Estimation of the rate of convergence of a variant of the method of intermediate problems

ESTIMATIONOF THE RATE OF CONVERGENCE OF A VARIANT OF THE METHOD OF INTERMEDIATE PROBLEMS* L.T. POZNYAK Leningrad (Received 16 May 19671

IT is well-known (see, for example, tll>, that Ritz’s method gives upper bounds for the eigenvalues of a selfadjoint positive definite operator with a discrete spectrum. In some cases lower limits for the eigenvalues of such an operator can be obtained by the intermediate problem method k-61. The joint application of both methods enables the eigenvalues sought to be confined within arbitrarily narrow intervals so that guaranteed results are obtained. This paper investig~es the rate of convergence of a version of the method of intermediate problems. This version was fast proposed by Aronszajn f31, but we will confine ourselves to a later treatment of it given in 153. A brief description of this version, necessary for an understanding of the subsequent reasoning, is given in section 1. More detailed information can be obtained in

[51. An estimate of the rate of convergence is obtained in Section 2. In the derivation of this estimate we have ~vestig~ some methods which are applied in [7, 81 in the investigation of the convergence of projection methods. 1. Aronszajn’s

variant. in the method of intermediate problems

The reasoning which follows is based on a separable Hilbert space H with scalar product (u, u) and norm 11u 11= (u, u?. If T is some operator in H, its domain of definition and domain of values will be denoted by D(T) and R(T) respectively.

* 2%. vychisl. Mat. mat. Fiz. 8, 5, 1117-1126,

246

1968.

Variant of the method of intermediate

problems

247

We also agree to consider the eigenvalues of each selfadjoint operator with discrete spectrum encountered later to be numbered in increasing order taking account of multiplicity. We now pass to a description of the variant of the method of intermediate problems proposed by Aronszajn. In H we consider the problem Au = hu

(f-1)

of the determination of the eigenvalues hi (i = 1, 2, . . . ) of the selfadjoint positive definite operator A with discrete spectrum. We will denote by ui the eigenelement of the operator A belonging to Xi. We suppose that the operator A can be represented as the sum of two operators A=

Ao+&

(1.2)

where A, is a selfadjoint positive definite operator similar to A (that is D(A,) = D(A) ), with known eigenvalues X i0 (i = 1, 2, . . . ), and B is a symmetric positive definite operator Pa, u) > /3ll41*,

0 < /3 = const,

u ED(B),

(1.3)

such that D(B) =D(Ao)

(1.4)

and (Bu, a) G b(Aou, u),

u E D(Ao),

0 < b = const.

It is then obvious that A, < A and the eigenvalues hi0 (i = 1, 2, . the eigenvalues hi (i = 1, 2,

(1.5) ) underbound

. . . )

ki0

G

li

(i =

1, 2,...).

(1.6)

With these conditions the method of intermediate problems enables us to obtain from, generally speaking, crude lower limits (1.6), arbitrarily close lower limits, by the construction of a sequence A, (i =,l, 2, . . . ) of selfadjoint

248

L. T. Poznyak

operators, satisfying the inequalities (n= 1, Z,...);

A,,
(1.7)

The A, h = 1, 2, . . . I are called intermediateoperators, A,, is the fundamental. According to Aronszajn the intermediateoperators are constructed as follows. In the linear set D(B) we introduce the new scalar product [u, u1~ = (Bu, VI and complete D(B) with respect to this scalar product into the Hilbert space H,. It is known [91, that it is possible to carry out this completion on account of the elements of the space H, where for any element u E H,

where I(- lls denotes the normin H, generated by the scalar product [u, v I,. In the set D(B) we select a linearly independent system ui (i = 1, 2, . . . ), complete in H,. We denote by Qn the operator, orthogonal in H,, of the projection onto the linear envelope V, of the first n elements of the sequence ui (i = 1, 2, . . . ). For an arbitraryelement u E H, we have Q,u

=

i

r1.9r

UiVi,

L-E!

where the coefficients ai are defined by the system of equations i

(vi, Buj)ai

= (u, Bvj)

(j=

1,2 ,...,

n).

