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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



N€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.



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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



■^ p(y) ^ (x,y) +q(y)(|) +r(y) — ^ = , -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 •'^ri^y^^y = ° '



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^ •
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 °

1
/ F(y)G(y,Ti,z)

(2.17) lim ^ # -^ dy = .



^ 27ri
p — > 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 — »• 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 (X»(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 ->^„) 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) = y— (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

Online LibraryArthur S PetersSome generalized eigenfunction expansions and uniqueness theorems → online text (page 1 of 2)