IC/80/108

INTERNATIONAL CENTRE FOR

THEORETICAL PHYSICS

THE METHOD OF MOMENTS AMD NESTED HILBERT SPACES

IN QUANTUM MECHANICS

E. Adeniyi Bangudu'

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION 1980 MIRAMARE-TRIESTE

IC/80/1Q8I. INTRODUCTION

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE METHOD OP MOMENTS AHD NESTED HILBERT SPACES IN QUANTUM MECHANICS •

E. Adeniyi Bangudu **

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

I t i s shown how the s t ructures of a nested Hilbert apace Hj, associated

v i th a given Hilbert space HQ, may be used to simplify our understanding of

the effects of parameters, whose values have to be chosen rather than determined

var ia t iona l ly , in the method of moments. The r e s u l t , as applied to non-

r e l a t i v i s t i c quartic o sc i l l a to r and helium atom, i s to associate the parameters

with sequences of Hilbert spaces, while the error of the method of moments

re la t ive to the var ia t ional method corresponds to a nesting operator of the

nested Hilbert space. Dif f icul t ies hindering similar in terpre ta t ions in

terms of rigged Hilbert space s t ructures are highlighted.

MIRAMARE - TRIESTE

August I960

• To be submitted for publicat ion.

• • Permanent address: Department of Mathematics, University of I l o r i n ,F.M.B. 1515, I l o r i n , Kwara S ta t e , Nigeria.

A number of variants of the RayleLgh-Hitz var ia t ional method,

par t i cu la r ly those employing non-symmetric matrices in the set of algebraic

equations,the solutions of which are expected to be good approximations to

the eigenvalues and eigenfunctions of the Schrodinger equations of quantum

systems, are here described as the method of moments. A prominent and un-

desirable feature of th i s method is i t s non-monotonie convergence to the

appropriate eigenvalues v i th increase in the number of basis functions, in

terms of which the approximate eigenfunctions are normally expanded. Since

the method ts supposed to be an a l ternat ive to the Rayleigh-Ritz var ia t ional

method, optimum values for the parameters invariably contained in the basis

functions can no longer be determined by the energy extreraum c r i t e r i o n , and

remain largely but Judiciously chosen; yet t h e i r values affect convergence.

Although convergence, under different values of these parameters, i s

interpreted as an indication of the s t a b i l i t y of the method, the aim of t h i s

paper, in re- invest igat ing the actual applications of the method, i s to remove

t h i s element of a rb i t ra r iness in the values assignable to the parameters.

The method of moments, in i t s most general fora, depends on the

ava i l ab i l i t y of basis vectors that span two different subspaces (the co-

ordinate and projective subspaces) of the same Hilbert space of the problem

being solved. The rigged and nested Hi l t e r t spaces are each defined from

families of Hilbert spaces; hence the novel idea of using them to attempt

the resolution of the doubt as to whether or not the basis vectors employed

in the examples (the quart ic osc i l l a to r and helium atom) actually sat isfy th i s

requirement. In t h i s approach, a well-defined correspondence between the

parameters and Hilbert spaces of a nested Hilbert space i s es tabl ished, under

the standard rea l iza t ion of a Hilbert space as a space of functions. The

nesting operators are also shown to be more effect ive and easily rea l izable -

indicators of the re la t ive error between the method of moments and the

var ia t ional method. Some unresolved d i f f i cu l t i e s are encountered with rigged

Hilbert space.

Sec.II reviews the different forms of the method of moments and give3

de ta i l s of I t s application to the quartie o sc i l l a to r and helium atom problems

as examples. Sec . I l l contains the relevant aspects of the construction of

a nested Hilbert space, par t icu lar ly that associated with a given Hilbert

space, and thus provides the necessary s t ructure in terms of which the method

of moments i s analysed in Sec.IV. The d i f f i cu l t i e s t o be resolved before a

rigged Hilbert space can be similarly applied to the method of moments i s

summarized in Sec.V.