I1.10)

f-i

Since V, c V,+, c H,, the corresponding projectors satisfy in H, the inequalities 0 <

Qn

sz Qn+iG E

(n = 1, 2,.

..)

(1.11)

where E is the identical operator. We now define the intermediateoperator A, by putting An =Ao+BQn,

or in more explicit form

(1.12)

Variant

of the method of intermediate

Anu=Aaic+~ CZiBVi, u

E

problems

D(A,).

249

[1.13)

i-l

The operator BQn is defined in H, and has there the representation BQ~u =

i CtiBvi =

i

i=l

i=l

i

bij(u,

Buj)Bvi,

11.14):

j-1

where bij (i, j = 1, 2, . . , , n) are elements of the matrix inverse to the Gram matrix of the elements uI, u?, . . . , u, in the space H,. It is obvious from the representation (1.14) that the operator BQn may be extended to the whole H space and will be a symmetric, completely continuous operator there. Therefore the operator A, just constructed, since it is the sum of a selfadjoint operator A, and a symmetric bounded operator BQ,, is selfsdjoint, and its domain of definition D(A,) = D(A,f. Using fl.llf, it is easy to verify that the operators A, (n = 1, 2, . . . ) satisfy the inequalities (1.7). The operator A, is obviously similar to A,, and consequently also has a discrete spectrum. Let Ai,, uin (i = 1, 2, . . . 1 be its eigenvalues and the corresponding eigenelements. By (1.7) the eigenvalues inequalities Ai0

G

hi,

of the operators A, (n = 1, 2, . . . 1 satisfy the

G

ai,

n+l

G J+i

(n = 1, 2,. . .)

(1.15)

and therefore give improved lower bounds which for large values of n differ arbitrarily little from hi The eigenvalues X, ti = 1, 2, . . . f of the operator A, are found as sol~ions of transcendental equations which are uniquely defined by the eigenvaluee A, (i = 1, 2, . . * ) and the systems tljo and Ui (i = 1, 2, . . . ). Indeed, the eigenvalues equation

of the operator A,, are, firstly, all the roots of the

det(ui + R&t, Bvj) cj=, = 0,

11.16)

L. T. Poznyak

250

where R, is the resolvent of the operator A,, secondly, those of the various eigenvalues hil, O‘Xi O’ Ai290 <. . . of the operator A, which satisfy the corresponding equatio&

det

R;Bvit Bvj 1:

i=n+h, , j=n

Bvj)i=n+l,j=l __________________,________~___________.________~------~--~~~~~~~ =o (1.17) (vi +

@vi,

i=n,

uiy+j-n-l,O)i=l,

j=l

iv+&n-l,0

3

j=n+h,

0

j=n+l

Here we denote by h, the multiplicity of hi

o as an eigenvalue of the operator

A,, and by R,,’ the resolvent of the part of ge operator A, situated in the orthogonal (in Z-I)complementto the eigen-subspace stretched by ui y,o, uiV+LO, . . . ) ui

v+h

Y

‘0.

Equations (1.161, (1.17) determinethe eigenvalues of the operator A,, together with their multiplicity, which is the same as the defect of the corresponding matrices in (1.161, (1.171. It must be mentionedthat in the case of an arbitrarychoice of the complete system ui (i = 1, 2, . . . ) the solution of equations (1.161, (1.17) is a fairly difficult computationalproblem. The trouble is that as a rule we do not know the closed expression for the resolvents RA and Ri, and therefore to calculate the scalar products (R$3ui, Bu) and RABui, Buj) we have to use the spectral representationsof R, and RA, so that, for example,

It is mentionedin [4l that the calculation is considerably simplified if it is possible to select ui satisfying the relations Bvi = uio

(i = 1, 2,. . .),

(1.18)

since in this case (R&vi, Bvjl ,= (Rubto, U~O)= 6tj / (ho -a), where ‘ii is the Kronecker delta. (The condition of completeness of the system ui (i = 1, 2, . . . ) in H, is here satisfied automatically 111.) Unfortunately,such a choice is possible only for simple operators B (for example, for multiplication operators), for which I3-I ui o is easily calculated. The convergence of the lower limits bin (i = 1, 2, . . . ) to hi as n -) 00with

Variant

of the method of intermediate

251

problems

the assumptions made at the beginning of the present section in respect of the operators A, A, and B, and the system ui (i = 1, 2, . . . ), were established in [51. We will estimate the rate of this convergence.

