Eigenfunction Expansions Associated
with Schroedinger Operators in Rn, n 4
DALE W. THOE
Communicated by M. M. SCHIrrrat
w 1. Introduction
In this paper we develop an eigenfunction expansion associated with the
Schroedinger operator -A+q(x), where A denotes the Laplace operator in
n-dimensional Euclidean space Rn and q (x), (xe Ra), is a real-valued potential function tending to zero at infinity. The eigenfunction equation under consideration
here is
(1.1)
--A~p(x)+q(x)(p(x)=12q~(x),
xeRn.
Equation (1.1) will be shown to have a hounded solution q~(x, ~) for each 04=
~eRn, with # = [ ~ [ 2 > 0 . The solution r
~) is called an eigenfunction of (1.1)
with eigen-number # = Ir 12, and is obtained as the unique solution of the integral
equation
(1.2)
q~(x, 4) = ei x. ~_ I FIr (l x - Y I) q (Y) 9 (Y, {) d y ,
Rn
where F Ir (r), (r > 0), is a particular fundamental solution of A u(x) + I { 12u(x) = 0,
(xeR,,). In the case # < 0 , (1.1) is known to be solvable only for a particular set of
values of /t, the eigenvalues of equation (1.1). We denote these values by /~k
( k = l , 2, ...) and call the associated solutions ek(x)eLZ(Rn) the eigenvectors of
(1.1). In terms of the eigenfunctions r
{) and eigenvectors ek(x), the expansion
formula to be proved reads
f(x)=(27r) -'/2 5 q~(x,~) f(~) d{+~.fk ek(x)
Rn
where
f(r
-"/2
I ~o(x,r
Rn
dx,
fk = I ~(x)f(x) dx.
Rn
In the cases n = 2 , 3 this expansion has been obtained by Ir~BE [1]. The methods
employed by him do not extend directly to higher dimensions, however, the main
difficulty arising from the fact that the resolvent kernel associated with the
Schroedinger operator is not of Carleman type in the case n =>4. We shall restrict
all considerations in this paper to the cases n > 4. While our proof can be modified
to handle the cases n = 2, 3, these cases do not fit naturally into the scheme presented here. This seems to be typical, see Rr_JTO [2] for a similar example.
There are two basic features which permit the proof to proceed for general
n > 4. The first of these was used by SHENK [3], and played a central role in his
23 Arch. Rational Mech. Anal., Vol. 26
336
D.W. THOE:
comprehensive study of scattering by a bounded obstacle in R,. Basically it allows
one to avoid the explicit determination of the resolvent kernel associated with the
Schroedinger operator; it appears here as Lemma 4.2. The second consists essentially of reducing the expansion problem to the case where the potential q has
compact support and was previously used by the author [4] in a study of the
classical wave equation; it appears here as Lemma 4.3.
As noted above, we consider potentials q which vanish at infinity. More precisely, we shall require that q satisfy the following assumptions:
O) q(x), (xER,,), is real-valued and locally H61der continuous except at a
finite number of singularities,
(ii) q~L 2 (R.) and behaves like
in + 1\
(1.3(ii))
O(Ixl-t-r-)-~),
(h>0),
at Ix l = oo, i.e., there exist constants h, M and A such that
I q ( x ) l < M l x l -(z-'~-)-h
for [xl>__A,
(iii) there exists a constant ct > 0 such that
(1.3 (iii))
M~(x, 6)= [1~-~1~
_
[q(y)]2 dy ~
I x - y l "-4+~]
(6 > 0)
satisfies
sup Mq(x, 6) = o(1)
at 6 = 0 .
x~Rn
Note that (ii) and (iii) imply that
sup Ms(x, 6)
x~R.
is bounded for 6 >0. Condition (iii) is called the strong Stummel condition by
REJTO [2]. The constants h, A and a will retain the meanings given in (1.3) throughout the sequel.
Let H o denote the self-adjoint Laplace operator in L 2 (Rn). For f e C~~(Rn) 1
set
HW) f(x) = - A f(x) + q (x) f ( x ) .
It is known that H ~~ has a unique self-adjoint extension [5], which we shall denote
by H, and that ~ ( H ) = ~ ( H o ) 2. F o r f e ~ ( H ) ,
(1.4)
H f=Ho f+ q f= --Af + q f.
For a > A let ~k,(x) be a smooth function of x~R,, satisfying 0<~k~< 1, ~k,= 1
for Ix I < a, ~k, = 0 for Ix I > a + 1, and set
(1.5)
q~(x) = r
q (x).
Clearly q~ enjoys all of the properties (1.3) in addition to having compact support.
1 c~o (Rn) denotes the class of infinitely differentiable complex-valued functions having
compact support.
z ~(T) denotes the domain of definition of the operator T.
Eigenfunction Expansions for Schroedinger Operators
337
Denote by H(a), ( a > A ) , the self-adjoint operator given by
(1.6)
H(a)f= - A f +q, f
forfe~(H(a))=~(Ho). It is evident from (1.3) that H(a) converges to H in the
strong operator topology as ~-~ co. We shall use the following facts concerning
the spectrum of H and H(a). A theorem of KATO [6] implies that the positive real
line contains no point spectra of either H or H(a). A recent result by REJTO [2]
shows that (iii) guarantees that on the negative real line, H and H(a) can have
only point spectra of finite multiplicity, i.e., the continuous spectrum of H (and
H(a)) is absent from the interval ( - or, 0).
A short description of the paper will now be given. w contains a list of the
properties of the fundamental solutions of A u(x) + tr u(x) = 0, (x~Rn). The nearfield and far-field behavior of these fundamental solutions has been used by R~JTO
[2] to establish the A-compactness of the operation of multiplication by q, and to
establish the absolute continuity of the positive spectrum of H [7]. We require
more specific information concerning the fundamental solutions in order to
obtain the expansion theorem, however. In w3 the existence of the eigenfunctions
~0(x, ~) is proved. The arguments follow very closely those given by IKEBE [1].
Finally, the eigenfunction expansion theorem is proved in w4.
w 2. Fundamental Solutions
Let K denote the open half-plane Re x > 0 and set
K•
~ E K : O<_+arg~c<~- .
For ~ satisfying 0 < ~ < 1 let
K~={xeg• ~<1~c1<~-1}.
For zeK set
G~ (z)=H(ol)(z),
G2 (z)=H(o2)(z),
G~(z)=z-l e +'z
and for n =>2
(2.1)
G~+2(z)=z_ 1 d G~(z).
