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IMM 368

June 1968

Courant Institute of

Mathematical Sciences

Some Generalized Eigenfunction

Expansions and Uniqueness Theorems

A. S. Peters

Prepared under Contract Nonr-285(55)

with the Office of Naval Research NR 062-160

Distribution of this document is unlimited.

New York University

COURANT IN9I-ITUTE - LIBRARY

pSl/VWcerSl. New York, NY. 100.7

Nâ‚¬W YORK UNIVERSITY

:OURANT INariTUTE - LIBRARY

15 1 Me4xer St. New York, N.Y. rOOI2

NR 062-160 IMM 568

June 1968

New York University

Courant Institute of Mathematical Sciences

SOME GENERALIZED EIGENFUNCTION EXPANSIONS

AND UNIQUENESS THEOREMS

A. S. Peters

This report represents results obtained at the Courant

Institute of Mathematical Sciences, New York University,

with the Office of Naval Research, Contract Nonr-285(55)

Reproduction in whole or in part is permitted for any

purpose of the United States Government.

Distribution of this document is unlimited.

Niw rcwK UNrveRsrrr

BOURANT INSTiTUTI . I laaAM-r'

Abstract

A generalized elgenfunction expansion method, Churchill's

method, and a transform method are used to investigate the unique-

ness of the solution of the equation

â– ^ p(y) ^ (x,y) +q(y)(|) +r(y) â€” ^ = , -00 < X < 00 ,

hx

subject to the boundary conditions

and

^^{x,0) +aoxx(x,0) +P^(})(x,0) =

(})y(x,l) +a^(t)^(x,l) +p^(j)(x,l) =

< y < 1

11

1 . Introduction

A. Welnsteln [1] showed that if

l-L sinh Jl\

n ^ n

^^(y)

This association, however, must be rejected because if we choose

f{y) = y and note that

^ (1)

1) - r y^.

y â€¢'^ri^y^^y = Â° '

n > 1 ,

2

we see that (1.21) forces us to associate y with the constant H.

This shows that the set of eigenf unctions (1.17) is inadequate for

expansion purposes. We therefore conclude from the foregoing

remarks that the applicability of Weinstein's method to the case

in hand depends on finding a set of functions (x (y)} which contains

^n

ii' (y)} and admits the expansion

00

f(y) = IZ Cn>^n^y^ â€¢

n=0

The major part of this report is concerned with the develop-

ment of expansions which allow an extension of Weinstein's method.

In Section 2, using a Polncare-Birkhof f formulation, we show how

the elgenfunction method can be applied to the equation

(1.22) ^ p(y) A (t)(x,y) + q(y )J]>^) dz = .

p->oo ^

1

These results imply that if F(y) is such that / F (y)dy exists,

then Â°

1

/ F(y)G(y,Ti,z)

(2.17) lim ^ # -^ dy = .

^ 27ri

p â€” > oo

C

Also if f(y) is a twice dif f erentiable function such that

1]

Lf(y) = F(y)

then

(2.18) f(Ti) = - lim 2^ ^ Q[G(y,Ti,z),f(y)]dz .

p^ oo

The formula (2,l8) is called the Poincare-Blrkhoff formula.

It was noted in a less general form by Poincare [2] during some

work on a special problem in partial differential equations. Then

Birkhoff [5] proved the formula for an ordinary n order boundary

value problem subject to certain regularity assumptions and with

the eigenparameter absent from the boundary conditions. Later,

Tamarkin [4] showed that the formula is valid for a wide class of

boundary value problems with the parameter present in the boundary

conditions. Since then, the formula has been proved by Wilder [5],

Langer [6], Rasulov [7] and others under less restrictive conditions

than those used by Tamarkin. The proofs of (2.l8), as given by the

authors noted above, depend upon explicit asymptotic estimates of

the behavior of eigenfunctions and eigenvalues as A â€” Â»â€¢ 00 . For a

proof of (2.18) with respect to the second order system (2,9)-

(2.11); and one which does not depend on specific asymptotic evalua-

tions, see Peters [8].