2. Estimation

of the rate of convergence of the approximate eigenvalues in Aronszajn’s version

Let the eigenvalue h, of the operator A considered have multiplicity J
<

1:

hk = ?bk+l= . . . = &,+1-i < aR+i.

We denote by U, the eigen-subspace belonging to h, of the operator A, stretched On UkY$+l, . . . , Uk+l_l 7 by P, the projector orthogonal in H onto the subspace ‘k’

Wefirstderiveaformulafortheerror We have (E-LhA-‘)Ujn

I,-hj*

= Uj, -.LhAn-iUjn

where T, = A;’

-

(j=k,

(j=k,

+ LkT,Uj,

A-‘, or, since

An-iujn

(E-kk&A-‘)Uj~

=

k+l,...,k+I--1).

kfl,...,

kfl-I),

hj,,-fujnr

hi, - ha a, ujn $- AkTnUjn.

=

(2.1)

rn

We take the scalar product of both sides of the last equation with PkUj,:

((E - akA-')

ujn,

PkUjn)

a,” - ak a, lUjnv Pkujn)+

=

ak(TnUjn,

Pkujn).

(2.2)

,n

Since ((E--+.~-‘)ujt~, (ajrc,

Pkujn)

PhU,n) = (Ujn, (E -?~.hkA-~)P,,u~,) =

(Pkujn,

P kujn)

=

hk-‘(APkujn,

= (Uj,,

0) = 0,

Pkujn),

(2.2) implies that (Tnujn,

hh -

hj,

=

hfnhk’

p-__l.

(APkUin.

The operator T, may be represented

in the form

PkUjn)

PRUjvx)

(2.3)

L. T. Poznyak

252

or in the form T, = A,-‘BQ(“)A-‘,

where Qcnf= E - Q,. Formulas (2.4), (2.5) are obtained directly from the relations A=Ao+B=Aij+BQ,+BQ(“)=A,+BQ(“), T, = A,-’

-A-’

= A-‘@

- A,)A,-’

= A,-‘(A

- A,)A-*.

We transform the numerator of the fraction on the right side of (2.3), using for T, the representation (2.4):

It is possible to pass to the scalar product in H, because C D(B) f

Pkajn En(A)

As a result we arrive at the required formula for the error hk - Ajn: bi, - hf,, = ii,,

[Q(“)wn, Q('%~jnb Wkujn,

Pkutn)-

(t=

k, k+1,

. . . . k+Z-l).

(2.6)

From this, by means of certain inequalities, we obtain the required estimate of therateofconvergenceof Xi, (i=k,k+1,...,k+Z-1) tohkasn+w. We proceed to the derivation of these inequalities. From the definition and properties of the operators A and 6 it is clear that B G A.

(2.7)

Therefore (APkujn, PkUj*) > (BPhUjn, PhUjn) = lPkUjnIB’.

(2.8)

The numerator of the fraction on the right side of (2.6) can be written in the form

Variant of the method of intermediate

where

p(k) =

E -

Pk

253

problems

(Q(s)P(k)u, ,, is meaningful, since ujn and PkUjn E D(A) c Hi).

By the Schwartz inequality in H,

Consequently,

WJ) We

eStiIIlatC.3 IQ(“)Ph)ujn

(2.1).