- 2 -

II. THE METHOD OF MOMKSTS

There are a number of approaches to the derivation of linear algebraic

equations for the solution of the Schrb'dinger equation

The hypervirial relations are introduced by Hirschfelder (i960) ,

and shown to "be sufficient for the determination of the state vectors. (An

operator W is described as hypervirial if it does not commute with the

Hamiltonian H hut satisfies the relation

ft (1)

of a quantum-mechanical system. Each starts vith some arbitrary set of complete

basis vectors

, (i = 0,1.2,...) (2 )

ia the Hilbert space Ho of the Hamiltonian operator _H_ of the system and

assumes the expansion of each state vector * as linear combination of the

set (2) in the form

* = ) ^ \ • (3)i = 0

The simplest approach ia to use (3) in (l) and take the scalar product,

in the Hilbert space H-, with each of the basis vectors. This leads to the set

of algebraic equations

T T

CO

where

(5)

(6)

U . J = 0,1,2,•••) -

The various state energies, € k , (k = 0,1,2,...) are obtained as roots of the

determinant equation

(7)

(* , [H, W] 1) = 0

In part icular , i f the ground state vector f- and energy[]

( 6 ) . )

are assumed to beg ,[2]known, Coulson (1965) suggested the expansion of excited state vectors as

in (3) and the use of a sequence {W } (j = 0 ,1 ,2 , . . . ) of linearly independent

hypervirial operators to give the set of algebraic equations^ (It), with 6

replaced by (E^ - 6) and the B. , MJJ now defined as- 6) and the B. , MJ

V (5a)

(6a)

[3]are no longer symmetric matrices. Schwartz's (196?) '"" proposal uses only oneoperator W, with no assumption on any s ta te vector nor energy, but with H,, ,

defined as

V =M J i = (6b)

•here H i s non-symmetric for a non-trivial choice of Vf. This approach ia

the f i r s t under the name, method of moments •

The trans-correlated polydetor approximation of Boys (19-69)

and Handy (19&9) transforms H to the form

H,~tc H S

Boys

(9)

where S3_= exp(L) and L_ ia a real function of the quantum system's co-ordinatesacting as operators, so that M . remains as in (6) but H J becomes

t h a t give a n o n - t r i v i a l . s o l u t i o n of (U) for the i n i t i a l l y unknown expansion

coe f f i c i en t s a . , and the corresponding s t a t e v e c t o r s , ¥ .

(5c)

and iB generally non-symmetric.

- 3 --k-

The Galerkin-Petrov approach (Bangudu, Jankcrwski and Dion (1973))

UMB elements of a second set Of bas i s vectors (x } ( j = 0 , 1 , 2 , . . . ) of the

same Bilbert space

the form

Hn to define the non-symmetric and symmetric Mj in

(5d)

<6d)

The different approaches are easily seen to be equivalent either to the use

of symmetric matrices or to the case in vhich the matrices are not symmetric.

In application, one chooses if^) ( i = 0 , l , . . . , N - l ) as the basis of a f inite (F

dimensional subspace Fj, (called the co-ordinate space) and, for the second

case, another set {XJJ (i = 0 ,1 , . . . ,H- l ) as the basis of a different subspace

GJJ (called the prtjective space) of the same Hilbert space H . The remaining

procedure of determining approximate solutions to (l) by solving (h) i s

theoretically similar to the Galerkin method or Galerking-type method involving

two subspaces (for the second case) of finding approximations to the weak

solution of boundary value problems. In practise, however, the solution of (7)

for the approximate energy values, which are of more significance in quantum

mechanics than approximate state vectors, i s usually sufficient to indicate

the effectiveness of either case.

The general form of the method of moments has received better attention

since it appears to be the closest alternative to the variational method which

cannot be accurately performed (because of the difficult integrals involved)j

particularly when trial basis vectors {+.) include correlation factors. In

such cases the (x.} set of basis vectors are to be selected to lead to more

manageable integrals for the matrix elements. Application of the method to

simple systems has provided fairly good results for the ground and the lower

excited state energy values * . The convergence of the energy values to

the "exact" values with increase in M (the dimension of the subspaces) depends

} and i t is always non-

monotonic; unlike those of the variational method that are always upper bounds

to the "exact" values.

expectedly on the choice of the basis sets {$ } ,

Jankowski (1976) *-9' proved a theorem that i f the basis sets

are orthonormal in their respective subspaces F , G and are such that

CV V = Miwhere (10)

0 ,1 , .

thethen the maximization of any one of^ suggested functions of the M^s (such as

their sum, the sum of. their squares sr their products) should lead to more

reliable results. The i l lustrative computations confirm this ; but

apart from the non-monotonic convergence, there s t i l l remains another unsatisfactory

aspect of the method of moments. Parameters, usually scale factors ,

are employed in the choice of the basis sets {$,} or {x,} (or what corresponds

to the lat ter) , and although their value* affect convergence there i s as yet no

way of determining optimum values for them. Instead, the fact that different,

though Judicious, ehoicesof values for the parameters lead to convergence

(at differing rates) is interpreted as an indication of the stability of the

method. There is thus an obvious need to remove this element of arbitrariness

from this method to make i t more acceptable.

