()()'())(sin()()(

xxxpyxxxyθρ

θρ=

=

Differential equation for ρ & θ :

))(cos())(sin())()()(()(/)('))((sin))()(())((cos)()('

1

221

xxxqxwxpxxxxqxwxxpxθθλρρ

θλθθ

+−=

−+=−

−

)/()()/()(

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12

BBarctgnbAAarctga

−=−−=πθ

θ

Boundary conditions for θ :

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The following disadvantages were found in the program package SLEIGN2:

• The interface of the program is organized on the base of console dialog. This approach considerably increase the time for defining the problem paramters.

• The program is unstable towards the input : if numbers are taken in the incorrect format (say, with comma instead of dot) or letters are taking instead of number. In this case the program is terminated and it is required to start work from the very beginning.

• Additional program MAKEPQW is required in order to create own examples.

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Solution in BARSIC (Business And Research Scientific Interactive Calculator)

• Numerical algorithms from SLEIGN2 remain unchanged.Subroutines of SLEIGN2 package (programming language FORTRAN) are compiled into ‘dll’ file (‘so’ files in case of OSLinux ) and then they are called from BARSIC programs.

• Additional functions for calculation of first and second derivatives were created (It’s necessary to write them in FORTRAN when SLEIGN2 is used directly)

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The comparison between SLEIGN2 and well known mathematical packages (Mathematica, Maple, COMSOL Multiphysics) shows that SLEIGN2 is more efficient from the point of view time of computation and numerical errors. Moreover, is not possible to solve problem with coupled conditions in Maple and Mathematica.

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References and resources:

1. Werener O. Amerin, Andress M. Hinz, David B. Pearson “Sturm – Liouville Theory. Past an Present”

2. J.D. Pryce “Numerical solution of Sturm-Liouville Problems” (Oxford University Press; 1993)

3. http://www.docstoc.com/docs/2594692/SLEIGN2

4. http://www.math.niu.edu/SL2

5. A. Devdariani, E. Dalimier “Dipole transition-matrix elements of the one-electron heterodiatomic quasimolecules”

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Thank you for attention!