1B in tETIIls

of IQ(“)PkUjn 1B. For this we turn to formula

We apply to both sides of (2.1) the operator Ptk) and use the per-mutability

of P’ ‘) with the operator

A -I(it

follows from the permutability of P, with A-‘)

and the property P’ k, P’ k, = P’ k):

(E

-

kkP(k)A--I)P(“)ajn

=

hjn

-

hk

P(‘)Ujn + hkP(')TnU,,.

(2.10)

hjn

The symmetric completely continuous operator P’ k, A -’ has the same eigenelements as the operator A -I, where $WA-‘ui

=

hi-‘% 0,

i # k, k + 1, . . . , k + I-

1;

i =

1.

k, k + 1, . . . , k + I-

Therefore, hi’ is not as eigenvalue of the operator P’ k, A-‘, so that there exists an inverse operator Gk = (E- hkP@)d-‘)-’ = hkei(hkmi& -P(k)A-l)-l. Its norm yk in

H

is easily calculated by spectral resolution and is given by hi

max t#A,k+t,....k+l_i

Ihi

-

hk

1 ’

Applying to both sides of (2.10) the operator G, and replacing ‘I’, by its expression (251, we find

P(Wujn

=

Ajn - hr

GkP(‘)Ujn + hkGnPc’)An-‘BQ(n)A-‘~j,.

(2.11)

ir-j*

The operator E - I,pWA-i Since PCk)ujn E D(A),

maps

D(A)onto

itself.

therefore GkP(‘)U.jnE D(A) c D(B) implies the estimate

L.

254

Ah - ah

fPfA)@j, 1B G

T. Poznyak

IGAWu5, 1

B

kiGAfA)P(A)A,-'BQ(n)A-'uln

+

f B.

(2.12)

bin

We estimate ) G,u js for u E D(A). We have Gn =

(E -

1 (E + &+P(~)A-‘G~)UIB

IGAul 5 =

&, lP(k)A-‘Gku~ g

p =

hA(AP(k)A-iGkU,

= hA%(GAU,

~.,pfOA-~)-~ g

/U\B

+ ~AIP(~)A-‘GAU~II,

.&A(BPcA)A-‘G~u,

P!A)A-‘GAU)‘I’Z

~#(A)A-‘G~u)%

E + ~&?(&)A-‘GA,

=

=

=

~A%(GAU,

P(A)A-iG~~)‘J~

hk (Gku, (GA -

&

P(A)A-‘G~u)‘fz

E)U)‘h

=

< ~A’~S(YA~ + y~)‘&\l.

Now taking into account (1.81, we finally obtain I&u~B

where

SA

=

1 -i- If#-'~A

hk2

f

~AIGIB,

(2.13)

u=f)W,

+ yk)).

Since P”’ u . and P(uA n -iBQ(n)A-%j, E D(A), both terms on the right side In of (2.12) can be estimated by formula (2.13). We obtain hA

ph)q,l

hjn

-

G

B

h

6A IP(‘)uja 1~ + hk&k IP(‘)An-‘BQ(n)A-‘~j,

18s

(2.14)

jn

Considering n to be so great that hj,-‘(LA (2.14) for lP(“)ujn I B: IP(k)U

1

<

1,

we

3rj, --(lip

the inequity

S&e

?‘~jnhA~A

c

“’ B .

hjnfdk

~-IP(‘)A,-‘BQ(‘)A-iuj, - h,,)&k

18.

The quantity k&jn - @A - &,)6nl-’ * 1 as n -) DO.Consequently, considered that for sufficiently great values of n IP(‘)Ujn 1

B

<

LIk8k IP(k)An-‘BQ(“)A-‘Ujn

We estimate 1Ptkf u \B for u E D(A).

Hence

We have

1 B.

it may be

(2.15)

Variant of the method of intermediate

IP(k)Uls f

We estimate )A;‘BujS

lull%

(f +t'(B-'hk))

255

problems

u ED(A).

(2.16)

for ZJE D(A). We have IA,,-‘Bullr2

= (BA,-‘Bu,

A,-1Bu).