2 . 1

a) The auartic oscillator[6]

Here the set {^t (i = 0 , 1 , 2 , . . . ) of Hermite functions, which

are eigecfunctions of the harmonic oscillator, is chosen as one of the

realizations of orthonormal basis for HQ. They are defined on the real line by

(11)

for the same HQ is obtained by scaling;

where

The orthonormal projective basis set

i . e . , * i Va L

$

(12)

where (l + b) is the scale factor with b taking any real value greater than

-1. The Ramiltonian of the system, in reduced units, is

) 1= T * (13)

-5-- 6 -

and completes the requirements fgr the method of moments, which is expected to

produce i t s test results for optimum value of b^since the M. ncmbers of (10)

are nov the overlap integrals, M. • (XJI +j)"

The XjW's are related to the t ^ W s through

where j i frb)J - l

{HO

This simplifies computation t>ut raises the question of the suitability of

expressing the solution of (l), with H given by (13),as a linear combination

of (11) or even (12) when (ll) and (12) are not so related as expected of

complete orthonoimal systems for !!„. It is easily verified, however, that

reduces to tvo simpler relations:

i ) the relation

(15)

which produces independent functions In H., and

i i ) the application of Gram-Schmidt orthogonaliiation on the x ^ * ) ' 3

(i » 0 ,1 .2 , . . . ) results in the orthonormal set (12).

Yet,in the light of nested Hilbert space associated to an orthooormal

basis of a given Hilbert space, i t is Eq.(l5) that raises the question

Do the sets U . ) and (x.} span the same Hilbert

space HQ as required by all the analysis of the method ^ * (Q)of moments?

- 7 -

b) The helium atom

In Hylleraas variables (s + r2 , t - rx - r £ , u where

0-4 t { « { M ' ) there are many possible choices of basis sets but typicalseta, vhieh produce good results ,are "correlation factor basis" as theco-ordinate basis set {4^} , realized in three-dimensional space as

^ ( s . t . u ) = e"OS s* t 2 n (1 + (16)

and "configuration interaction basis" as the projective basis set (x>} given

(17)

Bote: a, 6 are parameters; U,n = 0 ,1 ,2 , . . . ) and the same positive integralvalue function of Jt and n, 1 • i ( t ;n ) , is normally used to order bothsets (16) and (17).

The Hsalltonian H, in atomic units, is

J - x *_

J_(18)

However, the point of interest is the relation

Xi(s,t) Su)-1 (19)

betveen Xj» * >"i the correlation function ( l + Bu) that contains thedistinguishing parameter 6. (The parameter a is common to both sets{•t> and ix^} and is thus of no consequence in determining the differencebetween them. It is given a fixed value o • 1,85 ^3'10^ vhich compareswell with i ts variatianally derived value a = 1.860, Y • -0.260 fromcorrelated Kylleraas function e -1 f11 e~ a s

for ground state helium

atom,) Because of its similarity to (15), Eq.(19) equally raises question (Q).

- 8 -

i

III. NESTED HILBERT SPACE (23)

This section contains a brief review of the concepts and properties

of nested Hilbert space relevant for our purpose. The main theorem is proved

in the appendix; for other details see Grossman

A linear transformation E,. from an infinitely dimensional,do

separable Hilbert space into another Hilbert space Hd i3 called a

neating if it is bounded, injective, and its range is dense in H, (e.g. if

t(b'fe H. , then there exists one T( d )e H, such that + ( d ) = E.. * ( b ) ) . Its

0 g d dbadjoint t (E-. ). ,» defined bydo od

is also a nesting- Note that at the left-hand aide of (20) the scalar

product is in H and i t is in Hd at the right-hand side.