Ho(~) and Ho(2) are the Bessel functions of the third kind and order zero [8]. The
functions G~ (z) have the asymptotic behavior
1 --n
(2.2)
1 --n
dii G~ (z) = ( _ i)j d~ e • 'z [z - r - + O (1 z I - T ) ]
dz
as [z I -~ oo, uniformly for zeK • while
(2.3)
[ G~(z)l<C(ro) Iz[ z-",
0<lzl_-<r0.
The d~ in (2.2) are constants depending only upon the dimension n, and z ~ is
chosen to have Re z ~ > 0.
23*
338
D.W. THOE:
For
reeK • r>O and fixed n set
-
-2
F : (r) = a~ con 1 :
(2.4)
+
G. (s: r),
where co, denotes the surface area of the unit sphere in Rn3. F~ (r) is a fundamental solution of (A + r;2) u(x)= 0 in R. which has the asymptotic behavior
dj
(2.5)
l-n
~-~ F : (r)=(__ i ~)i c~
-l--n
e+'~'[r-2-+O(r---z--) ]
for each j>=0, uniformly for xEK~ as r--+~. In particular, given 5 > 0 , r o > 0
there exists a constant M(6, ro) such that
1--n
(2.6)
I F f (r)l
<=M(5, ro) r 2
uniformly for reK~, r>r o. For small r a similar estimate holds. Given 5 > 0 ,
ro > 0 there exists a constant M(5, ro) such that
IF~ (r) l < M (5, ro) r 2-n
(2.7)
uniformly for 0 < r < to, r ~ K ~ .
Let Zo, zleK. By (2.1)
F
where F is the line segment which joins z 1 to Zo. Applying the estimates (2.2) and
(2.3) for G,+ 2, one can obtain the following estimates for F : :
Let x, x', yeRn, with ]x-x'[< 89 Given constants 5 > 0 , ro>__l, there exists a
constant N(5, ro) such that
[Fff (Ix--yl)-Fff (lx'--yl)l
n--3
(2.8)
( N I x - x ' l ~=olX-ylE+k-"lx'-yl-k-1
if [ x - y l < r o ,
1
=
-1--.
Nix_x,l[ix_yl--2~+lx,
iflx-yl>ro,
--1-n
y[ 2 ]
uniformly for r ~ K f i .
By the same procedure one can prove the existence of a constant, which we
shall again denote by N(5, to), such that
(2.9)
N [ ~ - ~ : ' [ [ x - y l 2-n
/
[Fff(lx--yl)--Ffi(Ix--Yl)l=[Nl~_d[ I x - - y [ - 2 3-n
uniformly for n, ~'EK~.
3 The constant a~ is chosen so as to obtain the normalization
lim r " - I ~
r--~O
ar
Ff (r) = co: 1.
if [ x - y [ < r o ,
if
Ix--yl>ro,
Eigenfunction Expansions for Schroedinger Operators
339
Of primary importance to us is the fact that if r E K +, Im ~ > 0 then the resolvent operator R(rc 2, H o ) - ( x 2 I - H o ) -1 is an integral operator with kernel
- - F + ( [ x - y [ ) . Thus we have
(2.10)
R (x 2, Ho) f ( x ) = - ~ F+ (1x - y 1)f ( y ) d y
Rn
for K~K +, Im x > 0 .
w 3. Existence of Eigenfunctions
Let B denote the Banach space of complex-valued continuous functions f ( x )
defined on Rn and tending uniformly to 0 as Ix[ ~ oo, with the norm
llfll~=max If(x) l.
xERn
Define an operator T~ for f u n c t i o n s f ~ B by
(3.1)
T~ f ( x ) = -- ~ F f (1x - y 1) q (y) f ( y ) d y
Rn
where K e K ~.
For convenience in notation, we shall use the abbreviations
7" -- T~+,
F =_F~+
throughout the remainder of the paper, and will state theorems and lemmas
concerning the operator T~ only. The same results will apply equally well to the
operator T Z ; the only changes required in the proofs are the substitutions of
" - " for " + " in the appropriate subscript or superscript locations.
Lemma 3.1.
(a) T~ is a bounded linear operator on B to B, (xEK+).
(b) T~f (x) is a H6lder continuous function of xE Rn with exponent c~/2, uniformly
with respect to x ~ K ~ . The constant c~ appearing here is identical to that appearing
in (1.3).
Proof. Let f e B . We use the estimates (2.6) and (2.7) with ro = 1 and write T ~ f
in the form
T~f(x)=- S F~+(Ix-yI)q(Y)f(y)dy = - S R~
S(A)
S(a)'
=L(x)+I2(x),
and proceed to show t h a t / 1 (x) and 12 (x) are bounded and tend uniformly to 0
as I xl --, oo. Here and in the sequel S(x, r) denotes the ball of radius r and center
x, S(r)=S(O, r), and S(x, r ) ' = R n - S ( x , r) denotes the exterior of S(x, r).
F r o m (2.6) it follows immediately that
1--n
I x ( x ) = O ( I x l --2--)
as I x l ~ o o ,
340
D.W. THOE"
so that I1 (x) is certainly bounded for large [x[. Moreover, 11 (x) is bounded
uniformly in x~R., for (1.3 (iii)) and the Schwarz inequality show that
IIl(x)l=] j" F~(Ix-yl) q(y)f(y) dy I
S(A)
<_C Hf[]B [
S
Iq(y)l dy
S(A) s(,,,1)Ix-yl "-2
<C IlfHB[Mq(x, 1)+ ~
Iq(Y)l
S(A)
j"
s(a)~s(x,1)'
Iq(y)l dy ]
Ix--y[ . -2I
de]
IIflIB.
<c
To estimate
t-
Iz(x ), use
(2.6) and (2.7) with ro = 1 to obtain
112(x)1=1 $ F~(Ix-yl)q(y)f(y)dy]
S(A)"
<C [If liB [
-S
I(x,1),
S<A)'~
n+l+hdY
n-1 ~-
n+l dy
I
lYl--Z- ix_yl-r- s(a),~s(x,1)lyl-~--+hlx_yl.-2
']
= C IIf IIB [I; (x) + I~'(x)].
It is evident that 1~ and 1~' are bounded on each sphere Ix I < r < 0% and by comparing I~ (x) with the homogeneous function
ml(x)= I
.+1
dy
.-1
R. lYl--2-+h Ix-yl
2
it is found that II~ (x) I = ml (x) = O (1x I-h) as Ix I --' oo. In a similar manner, I~' (x)
is bounded on compact sets and is majorized by the homogeneous function
mz(x) = I
R.
