Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
727 Yoshimi Saito
Spectral Representations for SchrSdinger Operators With Long-Range Potentials
Springer-Verlag Berlin Heidelberq New York 19 7 9
Author Yoshimi Sait6 Department of Mathematics Osaka City University Sugimoto-cho, Sumiyoshi-ku Osaka/Japan
AMS Subject Classifications (1970): 35J 10, 35 P25, 4 7 A 4 0 ISBN 3-540-09514-4 Springer-Verlag Berlin Heidelberg NewYork ISBN 0 - 3 8 7 - 0 9 5 1 4 - 4 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Catalogingin PubhcationData Satt6,Yoshml. Spectral representahonsfor Schr6dingeroperatorswith long-rangepotentials. (Lecture notes in mathematics; 727) 81bhography:p. Includes index. I. Differential equations. Elliptic. 2. Schr6dmgeroperator.3. Scattering (Mathematics)4. Spectral theory (Mathematics) I. Trtle. II. Series: Lecture notes in mathemahcs(Berlin) : 727. QA3.L28 no. 727 [QA377J 510'.8s [515'.353] 79-15958 Th~s work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. ~:i by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE
The present lecture notes are based on the lectures given at the University of Utah during the f a l l quarter of 1978.
The main purpose of the lectures was to present a
complete and self-contained exposition of the spectral representation theory f o r Schr'6dinger operators with longrange potentials. I would l i k e to thank Professor Calvin H. Wilcox of the University of Utah for not only the opportunity to present these lectures but also his unceasing encouragement and helpful suggestions.
I would l i k e to thank Professor
W i l l i Jager of the University of Heidelberg, too, f o r useful discussions during the lectures.
Yoshimi Saito
TABLE OF CONTENTS INTRODUCTION CHAPTER I Limiting Absorption Principle §I §2 §3 §4
Preliminaries Main Theorem Uniqueness of the Radiative Function Proof of the Lemmas
I0 21 29 39
CHAPTER I I Asymptotic Behavior of the Radiative Function §5 §6 §7 §8
Construction of a Stationary Modifier An Estimate for the Radiative Function Proof of the Main Theorem . Some Properties of r-~oo l i m e -l!j v(r)
53 64 78 86
CHAPTER I I I Spectral Representation §9 §I0 §II §12
The Green Kernel The Eigenoperators ExpansionTheorem The General Short-range Case
I00 104 ll5 126
CONCLUDING REMARKS
135
REFERENCES
141
NOTATION INDEX
145
TUBJECT INDEX
149
INTRODUCTION In this series of lectures we shall be concerned with the spectral representations for Schr6dinger operators (0. I)
T = -A+Q(y)
on the N-dimensional Euclidean space RN .
, yN) c R N
I Y = (Yl 'Y2 . . . .
(o.2)
N
A
Here
,~2
(Laplacian)
~ --2j : l ~yj
and Q(y) , which is usually called a "potential",
is a real-valued functio~
on RN Let us explain the notion of spectral representation by recalling the usual Fourier transforms.
The Fourier transforms
FO± are defined by
N
(Fo+f)(~)_ = (27,i ~ l.i.m.R '~ I'Yl< Re~iY~f(y)dy
(0.3) in where
f c L 2 ( ~ ) , ,~ = (~i,~2 . . . .
in the mean and
yg
L2(~N~) ,
, KN) c ~N~ , l.i.m, means the l i m i t
is the inner product in
NN
i.e
N
(0.4) j=l yj~j Then i t is well-known that L2(~)
FO± are unitary operators from
and that the inverse operators
F~± are given by
L2(~~)
onto
N
(F~zF)(y) = (2.r) --2 l . i . m ,
f
R~
(o.5)
e ~iy~.r ~I j~u.~~
IL
in L2(~~) for
F E L2(R~) .
Here i t should be remarked that
e ±iy~
are solutions of
the equation (0.6)
(-A-I~I2)u : 0
that i s ,
e = y~
eigenvalue
(l~I 2 =
are the eigenfunctions
I~! 2 associated with
N~ ~J)2 j=l
(in a generalized sense) with the
-A .
Further, when f ( y )
is s u f f i c i e n t l y
smooth and rapidly decreasing, we have r I
F0~(_Af)(~ ) = ]~[2(F0 f)(~,)
(0.7)
or
{I
(-Af)(y)
= F~,(l~12F0±f)(y)
which means that the p a r t i a l
differential
the m u l t i p l i c a t i o n
I¢12x, by the Fourier transforms
operator
operator
Now we can find deeper and clearer r e l a t i o n s
-A
is transformed into
between the operator
and the Fourier transforms i f we consider the s e l f - a d j o i n t the Laplacian in L2(RN )
L2(~N) .