The polar decomposition of E_ , given bydo

(20)

db U db

where

,1/2(21)

is injectire and L1

their product E ,

is unitary. If E and are nestings, then

i s a nesting from E into H . E. = 1 is the identity

Let I lie a part ia l ly ordered set that is directed to the r ight , has

an order-reversing involution b*-*-b = b and.contains an element 0 such that

0 = 0. For every b 6 l , l e t V be a vector space. Define an equivalence

relation in the disjoint union U V. by writing • « T (where

+ l ' 6 V , Tl t U 6 V ) If and only i f there exists a d 3. a,b such that

E_30

Ed a

The set of classes to which is thus decomposed

forms a vector space denoted by V and called the algebraic inductive limit

of the vector spaces ¥. with respect to E« and I ;

I.e. V\, = [V Edb '

For every Toil, define the natural embedding operator of V into

(22)

b y

vhere Vx, • £ V • ( i . e . for every 4 fiV., E_ associates the1 0 0 ID »^i

t he(b)same + considered as an element of V_) and «al l $representative of if in Vb.

The algebraic inductive limit Hj = [ l^; E^ ; i ] of a family of

Hilbert spaces H, is called a nested Hilbert space i f the following

conditions are satisfied:

( i ) If b and d are any two elements of I , then there exists an

a 5 b,d such that

Ex Hla a. B I b H b A E I d H d (21*)

Cii) For every b « I , there exists a unitary mapping Ur- from H.

onto Hr such that

uoo(25)

(be I ; d > 6)

3.1 Heated Hilbert space HT associated with a given Hilbert space H-

Let {+| } (i = 0 ,1 ,2 , . . . 1 be the orthonormal basis of a. given Hilbert

space HQ and I the set of al l sequences of s t r i c t ly positive numbers. Hence

b S I implies b « {b(i}} Is a sequence of numbers with b( i ) > 0 for a l l i .

Defining the par t ia l order in I by d ^ b implies d(i) j ^ b ( i ) , and the order

reversing involution is defined by

for a l l I .

Hote t h a t for 0 = 6 t o o b t a i n , p ( i ) must be equa.1 t o one for a l l 1 .

(26)

For every b £ I l e t

combinations of the vectors

V be the vector space of all finite linear

+ } with s c a l a r product defined by

(21)

-9--10-

where +[0 ); are elects of

Denote by 11 , the Hilbert space obtained by completing V with

raepeot to the scalar product (27 ) , and by E ^ the natural embedding of

into H. for a l l d > b that belong to I . Observe, from (27 ) , that the

scalar product in }L has the form of the scalar product in H

where 4, ( 0 )

(2S)

a^ ^ ; tf> = y B $ ' are now elements of H ;

i ibut the sequence (b(i)} effects the possibility of defining the scalar

product between two functions both of which belong to H. while they need

not belong to H .

The algebraic inductive limit HT = [H. ; E,. ; I] is the nested HilbertI b db , . •. ,

space associated to H_. It follows from the relation *. = ET_ *; thatr-r\ 0 /_\ 1 ,. . ID 1

the vectors ^V"' = E +| in Kj have representation $\ in

orthonormal basis can be shown from (2T) and (28) to be the set

{•>'} = {b(i) +Jb)} , (i = 0 ,1 ,2 , . . . ) -

whose

(29)

From now on we make the simplifying

assumption that the order in I i s total ( i . e . a l l pairs of i t s elements are

comparable). In particulaty every other element i s comparable vith the zero

^lenent^ 0 » 0 . The scalar product of + € H. and y'°'ff HQ i s then a

bilinear functional B(+ , <p ) on 1^ and Ho; and for i t s correct

evaluation, we need the nesting £ „ or i t s adjoint (E.Q) b . so that

(30)

IV. THE EXAMPLES IN TERMS OF NESTED HILBERT SPACE

The preceding section allowed us to construct a nested Hilbert space,

H r from a given Hilbert space HQ- The family of Hilbert spaces, employed

in the construction, come in pairs 1^, Kg through the order-reversing

involution defined for al l elements bftl (the set of all totally ordered #

sequences of_strictly positive numbers); Just like the nestings E^ , ( E ^ ) ^

for any pair of elements d,be I vith d :> b. In this section we need

only the family b e l with b ^ J; and shall realize the

corresponding Hilbert space, 1^, as the space of classes of real valued

functions L ftRH,T) defined on N-dimensional space, measurable and square

integrable with respect to thtmeasure T, with the usual scalar product.