.+1+dy
lyl--r- hlx__yln-2
Thus
[I'2'(x)l<=me(x)=O(lxl Lr-ry-h)
as [ x l ~ o o .
As a consequence we have, certainly, I2 (x)= o(1) as Ix] ~ 0% which together with
the estimate for I1 (x) shows that T~f(x)is bounded and vanishes uniformly at oo.
In order to complete the proof of (a) and at the same time prove (b), consider
the difference
T~f ( x ) - T~f(x') = - I [F~([ x -
y 1)- F~ (1x' - y [)]
q (y) f(y) d y
Rn
where x, x'~R. satisfy ] x - x ' [ < 1. Apply the estimate (2.8) with r o = l and 6
chosen so as to have
5<l,~l <-}-,
The result is
(3.2)
0<5<1.
]T~f(x)-- T,,T(x')I <=CIx--x'] [13(x,x')+ I4(x,x')]
Eigenfunction Expansions for Schroedinger Operators
341
where
n-3
Ia(x,x')= Y'.
I
Iq(Y)l
Ix-ylP+2-~Ix'-yl-P-ldY
p=O S(x,1)
n-3
= E B,(x,x'),
p=0
--1-n
I4(x,x')=
--1-n
I Iq(Y)l[Ix-Yl--r-+lx'-yl
z ]dy.
S(x,*)'
Consider now a typical term Bp of I 3 , and suppose that it is the case that n - 2 - p
p + l , so that
n-3
O<p< 2
Now apply Schwarz's inequality, the estimate (1.3 (iii)) and a homogeniety argument to obtain the inequality
[~
Bp(x,x')<M~(x, 1) s~ I ) I x - y l
<=C
dy
]~
n-~-2p I x ' - y l 2p+2
ix_yln_~,_2plx,_yl2p+ z
< C Ix-x'l ~/z-1.
In the event that n - 2 - p < p + l we note that
proceed as above. Notice that the case
n-3
P= 2
Bp(x,x')=Bn-p-a(x', x),
and
can occur only if n is odd. The estimates for 13 (x, x') show that
Ix-x'lI3(x,x')=O(lx-x'l ~'/2)
at
x=x'.
The estimates for 14 are less involved; for example we need only notice that
by (1.3 (ii)),
S
[q(y)[dy <
=
s(x,,), ix_yl--f-
S
[q(y)id--Yr
s(A),~s(x.*)"ix_yl--r=Jl(x)+J2(x).
q-C
j
dy
n+l
n+l+h
S(x,1)',~S(A)"[yl--2-- ix_yl-'Z-
The integral J, (x) is bounded by
S Iq(Y)ldy.
S(A)
The integral J2 (X) can easily be estimated by Schwarz's inequality, with the result
dy
]~
dy
§
342
D.W. THOE:
This shows that ,lz(x ) is bounded in x e R . . The second term of I 4 can be handled
in a manner similar to the first term estimates. Indeed
S
Iq(Y)ldy<
n+l
S
Iq(Y)Idy
=
s(x,1), I x ' - y l 3"
n+l
stx,,,), I x ' - y l 2
since S(x', 89 S(x, 1), for we have been assuming all along that l x - x ' [ < 8 9 It
follows then that/4(x, x ' ) = O(1) at x = x ' , and so from (3.2)
ITs f ( x ) - T~ f(x') I < C Ix - x'l ~/2
if Ix - x ' l < 8 9 Here the constant C depends only upon q and the constant t5
selected at the beginning of the proof. This shows that the Htlder continuity is
uniform with respect to x~K~+, and completes the proof of the 1emma.
Remark. The above proof uses only the boundedness of the functions belonging
to B. Hence T~f(x) is a H61der continuous function of x belonging to B if o n l y f
is bounded, and
IT~f(x)l<llT~llRmaxlf(x)l,
(x~Rn).
XERn
The following lemma is an easy consequence of Lemma 3.1. For the proof
see IKEaE [I], Lemma 4.2.
Lemma 3.2. T~ is completely continuous.
We next apply the Riesz-Schauder theory of compact operators in a Banach
space to the operator T~. A summary of the results to be obtained is given in the
following
Lemma 3.3. (a) I f f ~ B satisfies
(3.3)
( I - T~) f =O
( x e K +)
( I - T~) f =g
( x ~ K +)
then f(x) - O.
(b) The equation
has a unique solution
f = ( I - T~)-1 g~B
for each g~B.
(c) The operators T~: B ~ B , (xEK+), depend continuously upon x and are
bounded uniformly in norm for reK~-, 6fixed.
(d) Let g ~ B be strongly continuous in x e K + for some 6, and let f ~ B be the
unique solution of
fx=T~f~+g~.
Then f~ is strongly continuous in rceK~'.
Proof. The proof of (a) in the case when Im ~2 =~0 follows exactly the proof
given by IKEBE in the 3-dimensional case (Ird~B~ [1], Lemma 4.5). If Im K2 ~=0,
then (3.3) can be written in the form (see (2.10))
(3.4)
~f= R (x2,1Ho) (q f ) ,
Eigenfunction Expansions for Schroedinger Operators
343
for qf~L2(Rn), and - F ~ is the resolvent kernel for Ho when x~K +, Im r 24=0.
Now (3.4) states t h a t f ~ L 2 (Rn) n ~ ( H o ) and satisfies (~c2 - H 0 ) f = qf, or equivalently, ( ~ 2 - H ) f = 0 . But this is clearly impossible if Imrc24:0, for H is selfadjoint and has real point-spectrum.
If x ~ K + is real, the proof of (a) is considerably more involved. However, it
is again possible to extend the results of POVZNER [9] and IKEBE for the 3-dimensional case to the general Rn-case, n > 4. Since the proof is quite lengthy, we shall
give only a sketch here. One begins by showing that i f f ~ B satisfies (3.3), then
l--n
f(x)=O(lx[ --~--)
as ] x ] ~ o o .
Consequently qf(x)=O(lxl -'-h) as Ixl--,oo. It then follows (see IKEBE [1],
Lemma 3.2) that
e ~~ Ix[
x- n
(3.5)
f(X)=--C.
.-1 S e - ~ x ' Y q(Y) f(y)dy+~
9
lxl--z-..
Here e. is a constant depending only upon n, while COx= x ]x I-1. A second consequence of the asymptotic behavior of q f and (3.3) is the radiation condition
(3.6)
For
lim ] x I-w-
i tc f
= 0.
the proof in the R 3 case see POVZNER [9], Chap. II, Lemma 2.