F0± •
realization
Let us define a symmetric operator
h0
-A of
in
by
~(h o) : C~(~ N)
(0.8) h0f where
P(h0)
self-adjoint h0 .
H0
is the domain of and i t s closure
is known to s a t i s f y
= -Af
h0 . H0 = h0
,
Then, as is well-known, is a unique s e l f - a d j o i n t
h0
is e s s e n t i a l l y
extension of
D(Ho) : {fe L 2 ( ~ ) / I ~ I 2 ( F o ± f ) ( ~ : ) e
Hof = -Af
( f e P(Ho) )
( = iieCT~:~))
L2(AE!)}
,
,
(o.9) Fo£(Hof)(~) = !~I2(Fo±f)(E)
Eo(B) = F~± X/~
where
H2(AN )
(0,~) , > : ~ EO(- )
tial
operator
,
/ B--
f u n c t i o n of
-A
is acting on f u n c t i o n s of
H2(AN)
and the m u l t i p l i c a t i o n
operator
Let us next consider the Schr6dinger operator Q(y)
is assumed to be a r e a l - v a l u e d ,
H0 .
The d i f f e r e r
T .
FO+ give u n i t a r y !~12x . Throughout t h i s
continuous f u n c t i o n on
A N and s a t i s f y (0. I0)
Q(y) ÷ 0
Then a symmetric operator
h
(IYl defined by
hf
= Tf
can be e a s i l y seen to be e s s e n t i a l l y H in
" ~=)
D(h) = CO(AN)
(o.11)
L2(AN ) .
self-adjoint
with a unique extension
We have f
D(H) = D(H O) (0.12)
J
Hf
= Tf
(f c D ( H ) )
,
in the d i s t r i b u t i o n
Thus i t can be seen that the Fourier transforms H0
in
= { ~ , o ~ N, , / I ~ 12 ~B}
denotes the spectral measure associated with
equivalence between
lecture
,
is the Sobolev space of the order ?~ B is an i n t e r v a l is the c h a r a c t e r i s t i c
and
sense.
FO±
(fcP(Ho))
Here the d i f f e r e n t i a l sense as in (0.9).
operator
T
in (0.12) should be taken in the d i s t r i b u t i o r
These facts w i l l
be shown in !;12 (Theorem 12.1).
aim is to construct the " g e n e r a l i z e d " operators from
L2(IR )
onto
Fourier transforms
where
f)(F~)e_L2(RN)} C~
F+(Hf)(~) = I&I2(F f)(F~)
E(B)
B and
associated with transforms
F
which are u n i t a r y
L2(._) and s a t i s f y
D(H) = { f C L 2 ( ~ ) / i ~ i 2 ( F
(0.13)
F,
Our f i n a l
(feP(H))
,
,
= F* + x / B - F+
×/~
are as in (0.9) and
H .
It will
E(.)
is the spectral measure
be also shown that the generalized Fourier
are constructed by the use of s o l u t i o n s of the equation
Tu = rCl2u .
Now l e t us state the exact conditions on the potential lecture.
Let us assume that
(0.14)
~: > 0 .
short-range p o t e n t i a l potential.
and when
(o.]5)
the p o t e n t i a l
0 < c < l
The more r a p i d l y diminishes T
In t h i s l e c t u r e a spectral
f o r the Schr~dinger operator additional
(JYi . . . . )
When ~ > 1
becomes the SchrSdinger operator easier.
in this
Q ( y ) satisfies
~(y) = O(Iyl -c)
with a constant
Q(y)
T
Q(y) ~(y)
to
-A
Q(y)
is c a l l e d a
is c a l l e d a long-range. at i n f i n i t y ,
and
T
the nearer
can be t r e a t e d the
r e p r e s e n t a t i o n theorem w i l l
with a long-range p o t e n t i a l
be shown Q(y)
with
conditions (DJo~)(Y) = O(IYl -j-'~;)
(lyi
....
, j
:
0,1,2
.....
m O)
,
where
Dj
is an a r b i t r a r y
determined by
Q(y) .
d e r i v a t i v e of
order and
Without these a d d i t i o n a l
s t r u c t u r e of the s e l f - a d j o i n t the one o f the s e l f - a d j o i n t with a p o t e n t i a l
j-th
realization realization
is a constant
conditions the spectral
H of
T
of
-A .