!*.l The quartic o sc i l l a to r

' In Eq.(l5) denote x jd> by t ^ U ) and ^ ( i ) by +[ 0 ) (x) so t h a t ,

in the notation of S e c . I l l , t h i s equation becomes

+[0)(

[- | (2b + b2) (15a)

$ +| }Identify $. as an element of 1^ t HQ and {+| } remains the set

of orthonormal basis of H , while the nestings E^ and i t s adjoint ( ^

are, .respectively, realized on It as

EbQ(i) = exp[- \ (2b + b2) X2] (31)

(32)

How, the integral

00

is obtained by taking scalar product either in Hn or in E , In practise,(b)

we s t i l l prefer to expand • 6 H. as linear combination of the orthonormalbasis vectors {,(,} ) of E, and take scalar product in H_; hence we need thenesting ( E ^ ) ^ at least.

is defined for al l real values of b. Therefore if,2,^ ,_-2

L (B,T), then we

can identify H. with H, = L2(K,u) = L2(IB,b"2 i) since the Lebesque measure

in t is related to the Lebesque me

b2(x) (i.e. dp(x) = b"2(i) di), where

V in t is related to the Lebesque measure in HQ through the function

b"2() (ie dp(x) = b"2( )

b(x) = exp[| (2b + ] (33)

-11-

-12-

Note that b(x) is a continuous function on JR, takes strictly positive values andis therefore an element of I since i t satisfies the total ordering in I althoughI 1B now a sequence of equal numbers, for fixed x.

Thus the direct answer to (Q) i s that R la not indentical with H».In fact for b >, 0, the assertion I 3 L i s a direct consequence of a lemmathat i f d,b are elements of I and d >. b then ETJ H, Zi E_ H. , where

Id d -** Ib D

Eja , Ej^ are, respectively, the embedding operators of Hd and H Into theresulting nested Hilbert space, Hj, of the family of Hilbert spaces. Further-more, the relations

[11]

show that H. is sufficiently spanned by the orthonormal basis of HQ. This is

expected because VL Is the completion of V , a vector space of all finite(0)

linear combinations of the vectors(0)} with scalar product defined by (27).

It confirms the nesting IL-fx) of (31) as a natural emcedding of BQ into tt( i . e . E. „ associates the same vector t< belonging to H, as an element of

V and that Hn C H. .

The effect of Grm-Schmidt orthogonalisation of the set of vectors(*i (x)} is to produce the orthonormal set (Xj(x)} ot (12)^but the integrals

^o»

where y = (l+b)x, show that (12) spans the Hilbert space H = L2(E,(l+b)T)which corresponds to the result of a simple choice of another element cfelgiven by *

= (1 + b)-1/2

with b > - 1 .

Again, e[ c ){x) = = x^x) ••

imply that H is spanned by the orthonormal basis HQ; also

(3<0

and

H contains or is contained in H^ according to whether c is greater or less

than b. The nesting E,- and Its adjoint (E Q ) Q are

cO

^o'oc

+ b ) 1 / 2

+ b)" 1 / £

(35)

(36)

Considerations baaed on Eq.(33) show that continuous functions takingstrictly positive values qualify as elements of I; but since elements oftype (3M are used in computations under consideration ' ', we restrict ourattention to this . The method of moments requires different subspaces of thesane Hilbert apace. The spaces Hc and HQ are not identical except when b => 0(making c - 1 - c by (26) and definition of HQ). Using finite sets fx.J^n ,^iU-O t h * t • " °P e r a t ive In H, and HQ, respectively, cannot therefore bestrictly interpreted as spanning different subspacea of the same Hilbert space.The facts that HQ c.Hc and both are effectively spanned by the set U*0 '}allow convergence of the method of moments calculations in the "mixed" Hilbertspaces BQ and He. Such calculations Invariably do not evaluate the bilinearfunctions B($ , Y > involved correctly - by not employing the adjointnesting operator (EcO>Oc - but this operator as identified In Eq.(36) happensto be a constant in the problem and therefore does not lead to error in thesolution of .the determinant equation (7). Yet, monotonically decreasing andcomparable convergence rate is demonstrably obtained when computations areperformed In th« sane Hilbert space. See Table 2 of Hef.6 reproduced below.Also, Eqs.(36), (30) and (10) give the K± numbers for HQ and H as

b) 1 / 2 for all

All the suggested functions

that of

[9]

But M. = 1 for all

then Identical with H as noted earlier.

of tha H±'B at tain the ir maximum values with

i when b - O, so that c • 1 and H ia

0

**.2 The helium atom

The same analysis for the luartic o s c i l l a t o r can be repeated for

the helium atom computations. Eq.' l?) defines Independent functions in

Denote their orthonormalized vers! ,n by KJO), M)f

(s,t,u) and the independent

f u n c t i o n s (1 + Bu) +! ( s . t . u ) by * J p ; ( s , t , u ) no t h a t E q . ( l 9 ) becomes

(19a)

- 1 3 -

Table 2 of Ref.6

Five lowest energy levels of even symmetry for the quarticoscillator for several b values and two expansion lengths (in reducedunits).