Since q a n d f are H t l d e r continuous, f is a solution of
-- Af + q f = x2 f
(3.7)
except perhaps at the singularities of q. Now from (3.7) we obtain
S [fAf-fAf]dx=O,
S(R)
SO that an application of Green's theorem and the radiation condition (3.6) leads
to the asymptotic result
(3.8)
[. If(x)12dSx=O(1)
as R ~ o o .
Ixl=g
In view of (3.5) we must then conclude that
1--n
(3.9)
f(x)=o(lxl -'z-)
as Ixl--,oo.
By a theorem of KATO, a f u n c t i o n f satisfying (3.3) and (3.9) must vanish identically outside of some sphere. Consequently f(x)=0 by the unique continuation
theorem for solutions of elliptic equations and the continuity of f (even at the
singular points of q).
A remark concerning the above application of Green's theorem is necessary,
f o r f does not necessarily satisfy (3.7) of the singularities of q. In order to verify
that (3.8) is a valid consequence of (3.7) even in the presence of singularities of q,
we form a new region from the domain of integration S(R) by deleting ,-spheres
344
D.W. THOE:
about each of the singularities of q (of which there are at most a finite number).
Let D~ denote the region thus formed; and let ~D, denote its boundary. Then
clearly
o_-D~s,-:.,J-.,:-..x-- 6~D~
s [:
0D~ consists of the surface [ x l = R and the surfaces [X--Xk[ =~,, {Xk}kN=1 being the
singularities of q. We assume e > 0 is so small that none of the surfaces [ x - x k [ =
intersect or contain any singularity of q other than the center Xk. We now integrate
(3.10) with respect to e over the interval [e/2, 5] and multiply by 2/~ to obtain
(3.11)
S
[f--~-~n-f-~--~n
" -] dSx= ~, 2/e
Ix l=R
k=l
~
f
-f
dx.
e/2<-lx-x~,l~_~
Estimates for the derivatives Ofit9 n can be obtained from the integral equation (3.3)
satisfied b y f . These estimates, together with the condition (1.3 (iii)) satisfied by
q, are sufficient to guarantee that each term in the sum (3.11) is o(1) as e--*0,
which justifies our use of Green's theorem.
Statement (b) follows from (a) by the Riesz-Schauder theory.
We now take up the proof of (c). Let fi be fixed, 0 < 6 < 1, and suppose that
f e b and x, x ' e K +. Let R satisfy R > A but otherwise be arbitrary, and write the
difference T,~f-- T,c,f in the form
T~f ( x ) - T~, f(x) = -- ~ (F~- F~,)([ x - y l) q (Y) f(Y) dy
Rn
(3.12)
=--
~ -- ~ =3a(x,R)+ J.c(x,R).
S(R)
S(g)"
Taking r o = 1 in (2.6) and (2.7), we find that
S(Rr~,S(x.1)" [x_yl--r-
S(R)'~S(x,1)
Ix--y[ n-2
= c6 IIf IIs ['ttRI) (x) + ~ > (x)].
Since R > A, (1.3 (ii)) produces the inequality
dy
i
,,+i+
s(m' lyl--Z- h l x _ y I 2
<~C R_h/2
I
s(A), lyl
dy
.-1.
2 [x_yl--z--
The last integral above is bounded on compact domains and can be shown to be
O ([ x l -h/2) as Ix I ~ oo (see the estimate for 12 in Lemma 3.1). Consequently,
suptl(R1)(x)=O(R -hI2)
as R--*oo.
x~Rn
A similar estimate holds for ~/~R
2), and thus for any given e > 0,
IJ4(x, g) l<e llfll~
as soon as R = R(e) is chosen sufficiently large. With R chosen and fixed, we turn
our attention to J3 (x, R). For this estimate we utilize (2.9) with ro = 1 and obtain
Eigenfunction Expansions for Schroedinger Operators
345
the estimate
IJ3(x) l=lJ~(x,e)l
ns
I x - y ' "-21q(y)ldyt< C ~ l x - ~ ' l Itfll~ [ s(R)Sx,1)
[ q ( y ) ,.d- ya ]
S
s(m~,s(x,1)" ix_yl-r-
__<c~ I ~r r'[ I[f IIB [Mq (x, 1) + ~ I q (Y) I d y]
S(R)
<=C(6,g) I x - ~ ' l IlflIB.
Returning to (3.12), it is now evident that for any given e> 0,
[[T~f-- T~, fllB<2e IlfllB
(3.13)
as soon as I x - x ' l is chosen sufficiently small, and the inequality (3.13) holds
uniformly in x, x' e K~.
Statement (d) relies upon (c) and the continuity of the mapping: S ~ S -1,
defined on the open set of regular elements in the Banach algebra formed by the
bounded linear operators on B.
Q.E.D.
We next introduce a family of operators defined on B and approximating T~.
For f e B , x e K + set
(3.14)
, T ~ f ( x ) = - ~ F ~ ( l x - y l ) q,(y) f ( y ) d y ,
Rn
where q, is given by (1.5). Since q~ has all of the properties possessed by q, ,T~
inherits all of the properties of T~ which have been established in Lemma 3.3.
Lemma 3.4. Let 6 be fixed, 0 < 6 < 1. Then
(a) ,T~ converges to T~ in the operator norm on B as tr ~ 0% uniformly in x e K +,
(b) ( I - , T ~ ) -1 converges to (I-T~) -1 as a--.oo, uniformly in x e K +.
Proof. Let f e B . Apply (2.6) and (2.7) with ro = 1 to the difference T ~ f - ~ T ~ f
to obtain
(3.15)
IT~ f ( x ) - ~T~ f ( x ) l
g
s(~)'~scx,,)
I q(Y)[ d y
I x - y l "-e
Iq(Y)
Idy_
s(~)'~s<x,1)' i x _ y l T
"1
,
Since a > A , we can substitute the estimate (1.3 (ii)) for q into (3.15). The resulting
functions of x thus obtained are bounded and have sup norms tending to 0 with
1/~ (see the estimate for t/~1~ in the previous lemma), uniformly in xeK~+.
For a proof of (b) see THOE [4], Lemma 3.7.
We are now able to show the existence of eigenfunctions of H. We take the
same course as Ird~BE(see [1], w and express the eigenfunctions q~(x, r (x, ~eR,),
in the form of a distorted plane wave.
(3.16)
~o (x, ~) = e ~x" ~ + w (x, O .