H0
m0
may be too d i f f e r e n t
from
The SchrOdinger operator
o f the form
(0.16)
~(y) = ~
1
sin lYl
,
f o r example, is beyond the scope o f t h i s l e c t u r e . The core of the method in t h i s l e c t u r e is to transform the SchrSdinger operator
T
efficients. with its space
i n t o an ordinary d i f f e r e n t i a l Let
I = (0,~)
inner product
L2(I,X)
and
( ' )x
which is a l l
operator w i t h operator-valued co-
X = L2(sN-I) and norm
X-valued
, SN-I
I Ix .
being the
(N-l)-sphere,
Let us introduce a H i l b e r t u(r)
functions
on
I : (0,~)
satisfying co
(0.17)
llUl120 = I
l u ( r ) 12 d r < ° ° 0
I t s inner product
( ' )0
(0.18)
x
is defined by
(u'v)°
Foo
=
Jo ( u ( r )
, v ( r ) ) x dr
N-I Then a m u l t i p l i c a t i o n
operator
U = r 2 x
defined by
N-I
L2(I~N ) ~ f ( y )
÷ r 2
f ( r ~ ) e L2(I,X )
(o.19) (r=lyl can be e a s i l y seen to be a u n i t a r y operator from
, ~: ]~T e s N-l) L2(~ N)
onto
L2(I,X)
.
Let
L
be an ordinary d i f f e r e n t i a l
(0.20)
d2
L .....
where, f o r each
dr 2
operator defined by
+ B ( ~" ) + C ( r )
IF ~ I )
r > 0 ,
(xcX)
(C (I')X ) ( i ,~) = Q(r,~,)× (:~)
(B(r)x)(:.)
(0.21)
= - ~ (-(ANX)(~) + .(N-1)(N-3.)4 x(~.,)) r
and
AN is a n o n - p o s i t i v e s e l f - a d j o i n t
operator on
SN-I .
operator called the Laplace-Beltrami
I f polar coordinates
(r,Ol,O 2, . . .
yj = rsinO 1 sinO 2 . . .
singj_ 1 cosOj
,ON_I)
are i n t r o -
duced by
(j = 1,2 . . . . . N-I) (0.22)
YN = rsin01 sin02 " ' " sin0N-2 sin@N-I
(r)0
then
,
, 0@ 01,82 . . . . 8N_2~ 2~r , 0 ~ ON_1 < 2~)
ANX can be w r i t t e n as (ANX)(@l,02 . . . . . 0N_ l )
(0.23)
N-I
j=l for
(sin,gl...singj_l)-2(sin@j)-N+j+~uj ~n'l ~{(singj)N-j-I~gT l~x.
x e C2(S N-I) (see, e . g . ,
(0.23) with the domain
Erd~lyi and a l .
C2(S N-I)
[I],
is e s s e n t i a l l y
p. 235). self-adjoint
The operator and i t s closure
is
~
The operators
~N"
operator
T
and
L
are combined by the use of the u n i t a r y
U in the f o l l o w i n g way
(0.24)
T',_.', = U-I LU#
f o r an a r b i t r a r y smooth function shall be concerned with tained f o r
L
will
Our operator
L
c, on
instead of
~N. T,
Thus, f o r the time being, we
and a f t e r t h a t the r e s u l t s ob-
be t r a n s l a t e d i n t o the r e s u l t s f o r
T
T.
corresponds to the t w o - p a r t i c l e problem on the quantum
mechanics and there are many works on the spectral representation theorem for
T
~(y).
with various degrees of the smallness assumption on the p o t e n t i a l Let
Q(y)
satisfy Q(y) = O(ly! -e)
(0.25)
In 1960, Ikebe I l l
e > 0). T
developed an eigenfunction expansion theory f o r N = 3
the case of s p a t i a l dimension eigenfunction
( l Y i ÷ ~,
@(y,~)
(y,~ e ~ 3 )
and
~ > 2.
in
He defined a generalized
as the s o l u t i o n of the Lippman-Schwinger
equation (0.26)
~ ( y , ~ ) = e lyc~
and by the use of associated with Kuroda [ I ] other hand,
~(y,~), T.
1 jC eilr!iy-z[ - 4-~. IR 3 lY-Zl
Q(z)cI~(z,~)dz,
he constructed the generalized Fourier transforms
The r e s u l t s of Ikebe [ I ] were extended by Thoe [ I ] and
to the case where J~ger r l ] - r 4 ]
N i s a r b i t r a r y and
c > ~(N + I ) .
On the
i n v e s t i g a t e d the p r o p e r t i e s of an o r d i n a r y d i f f e r -
e n t i a l operator with operator-valued c o e f f i c i e n t s and obtained a spectral representation theorem f o r i t . N
3
and
the case of
~ > ~. N~ 3
His r e s u l t s can be applied to the case of
Saito [ I ] , L 2 1 extended the r e s u l t s of J~ger to include and
c > I,
i.e.,
the general short-range case.
At
almost the same time S. Agmon obtained an eigenfunction expansion theorem f o r
general e l l i p t i c
operators in
IR
N
results can be seen in Agmon [ I ] . settled.
with short-range c o e f f i c i e n t s .
His
Thus, the short-range case has been
As for the long-range case, a f t e r the work of Dollard [I] which
deals with Coulomb potential
Q(y) = ~ c,
of
Saito [ I ] - [ 2 ] .
s > ½ along the l i n e of
Schr~dinger operator with as Saito [ 3 ] - [ 5 ] .