-0.5

o.ob'

0.5

1.0

2.0

5Enorgy

1.060O18.23553

34.oa70123.2*6389-57*

1.061317.47162

18.820452.906*

155-933

1.060357.44909

16.091526.714660.785*

1.060247.4J1 S5

16.2217!..«/i n .

1.035656.66191

20.3303l a .I n .

A R /

- 0.00035+ 0.778S3+ 17.825• 96.71B+351.65

+ 0.00095+• 0.01533• 2.5586• 26.380•118.0?

- 0.00001- 0.00661- 0.1705• 0.1861• 22.862

- 0.00012- 0.02385- O.OV10

-

-

- 0.02471- 0.79379+ 4.0685

- ,

10

1.060367.44488

17.723643.0658

105.331

1.060397.45574

16.276426.724943.7714

1.060367.45570

16.261926.523*37.8807

1.060367.45570

16.261926.530537.8988

1.060287.44980

15.253424.259433.0093

A"0.00000

- 0.01082+ 1.4162+16.537+67.403

+ 0.00002+ 0.00004+ 0.0146+ 0.1946+ 5.8484

0.000000.00000

+ 0.00001- 0.0051- 0.0423

0.000000.00000

• 0.0001+ 0.0020- 0.0242

- 0.00008- 0.00590-1.0034- 2.2691- 4.9137

- a/ 4 = En - E e x a C t , E e x a c t : very accurate results of Reid [12].b/ Results of the variational Ritz method.c/ im. - complex root.

Identify the set {+, } as the orthonorxal baHis of H • L 2 ( B 3 , T ) , the set(s) 2 3

{+) > as independent functions in HR " L (P , ( l + 8U)T), and the realizationof the nestings E and CEoo Os i n M

E e o(s,t ,u) = (1 + Bu)

Eeo)"e = (1 +

This is Justified because the function

B(u) = (1 + Bu)

(3T)

(38)

(39)

-1vith B ^ 0 is continuous in IB , and both B(u) and B(u) = TT—r »= (1 + 6u)

take strictly positive values, since 0 •* u - ; therefore B Is an element

of I (though B(u), like b(x) of the quartic oscillator, is a sequence of

equal numbers, for fixed u).

H« and H_ are not the ame Kilbert space except vhen 6 • 0 (and

0(u) = 1 • 0(u) for all u). Method of moments calculations in the mixed spaces

H- and H. converge because H is naturally embedded in Ho and both are(0)

sufficiently spanned by {+!• } . It is again significant that calculations with

B » 0.29 converge faster than with ft » 0.5 (1 e. when HQ ^a *=. H Q E'-

Table U of [10] corresponds to our choice of +. and %, as in EqB.{l6) and

(17), respectively, and shows that for H » 20, the helium atom ground state

energy value with B = 0.5 is -2.91UU5 a.u. j with 6 =• O.29 it is -2..90314 a.u.,

while its "exact" value is -a.903721* a.u. Mo doubt, the convergence would

have been comparably fast and monotonically decreasing vhen 6 » 0 (i.e.

when Ho becomeo identical with K ).

U.3 Remarks

is defined for

In both examples, and in fact generally, the situation remains unchangedwhen we u&e the nested Hilbert space H instead of the intermediate Hilbertspaces Hevery *l

when the range of EIQ is not the whole of H ( i . e . E . i s a proper nestingoperator), as long as we make the usual stipulation that our problem be solved

, because the nesting operator EIQ from HQ into

6H , continuous, has dense range in H and is injective. Even

in a given Hilbert space

intothen

•(as a aatnral embedding operator of H

causes the basis set (•. } of FL to span the effective part of H.

-15- -16-

^ = E inecessary for the problem. Recall that ^ = EI0 ii so-that formally

e. = I{i) £__ + . for al l i ; hence even in H_, the best results for our

problem are obtained when EIQ becomes the Identity nesting.