Here q~ is to be a bounded solution of the eigenfunction equation
(3.17)
- A~ (p(x, ~) + q (x) q~(x, ~) = ] ~ [z r (x, ~).
346
D.W. THOE"
Rather than deal with the differential equation (3.17), we consider instead the
corresponding integral equation
(3.18)
q~(x, 4) =e' x r
S FIr x - y l) q(Y) q~(Y, 4) d Y "
Rn
This integral equation cannot be solved in B as the inhomogeneous term e ix'r
does not belong to B. However, the iterated integral equation can be treated in B,
and to this end we introduce the function
p(x, 4 ) = -
(3.19)
S Fl~l(lx-yl) q(y)eir
dy.
Rn
By the remark following Lemma 3.1, it is clear that p(., 4)eB for each ~ e R n with
141>0, and p(x, 4) is Hrlder continuous in x~R,. If we put
w(x, 4) = q~(x, 4 ) - ei x. r
(3.20)
then (3.18) becomes
(3.21)
w(x, 4) =p(x, 4 ) - ~ FIr
x - y 1) q(y) w(y, 4) dy.
Rn
Now (3.21) has a unique solution w(., 4)sB for each 0 + ~ s R , , by Lemma 3.3.
Since p (x, 4) is Hrlder continuous, it follows from (3.21) that w, and consequently
q~, are Hrlder continuous. From the integral equation (3.18) we then find that q~
satisfies the eigenfunction equation (3.17).
We shall also require the eigenfunction for the self-adjoint operator H(cr).
Thus we also seek distorted plane waves q~,(x, 4) of the form
~o~(x, 4) = ei x. ~+ w~(x, 4)
(3.16)~
which satisfy the integral equation
(3.18),
q~,(x, 4 ) = e 'x" e_ ~ Flel(ix_yl) q,(y) opt(y, 4) dy
Rn
as well as the differential equation
(3.17),
- A~ q~,(x, 4) + q, (x) q~,(x, 4) = 1412 q~,(x, 4).
The function w,(., 4)~B appearing in (3.16), is obtained as the unique solution of
(3.21)~
w~(x, 4)=p,(x, 4 ) - ~ Flr
4)dy
Rn
with p,(-, 4)~B given by
(3.19),
p~(x, 4)= - ~ Flr
q,(y) e 'r'r dy;
Rn
such a solution w,~B exists for 0 + 4 ~ R , by Lemma 3.3.
We now sum up the above discussion in the following
Lemma 3.5. (a) There exists a unique solution w(., 4)~B of (3.21)for O+-4~R,,
4) defined by (3.16) is a solution of (3.17) and (3.18). Both
and the function r
Eigenfunction Expansions for Schroedinger Operators
347
r
~) and w(x, ~) are bounded and uniformly continuous in x and ~ for x ~ R . and
~ D , where D is any compact domain of R. which does not include the origin.
(b) Statement (a) holds equally well with q~ and w replaced by q~ and wr
respectively, and with the formula references in (a) subscripted with the symbol a,
(c) M a x l t p ( x , ~ ) - q ~ , ( x , ~ ) l ~ 0 as a ~ ,
a: E R n
uniformly with respect to ~ D , with D c 17. as in (a).
ProoL (a) It remains only to prove the uniform continuity of w(x, 0 in x
and ~. Since w can be written as
w(., ~ ) = ( I - - 711r 1 p ( . , ~)
it will follow from Lemma 3.3 (c), (d) that w is uniformly continuous in x e R , ,
~ D once we have established that p(., ~)~B is strongly continuous in ~eD.
To prove the strong continuity of p(., 0 , ~ED, it is first shown that given
e > 0, there exists a constant r 1 = rl (~) such that
(3.22)
Ip(x,~)-p(x,~')l<e,
x~S(rl)'
uniformly for ~, ~'eD. To see this, let
~l~l~+
for all ~D.
This is possible since D is compact and does not contain 0. Now employ the
estimates (2.6) and (2.7) with this choice of 6 and ro = 1 to obtain the inequality
ip(x, O l < C ~
[st~1 ) I[q(Y)l
dy
x - y l "-2
[q(y)[ d y ]
4-s~x,1)'
S ] x _ y l T,-1 9
Estimates for the integrals appearing here have already been obtained in Lemma
3.1 (see the estimates for 11 and 12), and show that these integrals vanish as
] x ] -~ 00, uniformly in all directions fox= ] x ] - 1x. Consequently (3.22) holds
uniformly in ~, ~ ' e D if x is restricted to lie outside a sufficiently large sphere
s(rl).
It remains to show that Ip(x, O - p ( x , ~')l is small with
in xeS(rl). To do so, write
I~-~'1, uniformly
p ( x , ~ ) - p ( x , ~')= - S (FteI-Fl~'l)(Ix-yl) q(Y) ei"r dY
Rn
- S FI~'I(IX-y[) q(Y)[ e'''r
(3.23)
dy.
Rn
Let r2 > 0 satisfy r 2 > Max [r 1 , A], but otherwise be arbitrary, and use the estimates
le,,.r
I'-"]
if lyl_<r 2
if ] y l > r 2 ,
348
D . W . THOE:
to obtain from (3.23) the estimate
Ip(x, 4)-p(x, 4')] <-<-S IFleI-FIe'II(Ix-yD
[q(Y)[
Rn
+r214-4'1
~
S(r2)
(3.24)
+2
~
S(r2)'
[FIe,I(Ix-yl)[
dy
Iq(Y)[
dy
]Fle,l(lx-y]) [ [q(y)ldy
=Js+J6+J7Estimates for the integral Js have already been obtained in the course of proving
Lemma 3.3 (see the estimates for Ja and J4). These estimates imply that for each
> 0 there exists a constant C(e, 6) such that
<
1J5(x,4,4,)l=~+c(~,6)[141-14'l]
for
xeR,, and ~141, 1 4 ' 1 ~ -1.
We next estimate the integral J7 (x, 4, 4'). Since r 2 > A we can write
(3.25)
IJT(x,4, 4')1
<C~
ely
+h
I
n+l+h
n--I
[s(,2)'~s(x,1)ly In+~
2 ay
i x _ y t , - 2 s(,2)'ns(x,x)'lyl-Z- Ix-yl 2
]
where we have employed (1.3 (ii)) and the inequalities (2.6) and (2.7) with r o = 1.