Saito [ 3 ] - [ 5 ]
treated the case
Ikebe [ 2 ] , [ 3 ]
treated the
c > ½ d i r e c t l y using e s s e n t i a l l y the same ideas
After these works, Saito [ 6 ] , [ 7 ]
showed a spectral
representation theorem for the SchrSdinger operator with a general longrange potential
Q(y)
with
E > O.
S. Agmon also obtains an eigenfunction
expansion theorem for general e l l i p t i c
operators with lonq-ranqe c o e f f i -
cients. Let us o u t l i n e the contents of this l e c t u r e . Throughout this l e c t u r e , the spatial dimension f o r each
r > O.
N is assume to be
As for the case of
N i>i 3.
N = 2,
Then we have
B(r) ~ 0
see Saito [6], ~5.
In Chapter I we shall show the l i m i t i n g absorption p r i n c i p l e for
L
which enables us to solve the equation
(0.27)
{ (L - k2)v = f ,
l for not only
k2
(d-%- ik)v ÷ 0
non-real, but also
In Chapter I I ,
k,
range, there exists an element
holds.
k2
real.
the asymptotic behavior of
equation (0.27) with real
(0.28)
(r ÷ ~),
will x
v(r) ~ e i r k x
v v
v(r),
be investigated.
When Q(y)
is short-
~ X such that the asymptotic r e l a t i o n (v ÷ ~)
But, in the case of the long-range p o t e n t i a l ,
modified in the following
the solution of the
(0.28) should be
(0.29) with
v(r) ~ e i r k - i ~ ( r " k ) ~ v ~v • X.
Chapter I I ,
Here the function
~(y,k),
which w i l l
be constructed in
is called a stationary modifier.
This asymptotic formula (0.29) w i l l representation theory for
L which w i l l
play an important role in spectral be developed in Chapter I I I .
The contents of this lecture are e s s e n t i a l l y given in the papers of Saito [ 3 ] - [ 7 ] ,
though they w i l l
contained form in t h i s l e c t u r e .
be developed into a more unified and s e l f But we have to assume in advance the
elements of functional analysis and theory of p a r t i a l
differential
equations:
to be more precise, the elemental properties of the H i l b e r t space and the Banach space, spectral decomposition theorem of s e l f - a d j o i n t operators, elemental knowledge of d i s t r i b u t i o n ,
etc.
CIIAPTER I LIMITING ABSORPTION PRINCIPLE ~I.
Preliminaries.
Let us begin w i t h i n t r o d u c i n g without
further
some n o t a t i o n s which w i l l
be employed
reference.
~:
real
numbers
~:
complex numbers
~+ = {k = k I + i k 2 c ~/k I { 0, k 2 > 0} . Rek:
the real
Imk:
the i m a g i n a r y p a r t o f
I
=
(0, ~)
=
part of
{r
c ~/
0
T = [0, ~) = { r ~ /
s ,~ ~ ,
J.
r
< ~}
.
norm
, ;x
and i n n e r p r o d u c t ( , )x
is the H i l b e r t
on an i n t e r v a l on
<
k.
0 ~ r < ~} .
X = L2(S N - I ) w i t h i t s L2,s(J,X),
k.
J
space o f a l l
such t h a t
The i n n e r p r o d u c t
(
(l+r)S!f(r)Ix '
")s,j
f(r)
X-valued f u n c t i o n s
is square i n t e g r a b l e
and norm
il II
are
s,J
d e f i n e d by (f'g)s,d
= ! (l+r)2S(f(r)' ;j
g(r))xdr
and llfll s,J = ,r~ll~f,fjs,j] respectively.
When s = 0
may be o m i t t e d as i n C0(J,X) = UC0(~j),
where
Qj = {y C-~RN/lyl G J}
I,B
L2(J,X), d
or
,
J = I
II II s
the s u b s c r i p t
and
H 0 , s ( J , X ) , s E~R, is the H i l b e r t
U = r
or
I
etc.
is an open i n t e r v a l N-I 2
0
x
in
I,
is given by
(o,18).
space o b t a i n e d by the c o m p l e t i o n o f
11 a pre-Hilbert ii $ II
B,s ,J
=
j
space
CO~(J X)
l¢'(r)':~
( l + r ) 2s
with i t s norm + rB~(r)(~(r)
+ i,(r)l
x
^)
dr] ~ ]
and inner product " (d'"#)B,S, J= j!j ( l + r ) 2 S { ( ~ ~ ' ( r ) , ' ~ ' ( r ) ) x + (B½(r)#(r) , B~(r)~J(r)) x + (:~(r),,~(r))x}dr where
B~(r) = ( B ( r ) ) ~ ~D
_
#'(r) --~ I
or
FB(J,X), £
s
When J = I
as in
3 > O, on
.