The non-monotonlc convergence of the method of moments together with

the question of best value for parameters usually present in the realization

of the basis vectors would appear to be simultaneously solved i f indeed the

seta (XJ^ •"* f+i J of vectors span the saute Hilbert space H_. This is

confirmed by the examples where in the attempt to span the same Hilbert space

H_ we obtain the best results, the chosen sets } become identical

and the method of moments reduces to the variatioRal method. All the difficulties

experienced inthe practical application of the method of moments thus appear

to be fundamentally connected with our inability to realize the same Hilbert

space Ho = L (B ,T) by two really distinct but measure-equivalent orthonorm&l

basis seta.

Where the method has to be,used, as in the examples considered,

( i .e . as an alternative to the variational method) (then the error

involved (relative to the error the variational method would have produced)

i s Identified with the difference between the identity nesting EQ0 and the

nesting ^^Sn'ob* w*1^cn *a easily determined from the respective measure T.

and TQ of the realizations of the Hilbert spaces H. and H_ > and is a more

practical form of evaluating errors of the method of moments than the M

numbers or any function of them. Eq.(36) illustrates this point. For• i/o implies

minimum relative errorwe have E00 - (Ec0)Qc = 1 - (1 + b) ' = 0, and this/thatb » 0 with H 5 Hn. H i H_ for any other allowed value of b. In fact,

C Q C .' U

as b -*•<=, the relative error tends to i t s nunr-imnni value 1. The same

deduction obtains from £4.(38) and both show that in general ^ o ' o b ' 1 ^ '

aB a function of the space of their realization, xeK , are continuous and

satisfy

0 <Ob

Just like the Mj numbers of Eq.(lO).

We conclude as follows:

1. The basis functions being employed in the application of the

method of moments are seen, through correspondence established between the

parameters they contain and the sequence of Hilbert spaces in nested Hilbert

space, not to span the same Hilbert space as required in theory.

- IT-

S'. Ttie.result's •are therefore always' lass accurate' than" this variationalmethod; the re la t ive error, which depends on the parameters, ia associated

with the difference between the corresponding nesting operators and tends to

zero with the values of the parameters that cause the bas i s se t s to span the

same Hilbert space.

3 . The question, as t o which of the two methods (variat ional or

moments) i s the bet ter one, i f conditions required by the l a t t e r in theory are

truely rea l i zed , thus appears open.

p .3 . Method of moments computations for the quartie o s c i l l a t o r , using two

dis t inc t but measure-invariant ort:-onormal basis se t s of the L ( B , T ) space,

i s now in progress and resul t s w i l l be published on completion.

V. NESTED VERSUS RIGGED HILBERT SPACE

. Interpretation of the method of moments in terns of rigged Hilbert

space appears, at f i r s t s i g h t , to be equally poss ib le . The theory of rigged

Hilbert space is based on topological concepts, for details see [ l^ ,15 r l £ ] ,

but i t i s briefly as follows.

Starting from a linear topological space V, in which a topology is

introduced by a norm, a count ably normed space V™ , is defined as a linear

topological space such that the neighbourhoods U .(0) of the zero element

are given by a denumerable number of comparable, compatible norm* with

M f ! * H f e * * • • > « | v | ^ . . . for a l l v « V ,

where U B(o) » (v||v|l •* j'> 6 > 0 and p are positive integers; and

which Is complete with respect to the topology given by U s ^ ) - A countably

normed Hilbert space H, , Is thus the topological limit of a sequence I,

of normed Hilbert spaces H , with norm given by scalar product, Uv|| • Af{ytv) ,

but in which the elements p e l are positive integers by definition, Instead

of real numbers as in nested Hilbert space. This creates the first difficulty.

To construct a rigged Hilbert space, a scalar product ( ' , ' ) i s farther

introduced Into Hy, to produce a nuclear countably normed Hilbert space

( s t i l l denoted by the same symbol H1o)>If in addition to the usual properties

of scalar product, this scalar product is such that llm gij1' * g^ with

respect to the topology in H>» n "

(1*0)

for all 6r f %

The second difficulty is connected with the fact that the completion of H^

with respect to the norm, l| || = 'v/t • , ' ) ' > produced by this scalar product,

i s what actually corresponds to the Hilbert space H of our problems.

Condition (itO) implies the existence of a continuous, l inear, one-to-one

operator T^{the properties of vhich are responsible for the name, nuclear)

mapping H^ into HQ; thus acting as the nesting operator ( EI 0 ) 0 I • 8e c a l i s e

the topology of Ity is stronger than the relative topology induced by H_ on

H,n , H^ is s t r i c t ly a subspace of H_. What should correspond to H&

(through the order.reversing involution on I) in a nested Hilbert space approach,

is H , here defined as the space of continuous antilinear functionala on H«,.