By the inequalities (1.5) and (1.7), I34 bA,. Therefore the preceding equation is continued as follows: (An-‘BURR*

G b(Bu,

A,-‘Bu)

= b[u, A,-‘Bu]

< b~u~~~Aa-‘B~~~,

whence IA,-‘Buls

u ED(A).

< bluls,

(2.17)

Estimatingthe right aide of (2.15) first by formula(2.16), and then by formula (2.17), we find IpCk)uj,lB < TAIQ(~)A-‘U~~I B,

where rk = 2b?‘,k(i+ T'[fi--'hk(YA'+ yA))l[f

(2.18)

+l@-‘hk)l*

Moreover, Q(“)A-‘Ujn

= Q’“)A-‘P,,nj,,

+ Q(“)A-WGi,,,, (2.19)

fQ(“)A-‘Ujn

1B < IQ(n)A-*P~~fn

1LX+ IQ(n)A-~P(k)uf,

1B.

Since R(d-1) = D(A) c D(B), therefore A” can also be considered as an operator in H,. Using the inequality (2.7) and the complete continuity of A ” in H, it is easy to show that it will be completely continuous in H, also. Noting that ‘_d-‘PA&n = b-*PA~jn, (2.19) csn be continued as follows:

taking account of the above, the inequality

We here replace j P’ ‘) @jn 1B by the estimate (2.18): (2.x!)

L. T. Poznyak

256

From the fact that the system ui (i = 1, 2, . . . 1 is complete in H, it follows that jQ(~)A“jB+oBsn-+X. ~nsidering n to be so great that zk /fN%-*] s c ‘/*, we find from (2.30). (2.21) Now replacing in (2.18) IP”)A-‘aj, I R by the estimate (2.211, we arrive at the requiredestimate:

Combining(2.6), (2.81, (2.9) and (2.22), we conclude that (j=k,k+i,...,k+Z--1),

~k-“3n
(2.23)

where dk = Xk{l + 4b[1+ ,‘(p-%)][I + -f(p-‘hk(~k’ + yk))]}. Formula(2.23) also gives an estimate of the rate of convergence of b3t~ 0 = k, k + 1% + - - t k + 2 - 1) Naturally, a more explicit dependence of this general estimate hk as n + m. on the subscript n can be obtained for a specific choice of the coordinate system ui (i = 1, 2, . . . ). to

We will first discuss a frequently encountered methodof choosing the coordinate system. We are also interested in this methodbecause its application in the solution of problem (1.1) by Ritz’s method enables us to obtain an explicit estimate of the rate of convergence of the approximateeigenv~ues obtained by this method, so that we can comp~ the methodof inte~~ia~ problems and Ritz’s method. .In fact, let F be a selfadjoint positive definite operator with discrete spectrum, with the domain of definition D(F) c D(B). Let o i and ui (i = 1, 2, . . . ) be the eigenvalues and eigenelements of the operator F: PV

f

=

OfVf,

vi ED(P)

c D(B)

(i = 1, 2,. . .).

We suppose that the eigenvalues wi f i = 1, 2, . , . 1 and eigenelements ui (i = 1, 2, . . f of the operator F are known, and that the systemUi (i = I, 2, . . . ) is complete in H,. Then it can be chosen as the coordinate system in the solution of (1.1) by the methodof intermediateproblems. l

(a >‘%I. For this it is first necessary We estimate ( Qfn) u I8 for u E II to show that D(Fa). c H, for a 3%. We denote by i the Friedrichs selfadjoint extension of the operator B. The inclusion D(F) c D(B) c D(B) implies the sequoia IIRVI! G +V+4l, v E D(F) (0 < r = const) UOI. The operators3 and

Variant

of

the

method

of

intermediate

257

problems

r*F satisfy all the conditions of Heinz’s theorem Ill], which, in p&icular, implies that D(F%)c D(R'/z) = Hs and also IlE’h II sg rllF%ull,

u

D(F%).