with
B(r)
or
given by (0.21) and
s = 0
we shall omit the subscript
11 liB, H~'B(I,X) etc.
is the set of a l l
H~'B(j,X),
,
anti-linear
continuous
functional
i.e.,
£: H~,B(j,X ) I
~ v
~<#~,v>e
~ ,
such that
III z"~i~i. F~,j = sup{ ~,'<'. ~ , ( l + r ) ' ~/>. . /8. . c CO(J,X )
F~(J,X)
is a Banach space with i t s norm
J = I I!! Cac(J,X),
the s u b s c r i p t
lllj, J
I
will
being an (open or closed) f(r)
When B : 0
be omitted as in
on
J
interval,
F(J,X),
such that
is the set of a l l f(r)
continuous on every compact i n t e r v a l
e x i s t s the weak d e r i v a t i v e f'(r)
or
[II [[!6,j-
I!! III~ etc.
X-valued functions absolutely
0
Jlq~IIB,J = 1 } <. co.
E L?(J',X)
f'(r)
is s t r o n g l y in
f o r almost a l l
f o r any compact i n t e r v a l
J' c J.
J
and there
r E J
with
oi
12
cJ(J,X)
is the set of a l l
X-valued functions on
continuous d e r i v a t i v e s . J
j
such t h a t Cl f u n c t i o n ~(Y,Z)
j
strong
is a non-negative i n t e g e r and
is the set of a l l
into
p(r)
on
T
We set
denoted simply by
I-.
bounded l i n e a r operators from , fly,z, where
~(Y,Y) = ~ ( Y )
Y and
and the norm of
Z
Y are
~(X)
W.
-A N + '.:(N-I)(N-3),
where
AN is the Laplace-Beltrani
operator in
operator
X).
D = D(A). D~ = D(A~2). C(~,3 . . . .
)
denotes a p o s i t i v e constant depending only on
But very often symbols i n d i c a t i n g
m, ~ . . . .
obvious dependence w i l l
be
omitted. Hj(~ N) L2
is the Sobolev space of the order functions with
L2
distribution
j,
i.e.,
derivatives
the set of a l l up to the j - t h
order, i n c l u s i v e . C~(~N), L2(~)), L2(~, ( l + l y l ) 2 S d y ) where Let
L
is
II It.
means the domain of
(as a s e l f - a d j o i n t
f(r)
f o r any r e a l - v a l u e d
having a compact support in
Z with i t s operator norm
Banach spaces.
AN =
X-valued functions
of e L2(I,X ) ( p f ~ H ~ ' B ( I , X ) )
is the Banach space of a l l
D(W)
having
is an (open or closed) i n t e r v a l .
L2(I,X)lo c (H~'B(I,X)loc)
A =
Here
J
etc., will
~) is a measuable set in
be the d i f f e r e n t i a l
The remainder of t h i s section w i l l
be employed as usual,
~N.
operator defined by
(0.20) and (0.21).
be devoted to showing some basic
p r o p e r t i e s of a (weak) s o l u t i o n of the equation
(L - k2)v = f.
To t h i s
13 end, we shall f i r s t
list
PROPOSITION l . l .
some properties of (i) Let
2j = (y c ~ N / ! y I ~ J}.
J
H~'B(I, X).
be an open interval
in
I
and let
Then
(I.I)
H~'B(j,X) = UHI(~j)
and (l • 2) where
(@' U'~)B,J : (~"~)l,::!j U is given by (0.18) and
( ' )Id.!j
(#, ~.~e Hl(r!j)), is the inner product of
Hl(i-!j), i . e . ,
~,.j
(ii)
l,B v ~ H0 (I,X
Let
v E Cac(T,X ) ,~ L2(I,X) derivative
(or
v' ~ L2(I,X )
for almost all
r e I
inner product
(or
v c: Cac(I,X )
L2(I,X)Ioc)
v' e L2(I,X)loc ).
(or
with
v (= H ' B ( l , X ) l o c ) .
B'2v e L2(I,X )
( ' )B and norm
II IIB of
(or
Then with the weak
Moreover
v(r) c D~~'
B1~v c L2(I,X)loc ).
H&'B(I,X)
have the following
form : i
(u,v) B
=
II
{(u'(r),v'(r))
x
+ (B~(r)u(r),B~(r)v(r))x
(1.4)
+ (u(r),v(r))xldr ;
(iii) (1.5)
Let
v c H~'B(I,X)Ioc . v(O)
Then we have the relations = o ,
The
14 I
(I .6
Iv(r)
- v ( s ) i x =<
'r-sl
~> livlb
IO,RI)
(r,s ~
B,(O,R)
with i-~ R , I~v II B,(O,R) = i.jo i "iv'(r)'2.× + !B~(r)v(r)I2x
(1.7
(I .8
i
!v(r) P_R_O_O_F. ( i )
X -< v~ IIV:IB,(O,R+I)
follows
:IU~:IB, J = I ~ : I I , ? j
CO(!~J)
holds for
about
sequence
of
l im ,s' = vI ~n
(1.9)
and
,,en
l im B~^'"n = v 2.