We are concerned only with x belonging to S(rl). It is clear that both integrals in
(3.25) tend to 0 as r2 ~ o% uniformly for xeS(rx). Thus it is possible to find
r2 > Max(r1, A) so that
IJ7(x, 4,4')1<~,
uniformly in xeS(rl), 4, ~'eD. With r2 chosen and fixed, we proceed to estimate
J6. Here it is only necessary to note that the integral
S IFle't(Ix-yl)q(Y)[ dy
S(r2)
is uniformly bounded for
Thus for x~S(rl),
x~S(rl), 4'~D
(see the estimates for 11 in Lemma 3.1).
]J6(x, 4, 4')[<M~
1141-1~'11.
Returning now to (3.24) it is evident that for
xES(rl),
Ip(x,O-p(x, 4')l<=2e-l-c(e,5) ]l~l-14']
]<3e
if 14 - 4'] is sufficiently small and 4, 4'e D. In view of (3.22), [p (x, 4 ) - P (x, 4')[ < 3 e
uniformly in x ~ R , if [4 - 4' [ < q (e), 4, 4'eD, which is the statement of strong
continuity of p(., 4)~B, 4eD. This proves (a) and (b).
Eigenfunction Expansions for Schroedinger Operators
349
To prove (c), note that
~0(., r
~0~(., ~) = w(., 4 ) - w~(., r
= ( I - ~r
p(., r
( I - ~ l ) -~ po(., r
= [ ( I - ~r
- (I--~r
P(', r
+ ( I - - ~ l ) - ~ [P(', 4) - p,(., r
and so by Lemma 3.3(d) it is sufficient to prove that
(3.26)
IIP(', r
r
-~0
as o-~ o% uniformly in ~eD. But since E(x, O = e ix'r is bounded uniformly in
x e R , , ~ D , (3.26) follows immediately from the remark following Lemma 3.1,
for
IIp(', 0 - p~(', ~)lIB = t[(Tic I --,TI~ t) E(-, ~)[In < 1[TIr ~TIr II ~ 0
as a ~ ~ , uniformly in r eD.
Q.E.D.
The integral equation (3.18), for q~ (x, ~) involves the operator ,T~ only for
real values of x, i.e. x= [~1>0. Introduce the complex parameter x by setting
(3.27)
p,(x, ~,x)= - S F~(lx-y[) q,(y) eir'r
Rn
and denote by wo(-, ~, ~c)~B the unique solution of
(3.28)
w,(.,~,x)=,T~w~(.,r162
for x s K +.
Finally, set
(3.29)
q~(x,4,x)=e~X'r
x),
tceK+.
The following result is an easy consequence of the preceeding lemmas.
Lemma 3.6. (a) Given 6, 0 < 6 < 1, there exists a constant M ~ > 0 such that
Iw=(x,#,~c)l<=Mn
for x, 4eRn, x ~ K +, and trfixed.
(b) w~(., 4, x)~B is strongly continuous in x~K+ for each 4~Rn.
(c) w~(x, r Ir
4), 4~:0.
It is useful to regard the function F~(lx-yl)q~(y) as a kernel defining an
integral operator in L2(f2), where f 2 = t 2 ( q , ) c R , is a bounded domain which
contains supp(q,) in its interior. Introduce in L 2 (f2) the integral operator S~=
S~(tr) by setting
(3.30)
S~ f(x) = -- S F~([ x - y 1) q~ (y) f(y) d y
for f e L 2 (f2), x e K +. The compactness of supp (q,) plays the central role in what
follows.
350
D.W. THOE:
Lemma 3.7.
(a) S~: L2(C2) ~L2(s
(b) The equation
is compact and depends continuously on KeK +.
(I-S~)f=O
(3.31)
( x ~ K +)
has the unique solution f = 0 in L 2 (f2).
(c) The inverse (I-S~) -1 exists for x e K + and
II(I-S~)-lll<C~,
x e K +.
Proof. (a) Let r e K +. Choose 6, 0 < 6 < 11 so small that xeK6+. From (2.7) we
obtain the existence of a constant C(6, f2) such that
IF~(lx- yl) [<C(6,O) I x - yl z-",
(3.32)
uniformly in x~K~+, (x, y)~f2 x C2.
Thus
Is~f(x) lz -_<[C(6,0)] 2 I f [q(y)f(y)[ d e ] 2
U
(3.33)
< [C(6, ~) Mq(x, diam(~))] 2 I If(Y)12 dy
=
~ Ix-yl"-"
< [M(6, ~)] 2~ [f(Y)lZdy
=
Ix-yl
Integrating (3.33) with respect to x over the domain t2 and interchanging the
order of integration shows that S~ is indeed a bounded operator (clearly linear)
on L2 (t2) with [1S~ ][__<N(6, t2) if ~:eK~+.
The compactness of S~ is obtained from (1.3 (iii)). For fl>0 let S~: L2(t2)
L 2 (s be the integral operator with kernel s~ (x, y, fl) given by
s~(x,y, fl)={O
F~(]x-y[)q~(y)
if [ x - y l < f l ,
if ]x-y]>fl.
S~ is a Hilbert-Schmidt class operator on L 2 (~) for fl > 0 and S~ ~ S~ as fl ~ 0.
In fact, if x~t2 then
[S~ - S~] f(x) = -
~ F~([ x - y [) q, (y) f(y) d y,
Ix-yl--#
SO that the inequality (3.32) and the Schwarz inequality show
(3.34)
][S,,-S~]f(x)12<[C(6,f2)supM~(x, fl)]2~
[f(Y)12dy
Integration of (3.34) with respect to x over s and an appeal to (1.3 (iii)) shows
that
II
I1_-<c ( 6 ,
sap Me(x, fl) o(fl)
=
JCEO
at fl = O, which proves that S~ is compact.
Eigenfunction Expansions for Schroedinger Operators
351
Consideration of the estimate (2.9) leads readily to the continuous dependence
of S~ upon x e K +, and completes the proof of (a).
For the proof of (b) it will suffice to prove that i f f satisfies (3.31) t h e n f is
bounded, for in this case this equation may be written in the formf=oTJ (see
the remark following the proof of Lemma 3.1). We can then conclude t h a t f ( x ) - 0
by Lemma 3.3 (a). Suppose then, that feL 2(t2) satisfies (3.31). By (3.33) it follows
that
If(y) l2 dy
(3.35)
If(x)12<C~ Ix-yl"-"'
xe~2.
Inserting the estimate (3.35) forf(x) into the integral in (3.35) and interchanging
the order of integration leads to the improved estimate
If(y)12 dy
If(x)12<CJ ix_yl,-2~,
x~f2.