Because H,. = H {i .e . 0 = 0x) we havs the rigped Hilbert space as the t r ip le t

of spaces C Hfl C H The lack of symmetry between and H ^ not-

with H. of nested Hilbert space implies

)

withstandingjidentification ofthat only T (corresponding to the nesting (ETn).-,T), the nuclear operators T

(corresponding to integral n 4 0 and _T (corresponding to(E- ) t ) for integral n,JJ with n » I - 1 and I f 1, are defined;

irrespective of the linear topologieal space V, with which we s t a r t .

Recall the significance of the elements b « I in Eq.. (29) and the role

of the nesting (E.0)01) for a l l real values of b e l in Subsec.l*.3" The

corresponding quantities JT? remain undefined for non-integral values of p

in the rigged Hilbert space interpretation; and i t is the source of the th i rd

difficulty. Finally, IL, C- Kg, whereas we need the possibil i ty of H«5 HQ,

when the best results are obtained in the examples considered, although the

method of momenta then reduces to the variations! method. Even if these

diff icult ies are somehow overcomed, interpretation of the method of moments

in terms of rigged Hilbert space will apparently not ~bs as simple nor

comprehensive as i s the case with nested Hilbert space interpretation.

APPENDIX

This appendix contains the proof, by adaptation of the methods of

Grossmann (1966), of the main

Theorem: The algebraic induction Limit H- =

Proof: F i r s t , we prove a

is a nested Hilbert space.

Lemma: Suppose b and d are any two elements of I and a i sdefined as a( i ) = a in (b( i ) , d(i)) for a l l i . If a. is a sequence of

a. $, and \ a, $. are strongly

convergent in H, and H , respectively, then the series \ a, t . ^3

«lso stroftftly eonvergent. i

Proof: From (29) j| •:'b J = b - 1{i) and the set {^b '} is orthogonal.Therefore, i f E' denotes summation over a f ini te linear combination of theset of vectors, then

Hence

ACKHOWLEDQffiHTS

The author wishes to thank Professor F. Zanolin for useful discussions

and Professor G.C. Ohirardi for reading the manuscript. He also expresses

his thanks to Professor Abdus Salam, the International Atomic Energy Agency

and UNESCO for hospitali ty at the International Centre for Theoretical Physics,

Trieste .

J ai +|b' and of ) a± $^

i ^—t , > is of ) o, +, are

1 T^ (a)But H is complete, hence the series > a ^: ' is strongly

. ' . If the par t ia l sums of

sequences, then the par t ia l sums

are C&uehy

also Cauchy sequences,

convergent.

-19--20-

IText, suppose f 6 Eid V a r c n a t u r a l

embeddings of HM H,, respectively, Into H_- From C27), (29) and the lemma•bore, i t follova that the strongly convergent expansions +

nd the lemma

and • • \ a, i\ are rectors in R^ and H^, respectively, and the

i

aeries

i

i

i s strongly convergent in Hft. But

i

y °t ^ a *i ' " / a i *i " * ; a h o w l nB t h a t • should be denoted

by t is the representation of $ ' in H . Hence t E. H andE. H

^ a Ha 2 EIb "b EId Hd *

Eq.(2l4) is thus also satisfied.

To satisfy Eq.(25), consider the vectors in H-, given 'by

*i

and also the vectors in S^ and Hfl, given, respectively, by

e<» -b(i)+<IJ ; . J ^ - o W 1 *i i 1 Ti

for every b , d f l , w i t h d > b.

From (A.I) and (A.2), define the nesting E_ bydb

db t

i t follows from (20), (29) and (A.3) that

mDefine a. unitary napping It-. from 11 onto Kj- as

it (b) , (5)

by linear extension and closure.

(A.2)

CA.3)

(A.M

(A.5)

-21-

Since 5 » 0, then by (A-5),

and (A.U), i t fellcys that

UgQ • 1. Also from (26), (A.5) ( U-3)

and that

shoving that

for every i . Hence Eq.(25) i s satisfied. •

Note that a general element t> * I can nov be taken as b(x) of (33),c of (3l») or $ of (39) and each satisfies the definitions in the proof ofthe theorem.

-22-

CURRENT ICTP PREPRINTS AND INTERHAL REPORTS

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m, m