I/,~n(r) - C;m(r)ix < = Jo
i t follows
that the sequence
r ~ [O,R]
with an a r b i t r a r y
is an X-valued,
continuous
~.~n(r)} positive
function
i f and only i f
converges to
there e x i s t s a .,,, i {'~n"
f6n}' v
By the
in
fB !f.in.~ are
L2(I,X).
l~'n(t)
- ~m(t) xdt
is convergent in
I.
Therefore Letting
X uniformly
(I~n(r)-~n(S)'X)x
we a r r i v e at
!
= Js ( ~ ' n ( t ) ' x ) x d t
for
v ( r ) = s - l i m #n(r) n ....
in the r e l a t i o n
rr
(I.I0)
We
From the estimate
R. on
+ B(r)
because the statement
such that the sequences
L2(I,X )
and
v e HIo'B(I,X),
v c H~'B(I,X)
(#n'i" c CO(I,X )
space
d2 L0 = ~
where we set
can be shown in quite a s i m i l a r way.
H~'B(I,X)
Cauchy sequences in set
only for
is the H i l b e r t
The second f a c t can be e a s i l y
(-';)c = U-II_oU¢ ,
v E H~'B(I,X)Ioc
definition
HI(':-SJ)
by the norm (1.3) and that the r e l a t i o n
~ c CO(~':j).
obtained from the r e l a t i o n Let us show ( i i )
(r • [O,R])
from two facts that
obtained by completion of
} -: ~ i v ( r ) I2, dri½,
(x c X, 0 _<_ s < r < ~o)
15
i.r (I.II)
(v(r)
whence f o l l o w s L2(I,X). set
that
Since
e
of
r c I - e.
I
- v(s),
X)x=Js
v .e C.a~(-[,X)
and the weak d e r i v a t i v e
Ba""'n converges. to such t h a t
Therefore
(vl(t),x)xdt,
v2
in
B~2(r)~n(r)
converges to
v ( r ) c: D~2 w i t h
B~'(r)v(r)
because o f the closedness of the o p e r a t o r
in
= v2(r)
X
for
In a s i m i l a r
2i r]
way we can show the f i r s t
Finally, T
n
us show ( i i i ) .
such t h a t
.....
,
~(r)
= I
whence f o l l o w s
(r S R + I),
{On. c C' ( l , X )
(1.6)
such t h a t
p(r)
Then, s i n c e llov - ,nllB ~ 0
that
Then, as is seen i n the p r o o f o f ( i i ) , (1.5)
C~ f u n c t i o n
= 0 ( r > R + 2).
llv - 9 n l I B , ( O , R + I ) +
r e [0, R + I ] .
o f ( 1 . 4 ) which is r e l a t e d
Take a r e a l - v a l u e d
there exists
(1.13)
n = 1,2 . . . .
relation
.
(u,v)B.
let
ov ~ H ' B ( I , X ) ,
follows
and ( 1 . 8 )
0
:>n(r)
(n ~ ~)
.
converges
from the f a c t t h a t
v(r)
Cn(O) = 0
are o b t a i n e d by l e t t i n g
n ....
in
f : : l ~ n ( r ) - ~ n ( S ) IX
I
's r ¢ ' -" ( t ) dt X L~
Ir-sl
in the
il~nll B,(O,R)
(0 < r ,
s < R)
X
for all
relations
and
for
Thus we o b t a i n
-I '
the i n n e r p r o d u c t
(I.14)
a null
r c. I - e
j
n~
as
there exists v2(r)
B;~(r).
in
llvli B = lim J[#n!IB = i-" i :v ' (r)Ix 2 + . B ' ~ ( r ) v ( r ) . 2.X + iv(r)! x d
(I.12)
on
L2(I,X),
v' = v I
for
16
:'Vn .... ( r )
(I .15)
2X ~ 2 ( l C n ( r ) - : : ; n ( t ) , 2X ~ , :;:n (t)12'X)
__.
respectively, i.;,n
(t)Ix
=
where
The i n t e r i o r
2,¢n112
(0 < r ,< R) = =
has been taken to s a t i s f y
'X"
Q.E.D.
e s t i m a t e f o r the o p e r a t o r
on ~N.
Let
L
exists
and
~ • F(I,X).