The formula (2.l8) leads to the expansion of f(y) into an

infinite sum of residues. If C is a circle with center at A^

containing no other eigenvalue we have

12

(2.19)

00 -, r

f(y) = -) p^ r Q[G(t,y,z),f(t)]

n=0

dz

n

00

5

Q

^ 7^ G(t,y,z)clz,f(t)

n

If (XÂ»(z) has a zero of order k at z = A , the corresponding term in

the expansion (2.19) is

(2.20)

Q[?^(t,y,n),f(t)]

where 9^(t,y,n)/(z ->^â€ž) comes from the Laurent expansion of

G(t,y,z) for the neighborhood of z = A . The function 9 can be

obtained by substituting the expansion

00

G(t,y,z) = yâ€” (z - A^)* 0_^(t,y,n) + T~ [z - A^)* 9^(t,y,n)

in the equation

LG(t,y,z) -A^r(t)G - (z -A^)rG = 5(t - y)

and the boundary conditions

G^(0,y,z) +PQG(0,y,z) = a^A^G(0,y , z) + a^(z - A^)G(0,y,z) ,

G^(l,y,z) + p^G(l,y,z) = a^A^G(l,y,z) + a^(z -A^)G(l,y,z)

which define G(t,y,z). We find from these equations, after

equating coefficients of like powers of (z -A ), that the circum-

flexed quantities must satisfy

(2.21) \{t,y,n) = 7^^^(t) ,

(2.22) (L- A^r)e^ = re^ + 5(t -y) ,

(2.23) (L-V^'^j = ^^J+1 ' ^ = l,2,...,(k-l) ,

with the boundary conditions

(2.24)

9j^(l,y,n) +P-^9j(l,y,n) = a^A^Q^ (l,y,n) + a^9j^^(l,y,n) ;

to be satisfied for

J = 0,1,2,. ..,(k-l) .

a*

The above equations can be satisfied only if the functions 9.(t,y)

satisfy the compatbility conditions

(2.25)

IMM 368

June 1968

Courant Institute of

Mathematical Sciences

Some Generalized Eigenfunction

Expansions and Uniqueness Theorems

A. S. Peters

Prepared under Contract Nonr-285(55)

with the Office of Naval Research NR 062-160

Distribution of this document is unlimited.

New York University

COURANT IN9I-ITUTE - LIBRARY

pSl/VWcerSl. New York, NY. 100.7

Nâ‚¬W YORK UNIVERSITY

:OURANT INariTUTE - LIBRARY

15 1 Me4xer St. New York, N.Y. rOOI2

NR 062-160 IMM 568

June 1968

New York University

Courant Institute of Mathematical Sciences

SOME GENERALIZED EIGENFUNCTION EXPANSIONS

AND UNIQUENESS THEOREMS

A. S. Peters

This report represents results obtained at the Courant

Institute of Mathematical Sciences, New York University,

with the Office of Naval Research, Contract Nonr-285(55)

Reproduction in whole or in part is permitted for any

purpose of the United States Government.

Distribution of this document is unlimited.

Niw rcwK UNrveRsrrr

BOURANT INSTiTUTI . I laaAM-r'

Abstract

A generalized elgenfunction expansion method, Churchill's

method, and a transform method are used to investigate the unique-

ness of the solution of the equation

â– ^ p(y) ^ (x,y) +q(y)(|) +r(y) â€” ^ = , -00 < X < 00 ,

hx

subject to the boundary conditions

and

^^{x,0) +aoxx(x,0) +P^(})(x,0) =

(})y(x,l) +a^(t)^(x,l) +p^(j)(x,l) =

< y < 1

11

1 . Introduction

A. Welnsteln [1] showed that if

l-L sinh Jl\

n ^ n

^^(y)

This association, however, must be rejected because if we choose

f{y) = y and note that

^ (1)

1) - r y^.

y â€¢'^ri^y^^y = Â° '

n > 1 ,

2

we see that (1.21) forces us to associate y with the constant H.