Repeat this process k times, where k is the least integer satisfying k>ne, to obtain
the boundedness off(x) for xsl2. From the integral equation (3.31) satisfied b y f
it is readily seen thatf(x) is bounded for x outside of g2. Thusf(x) is uniformly
bounded in xeR~ and, as we noted above, this proves (b).
The proof of (c) is clear.
w4. The Eigenfunetion Expansion Theorem
The Fourier transform, defined by
a~(~)=(2rc) -'/2 I
e-'X'r
dx
Rn
f o r f e Cd~(R.) provides a spectral representation for the operator Ho = - A , since
under the extended (unitary) mapping:
L2(R.)~f ~
~L2(R.),
H o is represented by multiplication by [~l 2, while the Fourier inversion formula
f(x)=l, i.m.(210-"/2 [.A(O e'X'r d~
Rn
representsfeL 2 (R,) in terms of the (generalized) eigenfunctions e ~x"r of H0.
We will now prove a similar expansion theorem for the operator H which
incorporates the eigenfunctions ~p(x, ~) given by (3.18) in place of the eigenfunctions e ~x"r of H0.
By the spectral theorem for self-adjoint operators there exists a resolution of
the identity {Ex} such that
H= S 2 dE~,
and we may choose the projections E~ to have Ea+ =E~.
It is known [2] that on the negative real line the continuous spectrum of H is
absent, and that the negative spectrum, if it exists, is discrete and has finite multiplicity, that is, the negative point spectrum forms an isolated point set having no
24
Arch. Rational Mech. AnaL, Vol. 26
352
D.W. THOE:
limit point other than the origin 0. Let {Pk}, #k <0, denote these eigenvalues
counted according to their multiplicity, and denote by {ek (X)} the orthonormalized
eigenvectors associated with {Pk}" The subspace EoL 2 (R,) is spanned by {ek), and
if feL(Rn), then
E o f ( x ) = E l k ek(X),
IlEo/ll2=~ IJ~l2,
where
J~=(f, ek)L=(a.)= S f(x) ek(X) dx
Rn
is the generalized Fourier coefficient off.
Let P = I - E o be the projection associated with the positive spectrum of H.
A theorem of KATO implies that H has no positive point spectrum [6]. Thus the
subspace PL 2 (Rn) is a continuous subspace, and
(4.1)
E~=Ex+ =Ex_,
2>0.
It will follow from the theorem below that PL 2 is actually an absolutely continuous subspace.
Define the generalized Fourier transform ~(~) of a function ~ C~~(Rn) by
setting
(4.2)
if({) =(2rt) -"/2 5 ~(x) q~(x, 4) dx,
Rn
with q~(x, {) given by (3.18). Lemma 3.5 shows that if({) is defined and continuous for 04: { e R,.
Theorem 1 (Eigenfunetion Expansions for H). (a) If ~ e C~ (R,), then ~ (~) is
in L z (R,) and
IIer 2= 51ff(012dg.
Rn
The map ~ ~ can be extended to a unitary map of PL2(R,) onto L2(R~). If f,
geL 2 (Rn) then
llfll == IIEofll =+ IIPflI2=~ I~12+ S If(~)l z d~,
Rn
I
Rn
f(x) = I. i.m. ~ f({) q9(x, ~) d { + Z f~ ek(x).
Rn
(b) The part of H in PL a is unitarily equivalent to Ho.
(c) f e ~ ( H ) if and only if [r Iz f ( O e L z (R~) and Z# 2 Ifk [z < o0. For f e ~ ( H )
we have
H f(x) = I. i.m. 5 [{ [2f ( O q~(x, {) d ~ + ~ #k fk ek(X).
Rn
The proof will be broken into several steps. The first of these deals with the
operators
H(tr) = S 2 dEa(tr)
which approximate H (see (1.6)).
Eigenfunction Expansions for Schroedinger Operators
353
For x ~ K + and ~k~Cg(R.) set
W~(x)=W~(x,~b)=(2n) -"/2 ~ q70(0(1~12-~2)-~ q~(x,r
(4.3)
d~
Rn
and
(4.4)
off~({) =(2n) -*12 ~ ~b(x) 9,(x, r re) dx.
Rn
Since fro({) vanishes rapidly for I@l--,oo, Lemma 3.5 implies that W,(x) is
bounded and continuous in x e R,.
Lemma 4.1. For each OeCd~
M(6, ~,, a) such that
(4.5)
and 6, 0 < 6 < 1 ,
j" I +ff,,(~) 12d ~__<M(6, ~p, a),
there exists a constant
x~K~.
Rn
Choose a bounded domain f2=f2(a, ~p), f2 ~R~, so that supp(~b)w
supp(qr is contained in the interior of f2. From the definitions of fro and o ~ we
have
(4.6)
.ff~(r
-"/z I ~k(x) w.(x, ~, ~) dx.
Proof.
t~
Let rl~LZ(R,) be such that qo(~)~C~(R~). By (4.6) and the Fubini Theorem,
(4.7)
~ ~o(~)[-~ff~(r162
d ~ = ~ ~b(x) U~(x)dx
Rn
42
where
U~(x)=(2a) -"/2 I ~o(~) w~(x, {,x) dr
(4.8)
Us(x) is bounded in x~f2, x~K~, and therefore UrrL2(g2). Inserting the expression (3.28) for w, into (4.8) and applying the Fubini Theorem to interchange
the order of integration leads to the equation
U~(x) = - ~ F~ ([ x - y 1) qo (y) U~(y) d y - ~ F~ (1 x - y D q~ (Y) r/(y) d y
t2
t2
=s~ vAx) + S~n(x).
Thus U~~L 2 (f2) satisfies
us (x) = (I - s~)- 1 s+ n (x)
and so by virtue of Lemma 3.7,
II U~ [I,2r ~ C(6, 42) IIq llL2<m
(4.9)
C(6, ~) II~ IIL2<R.>,
uniformly in ~c~K~+. Returning to (4.7) and incorporating the estimate (4.9) with
Schwarz inequality leads to
I(~o, ~ 24*
~7o)1=<c(6, f2) 11~
IIL2(R.) II,7 I]L2r
-
354
D.W. THOE:
In view of the arbitrariness of ~/, this implies that ~ - ~ o
uniformly in ~ K +.
~L2 (Rn), and that
Q.E.D.
Lemma 4.2. W~(x) given by (4.3) with Im ~>0 satisfies
(a) W~= R ( x 2, H(a)) ~.
(b) JR@ 2, H(a)) ~k]o(4) =(1 ~ 12_~2)-
'or (~).
Proof. Since W,,(x) is bounded and supp(q.) is compact, q.W,,~L2(Rn).