E

(2.24)

For a > ‘/i all the more is !XFaX)c H,, and the required inclusion is proved. We denote by E, the projector orthogonal in H, onto the subspace V,, and let

Etn)= E - E,. It is obvious that for any u E H, Q(“)E,,v

QWu = Q(n),q"Jv.

= 0,

Therefore lQ(“)ula = ~Q(“LP)u(~ < (EWzlB = jIB’/,E’“,ull It follows from (2.24) that the operator B”F-’

IQ-1

s

<

IIEW-‘/zll

is bounded in H. Hence IIFY~E(~)~I~.

The operator E(“) may be transposed with any power of the operator F. tinsequently, F%,?+)u = F%-~EE~W~W~V and (QWul

The bounded selfadjoint operator F, and

B

<

IIF'/z-aE(n)ll j(E('WuII.

IlE’/nF-‘hII

operator F ‘-aEE(n) has the same eigenelements

j”/,-orEC”)~~

i =

9

=

2, . . . ,

n;

i > n.

O’/+%Q

i

Therefore,

1,

as the

@?$? is its greatest eigenvalue,

so that

%-= IIF%-=E(n)ll= a,,+~ . Since llg(~~=~ll-+ o as n -) m, we finally obtain

IQ(n)ulB = o(o,,%-=) (u=D(F=),a 2 *h). Hence, if ur c D(P)

(a 2 f12), the

h* -ij,

estimate (2.23) has the order =

0(On’-2a).

(2.25)

L. T. Poznyak

258

We now also assume that the operator F is similar to A, D(F) = D(A) c H,. Then, by formula (2.251, the rate of convergence of hjn and X, has order not lower than 0,-l. We will solve problem (1.1) by Ritz’s method, choosing ui (i = 1, 2, . , . 1 as the coordinate system. We denote by &,, (j = k, k + 1,. . . , k + I - 1) the approxiobtained by Ritz’s method mations to the eigenvalue ?U (= &.s = . . . = k+~d, at the n-th step. By [71, ‘jn converges to X, at a rate nor less than ail. Moreover, if U, c D(F9 for some a > 1, the estimate of this rate is again of order that is, it is again the same as (2.25). As a second example where the general estimate (2.23) may be carried out, we consider the case where the system ui (i = 1, 2, . . . ) satisfies the relations (1.18). We first show that every element u E f)(B) is representable

as a series

convergent in H,. We denote by I, the projector, orthogonal in H, to the subspace stretched on ulo, u?,,, . . . , u, . We denote by S,LJ the m-th partial sum of the series (2.28). We have m

?n

= B

B&v

is,

js,u1

Bk

=

(B&v, S,u) =

Pv, uio)uio = ImBv,

v = WL

(I,& B-?T,Bv)-(Bvv) = [~]a*.

On the other hand, for any w E H, &lVlB~ = (mmv, &v) = (h,&

B-'Z,Bv)-(Bv, v)

=

[v]g.

m*m

which proves the cpnvergen~e of the series (2.28) in H,. Moreover, for any u E D(B), it is obvious that Q(~)s,v = 0, QW,~(+, = QW)~, where

m+ = i [u,u*& i-.n+i

Therefore,

vi.

Variant

of the method of intermediate

problems

259

that S(~)V= B-*Z(n)ZBu, v E D(B), The boundedness of the operator ?!?-I implies ’ and the preceding inequality may be continued as follows: where I(~I = E-I,,

Let u E D(B) be such that Bu E INA:) for some a $0. Reasoning as in the first example, we arrive at the conclusion that fQ(“)~IB= o&o-=) Therefore,

if B(UA) c D(Aa)

(VE+z.)(B), BLIEL)(Aij=), a > 0).

(a 2 0), -2r

hk -

We apply this result to a differenti~

hjn

=

(2.27)

o(h,o ).

operator.