C = C(k,R)
is shown in the f o l l o w i n g :
satisfies
( v , (L-k2).i)O = <~', ':.> k • ~
L
be as above and l e t
v e H0I ' B ( I , X ) loc
Let
(1.16) with
-
,(O,R+l )
[ r , r+l l
l~n(s)l
PROPOSITION 1.2. function
ds + i r+l
t •
min r<s~r+l
l%(s)l~ ds)
2(',r-tl JrF t I /~:nf ( S )Ix2
Let
!~(y)
be a continuous
the equation
(::, ~ C ~ ( I , X ) )
R be a p o s i t i v e
number.
Then there
such t h a t
"v" B,IO,RI __
The constant set of
C(k,R)
(k,R)
moves in a bounded
~ x i.
PROOF. Since ~d d-r ( v ( r ) ,
(I.18)
is bounded when the p a i r
i
i(v(r),
f o r almost a l l
. l ,B,
v (- mO
k I , X ) l o c,
¢'(r)) x = (v'(r),
we have by ( i i )
r c I.
X + (v(r),
,~'(r))
B(r)?(r)) X = (B~(r)v(r),
of P r o p o s i t i o n
(~,I'(r)) x
B2(r)¢(r)) X
Therefore from (1.16)
it
f o l l o w s by p a r t i a l
l.l
17 integration
that
IOR+I {( v ' ( r ) , ~ ' ( r ) )
(1.19)
+ (v(r), f o r any
,:~ e CO(I,X ).
such t h a t C~
~..S.(r) = 0
(C(r)-k2)~!~(r))x~dr = <~', ~>
Set
o =
llv - #nJlB,(O,R+l)
function
on
[0, R+2]
~0
,v I 2 x +
R+l ~
where
n ~ ~, and 0 ~ ;;: < l ,
~,;J(r)
:~;n ~- Co((O'R+2)'X) is a r e a l - v a l u e d
#(r) = l
n .... ,
IB~V,x2+ iv,~imr + 2
(0 L-: r j R),
we have
f0 ~,,~'(v',v)xdr
R+I
:
= ] 0 (l.17)
as
Then, l e t t i n g
fo 2{, ,
(1.20)
in ( l . 1 9 ) ,
such t h a t
(R+I < r < R+2).
~. Ba(r)¢.(r))x
x + ( B~" 2(r)v(r),
~ 2 ( v , ( k 2 + l - C ) v ) x d r + <0,', ~2v> .
is obtained from ( I . 2 0 )
by the use o f the Schwarz i n e q u a l i t y . Q.E.D.
PROPOSITION 1.3 ( r e g u l a r i t y 1.2.
Let
v E L2(I,X)loc
with
k • ~2
and
(I)
Let
L
be as in P r o p o s i t i o n
be a "weak" s o l u t i o n of the equation
f E L2(I,X)loc,
(1.21) for all
theorem).
i.e.,
let
v
(L-k2)v = f
satisfy
( v , (L-k2)~,~)O = ( f ' ~ ) O ~ • CO(I,X).
Then
v
satisfies
v • H ~ ' B ( I , X ) I o c {~ C I ( I , X ) .
f o r almost a l l
r • I
with
following
v ( r ) ~ D~
Bv (= L 2 ( ( a , b ) , X )
(I) ~ (4):
f o r each
f o r any
r ~ O.
v(r) • D
0 < a < b < ~. I,.
(2)
v' • Cac(I,X)
f o r almost a l l
r • I.
w i t h the weak d e r i v a t i v e B~v ' c Lp((a,b),X)
for' any
v"
and
v'(r)
0 < a < b < ~.
E D"2
18
(3)
B~v c Cac(I,X )
(B:.~(r)v(r)),
(1.22)
holds f o r almost a l l (4)
f o r almost a l l
+ B(r)v(r)
v = u-lv
and
and from ( l . 2 1 )
L2(~N)Io c (1.24)
(v, where
well-known,
in
(T-
( , )L2
(1.24)
f = u-lf.
Then
k2)g)L2 = (7, ~)L2
= f(r)
as
L2((a,b),X )
v, f
belong to
(4 e C~(]R'q))
denotes the i n n e r product of
implies
H2(~RN ) l o c
.I,B H0 ( I , X ) l o c
obtain
- k2v(r)
the r e l a t i o n
that
Therefore there exists
~n ~ v
+ C(r)v(r)
r e I. Set
PROOF.
L1j).
+ B:~(r)v,(r
r • I.
-v"(r)
follows,
= _ l_ B:Z(r)v(r) r
(B~'v)
We have
(1.23)
in
w i t h the weak d e r i v a t i v e
as
n ......
~} c H2(]RN)loc
a sequence
n , ~. and
Set
{¢qn} '
(0 < a < b < co).
L2(~RN).