This shows that the set of eigenf unctions (1.17) is inadequate for

expansion purposes. We therefore conclude from the foregoing

remarks that the applicability of Weinstein's method to the case

in hand depends on finding a set of functions (x (y)} which contains

^n

ii' (y)} and admits the expansion

00

f(y) = IZ Cn>^n^y^ â€¢

n=0

The major part of this report is concerned with the develop-

ment of expansions which allow an extension of Weinstein's method.

In Section 2, using a Polncare-Birkhof f formulation, we show how

the elgenfunction method can be applied to the equation

(1.22) ^ p(y) A (t)(x,y) + q(y )J]>^) dz = .

p->oo ^

1

These results imply that if F(y) is such that / F (y)dy exists,

then Â°

1

/ F(y)G(y,Ti,z)

(2.17) lim ^ # -^ dy = .

^ 27ri

p â€” > oo

C

Also if f(y) is a twice dif f erentiable function such that

1]

Lf(y) = F(y)

then

(2.18) f(Ti) = - lim 2^ ^ Q[G(y,Ti,z),f(y)]dz .

p^ oo

The formula (2,l8) is called the Poincare-Blrkhoff formula.

It was noted in a less general form by Poincare [2] during some

work on a special problem in partial differential equations. Then

Birkhoff [5] proved the formula for an ordinary n order boundary

value problem subject to certain regularity assumptions and with

the eigenparameter absent from the boundary conditions. Later,

Tamarkin [4] showed that the formula is valid for a wide class of

boundary value problems with the parameter present in the boundary

conditions. Since then, the formula has been proved by Wilder [5],

Langer [6], Rasulov [7] and others under less restrictive conditions

than those used by Tamarkin. The proofs of (2.l8), as given by the

authors noted above, depend upon explicit asymptotic estimates of

the behavior of eigenfunctions and eigenvalues as A â€” Â»â€¢ 00 . For a

proof of (2.18) with respect to the second order system (2,9)-

(2.11); and one which does not depend on specific asymptotic evalua-

tions, see Peters [8].

The formula (2.l8) leads to the expansion of f(y) into an

infinite sum of residues. If C is a circle with center at A^

containing no other eigenvalue we have

12

(2.19)

00 -, r

f(y) = -) p^ r Q[G(t,y,z),f(t)]

n=0

dz

n

00

5

Q

^ 7^ G(t,y,z)clz,f(t)

n

If (XÂ»(z) has a zero of order k at z = A , the corresponding term in

the expansion (2.19) is

(2.20)

Q[?^(t,y,n),f(t)]

where 9^(t,y,n)/(z ->^â€ž) comes from the Laurent expansion of

G(t,y,z) for the neighborhood of z = A . The function 9 can be

obtained by substituting the expansion

00

G(t,y,z) = yâ€” (z - A^)* 0_^(t,y,n) + T~ [z - A^)* 9^(t,y,n)

in the equation

LG(t,y,z) -A^r(t)G - (z -A^)rG = 5(t - y)

and the boundary conditions

G^(0,y,z) +PQG(0,y,z) = a^A^G(0,y , z) + a^(z - A^)G(0,y,z) ,

G^(l,y,z) + p^G(l,y,z) = a^A^G(l,y,z) + a^(z -A^)G(l,y,z)

which define G(t,y,z). We find from these equations, after

equating coefficients of like powers of (z -A ), that the circum-

flexed quantities must satisfy

(2.21) \{t,y,n) = 7^^^(t) ,

(2.22) (L- A^r)e^ = re^ + 5(t -y) ,

(2.23) (L-V^'^j = ^^J+1 ' ^ = l,2,...,(k-l) ,

with the boundary conditions

(2.24)

9j^(l,y,n) +P-^9j(l,y,n) = a^A^Q^ (l,y,n) + a^9j^^(l,y,n) ;

to be satisfied for

J = 0,1,2,. ..,(k-l) .

a*

The above equations can be satisfied only if the functions 9.(t,y)

satisfy the compatbility conditions

(2.25)

1 2

Online Library → Arthur S Peters → Some generalized eigenfunction expansions and uniqueness theorems → online text (page 1 of 2)