Substituting the expression (3.28) for w. into the integral (4.3), it follows by the
Fubini theorem and the Fourier inversion formula that
W~(x) = - (2r0 -"/2 S ffo(~) (I r 12- x2)- ~e' x. r d e - S F.(I x - y I) q,(y) W,(y) dy
Rn
Rn
= [R (x 2, Ho) ~k+ R (x 2, no) q, W.] (x).
This last equation shows that W,~(Ho)=~(H(a)), and that W. satisfies
(x~-Ho) W.=O +q, W.
or equivalently,
W. =0,
which proves (a).
Now l e t f e C~ (R,) and notice that by (a)
~ R(~ 2,H(a)) ip(x) f(x) dx =
(4.10)
Rn
~ ~k(x) W,,(x, f) dx.
Rn
Upon substituting for W,,(x, f) in (4.10) and interchanging the order of integration one finds the equation
(4.11)
~ R@2,H(a))~p(x)f(x)dx= ~ (l~12-~2)-~,ff,(~)j~(~)d~.
Rn
Rn
The interchange of order of integration is valid since ~p,(x, 4, x) is bounded for
x, ~ R , and fixed re. It now follows easily from the Parseval identity and the
arbitrariness o f f in (4.11) that (b) holds true.
Q.E.D.
The next result provides the key step needed for the proof of Theorem 1.
Lemma 4.3. If ~k~ Cd~(R~) then
(4.12)
((Eb--Ea)r
a<lr
~
1r
d~
for 0 < a < b and ~ given by (4.2). Recall that {Ez} is the spectral resolution of the
identity belonging to H.
Proof. We first obtain the equation
(4.12),,
((Eb(~)--EZ~)) ~, ~,)=
[.
a<lCl2<b
14.(4) Is d~
Eigenfunction Expansions for Schroedinger Operators
where ft. is given by
(4.13)
355
f f . ( 0 = ( 2 n ) - " / 2 ~ ~k(x)(P.(x,O d4.
Rn
Formula (4.12) will then follow directly from (4.12)., for by Lemma 3.5(c)
ft. (4) converges to ~ (4) as tr --* oo, uniformly in 4 satisfying 0 < a < ] 4 [2 < b, while
(E~(o)- eo(~)) ~ -, (E~- go) 0
as a ~ oo by a Theorem of RELLICH [10]. Thus
((Eb-E.) ~k,~k)= lim ((Eb(a)-E~(a)) ~k,~k)
r
oo
= lim
~
1g7~(4)12d4
~--, oo a < l r
=
.[
1ff(r162
a<lr
We turn now to the proof of (4.12).. By using the resolvent loop-integral formula
[13] and Lemma 4.2(b) we obtain
b
(E~,(a)g/-E.(a) ~, r =lim __e .[ IIR(#- i ~, H(a)) ~ [[2 d#
es
7~ a
b
=lira --e ~ d# $
~,[0 7~
[(1412-~)2+a~3 -t Io~7~(~)12d4.
Rn
a
Here we have written x in the form x = V~--+-~ with/~ >0, ~_~0 and Im x >0.
Pick ~ > 0 so small that 0 < 5 < a < b < ~- ~ and choose eo > 0 so that x ~ K~+ for
a<=#<b, 0 < e < % . Now Lemma 3.6 asserts that , ~ ( 0 is continuous in x e K +
for each 4, and that
(4.14)
for
I~ ( r
< M ( 5, O)
4eR., a<=l~<=b,O<=e<=ao.Therefore (see [11])
b
lim ---~~ [(I 412_/~) 2+ 82] -~ [off~(4) 12 d/~
8~0
~
[ lim l ~ ( O I 2 = l ~ ( O I 2
if a < 1 4 [ 2 < b ,
_ l K~Ir
-[~..>o-,-
if 0 < 1 4 l Z < a or 1412>b.
Now (4.14), Fubini's Theorem, the bounded convergence theorem and our choice
of c5 show that
b
l i m a I d/~
~$0 7Z a
I
I-(I ~ 12-#):~ +,:]- ~ I ~.~,,('D I2 de
1~12<6 - 1
b
=lim
S
eJ,o I~12<~ - ~
=
S
a<lr
dCeS
~z a
IV;o(r162
[(I r 12-~)~ +~2]-11 ~ff~(r 12
356
D.W. THOE: Eigenfunction Expansions for Schroedinger Operators
It r e m a i n s o n l y to show t h a t
b
lim---~ S d/t
S
[ ( l ~ 1 2 - k t ) 2 + e 2 ] -1 I ~ , , ( ~ ) l z d ~ = 0 ,
e.Lo 7~ a
1r
-l
b u t this is a n i m m e d i a t e consequence of L e m m a 4.1.
Q.E.D.
W e r e t u r n n o w to the p r o o f of T h e o r e m 1. T h e basis for the p r o o f is f o r m u l a
(4.12) of L e m m a 4.3. W i t h this f o r m u l a the p r o o f given b y II(EBE (see [1], pages
2 4 - - 3 1 ) applies directly to the situation here, with only the simplest of m o d i f i c a tions. W e refer the r e a d e r to IIO~BE'S p a p e r f o r the r e m a i n i n g details of the proof,
a n d only r e m a r k t h a t the wave o p e r a t o r s W+ required b y Ir-~BE for the p r o o f of
the u n i t a r i t y of the m a p
pL2~f---,f~L 2
are k n o w n to exist for the potentials u n d e r c o n s i d e r a t i o n here (see [12]).
Added in Proof. We have been informed by S.T. KURODA of a generalization of Ir.EBE'S
expansion theorem to arbitrary dimensions, based upon an abstract stationary approach to
perturbation of continuous spectra.
References
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[8] WATSON,G., The Theory of Bessel Functions, p. 73. Cambridge: University Press 1922.
[9] POVZNER,A. YA., On the expansions of arbitrary functions in terms of the eigenfunctions
of the operator -- A u + c u. Mat. Sbornik 32 (74), 109-- 156 (1953).
[10] RaEsz, F., & B. yon Sz.-NAGY, Functional Analysis, p. 369. New York: F. Ungar Publ.
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[11] TITCmaARSn,E., Introduction to the Theory of Fourier Integrals, p. 31. London: Oxford
University Press 1937.
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[13] STONE,M., Linear Transformations in Hilbert Space, p. 183. New York: American Mathematical Society 1932.
Division of Mathematical Sciences
Purdue University, Lafayette, Indiana
(Received April 17, 1967)