Let fl be a finite domain in the xy-plane with boundary C p(x, y), 9(x, y), y) be functions continuous in the closed region 6 = Q + r; and pot co positive constants. We suppose that P(S, y) z=-PO, dp/dx, dp/dy, dq/dx, dq/dy, ch,

Qf5

Y)

>

PO,

C(%

Y)

2

co*

In the Hilbert space L,(Q) we consider the selfadjoint positive definite operators A, A,, and B, generated respectively by the differential expressions -k(P$

~-~~~~~+c~, -~[(,-,o)~]-~[(Y-Po)~

-~o(~+-~~),

1

+cu

on the set M of twice continuously differentiable functions vanishing on r. It is obvious that A, and B are similar operators and A = A, + B. If the domain R is such that the eigenvalues and eigenfunctions of the operator A, are known for it (for example, a circle or rectangle), the method of intermediate problems can be applied to determine the eigenvalues of the operator A. As the coordinate system ui (i = 1, 2, . . . ) we take the eigenfunctions uio (i = 1, 2, . . . ) of the operator A,. Since A and A, are similar, by (2.25) 1Lb- hi= = o&o-9. It is known that the eigenvalues of the operator A, have the asymptotic behaviour h,o - n. Consequently hk - hjn = o(n-‘).

260

L. T. Poznyak

Now in addition to the previous assumptions, let P(Z, Y) = !I(? pi = const (>

Y) =

and c(x, y) be continuously differentiable in CL In this case it

PO)

is better to expand A into a sum in a different way, choosing as A, and I3 operators generated by the expressions pl(dWdz2 + CPU/&#) and c(x, y) on the same set M. It is more advantageous because this expansion makes it possible to choose a coordinate system satisfying the relations (1.18). ui = Wo/C(I, !/) (i = 1, 2,. . .). (2.27),

kk-hjn

Since

B zzio =

c(z,

In fact we have to take y)uro ~D(Ao’ln)

=

I?‘r~(Q),

by

= o(n-‘).

If c(x, y) also has second derivatives continuous in Q, then $uio E D(A,), = o(n-2). by (2.27), AR-ai,

and

Translated by J. Berry REFERENCES 1.

MIKHLIN, S. G. Variational methods in mathematical physics v matematicheskoi fizikel. Gostekhizdat, Moscow, 195’7.

(Variatsionnye

2.

ARONSZAJN, N. The Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I. Operators in Hilbert space. Proc. nut. Acad. Sci. U.S.A. 34, 10, 474-480, 1946.

3.

ARONSZAJN, operators. Stillwater,

4.

BAZLEY, N. W. Lower bounds for eigenvalues with application Proc. Nat. Acad. Sci. U.S.A. 45, 6, 859-653, 1959.

5.

BAZLEY, N. W. and FOX, D. W. Truncations in the method of intermediate problems for lower bounds to eigenvalues. J. Res. nat. Bur. Stand. 65B, 2, 105-111, 1961.

6.

BAZLEY, N. W. and FOX, D. W. Lower bounds to eigenvalues u~riig operator decomposition of the form B*B. Arch.Rat. mech. Anal. 10,4. 352-360, 1962.

7.

VAINIKKO, G. M. Asymptotic estimates of the error of projection methods in the problem of eigenvalues. Zh. vychisl. Mat. mat. Fiz. 4. 3, 405-425, 1964.

8.

VAINIKKO, G. M. Estimates of the error of the Bubnov-Galerkin method in the eigenvalue problem. Z/r. uychisl. Mat. mat. Fiz. 5, 4, 587-607, 1965.

9.

SMIRNOV, V. I. Course of higher mathematics Fizmargiz, Moscow, 1959.

N. Approximation methods for eigenvalues of completely Proc. Symposium on Spectral Theory and Differ. Problems Okla, 179-192, 1951.

10.

RELLICH, 1942.

11.

HEINZ, E. Beitrage zur Stomngstheorie 4, 415-438, 1951.

F.

Storungstheorie

(Kurs vysshei

der Spektralzerlegung.

metody

continuous (July 19591.

to the helium atom.

matematikil.

Vol. 7.

Math. Ann. 118, 4, 462-484,

der Spektralzerlegung.

Math. Ann. 123,