Ikebe-Kato
(see, e . g . ,
{~n ~ c C~(]RN) ~n = U~n"
- B ' 2~ ~ i " {Bd'~n}' .•. . . ~n
such t h a t
Then
"'n ....
are a l l
From these r e l a t i o n s
As is
v
in
Cauchy sequences
we can e a s i l y
(1) ~ (4). Q.E.D.
Now t h a t P r o p o s i t i o n one more f u n c t i o n the set of a l l (b):
space.
functions
1.3 has been e s t a b l i s h e d , Let v
J on
let
be an open i n t e r v a l J
which s a t i s f y
in
us i n t r o d u c e I.
the f o l l o w i n g
D(J)
denotes
(a) and
19
(a)
v • CI(j,X). r • J
v(r) • D
L2(J',X) (b)
v',
v ( r ) • D~ and
f o r each ~ D 2.
Bv
and
f o r any compact i n t e r v a l
J'
in
B2v • Cac(J,X)
v'(r)
f o r almost a l l follows
the e q u a t i o n
open i n t e r v a l point
B'~.~/Vl
+
v c L2(l,X)loc
H0l ' B ( I ' X) loc c~ O(1) "
theorem f o r the o p e r a t o r
L
takes the
form.
PROPOSITION 1.4. satisfy
belong to
J.
1.3 t h a t the s o l u t i o n
belongs to
The unique c o n t i n u a t i o n following
B~v '
r • J.
from P r o p o s i t i o n
o f the e q u a t i o n ( l . 2 1 )
For almost a l l
with
(B~v), . . . . 1 B'2v i" r
It
r • J.
in
r 0 • J. PROOF.
is a solution y • {y • ~ N / l l y continuation to show t h a t
I. Then
Set
Let
L
be as i n P r o p o s i t i o n
(L-k2)v = 0
on
J
Suppose t h a t
v(r)
v(r)
J.
= 0
v = U-Iv
on
(T-k2)v = 0
I - ro~r < T~} w i t h some theorem f o r e l l i p t i c on
= 0
k • ~,
where
{y • ~ N / I y
and
rl > O.
operators
v • D(J) J
is an
in a neighborhood o f some
as in the p r o o f o f P r o p o s i t i o n
o f the e q u a t i o n
v e 0
with
1.2 and l e t
v(y) = 0
1.3.
Then
for
Thus we can a p p l y the unique (see, e . g . ,
Aronszajn
[I])
I • J } , which completes the p r o o f . Q.E.D.
Finally theorem.
we s h a l l
show a p r o p o s i t i o n
which is a v e r s i o n o f the R e l i c h
20 PROEQS{.TI_0_N_!.5. Let bounded open interval Then
{v n}
is a r e l a t i v e l y
exists a subsequence sequence of
in
~u . m}
L I.
be as in Proposition 1.2 and l e t Let
{v n}
be a bounded sequence in
compact sequence in
L2(J,×),
of
{u m}
{v n}
such that
v- n
a bounded sequence in
H~'B(j,X) there
is a Cauchy
U-Ivn .
, Then, by ( i ) of Prooosition l.l,
Hl(q J)
with
Qj = {y e ~N/ly I c J}.
from the Rellich theorem that there exists a subsequence
{u m}
i.e.,
be a
L2(J,X ).
PROOF. Set
such that
J
{~m }
is a Cauchy sequence in
is a subsequence of
{v n}
L2(aj).
Set
{ V n ~~ is
I t follows
{Um]
of
um = Uum.
and is a Cauchy sequence in
{~n } Then
L2(J,x). Q.E.D.
21
§2.
Main Theorem.
We shall now state and prove the l i m i t i n g absorption p r i n c i p l e for the operator (2.1)
d2
L -
dr 2
+ B(r) + C(r)
which is given by (0.20) and (0.21).
The potential
Q(y)
is assumed to
s a t i s f y the following ASSUMPTION 2.1
(Q) Q(y)
can be decomposed as
Q(y) =Q o(y) + Ql(y)
Q1 are real-valued functions on ~N with (0.0)
% c CI(~RN)
N ~ 3.
and
',DJQo(y) I < co(l + lyl) -j-c
(2.2)
such that QO' and
with constants
c O > 0 and
(y c R N, j = O, l)
0 < c ~ I,
where
Dj
denotes an
a r b i t r a r y d e r i v a t i v e of j - t h order. (Ql)
O~1 ~- cOoRN)
(2.3)
and IQI(W)I-_< co(l + :Yl)
with a constant
~RN) (y e_-
-el
c I > 1 and the same constant
c O as in
We set IC(r) = Co(r ) + Cl(r)
,
(2.4) ICj(r)
= Q~(rw)x
Then, by (Qo) and (QI), we have
(j : O, I) .
(0~0).
22 {"Co(r)il < c o ( l + r ) -~: I
=
llC~(r)il
(2.5)
Cl(r)" where
II I[
'
< c o ( l + r ) II-~: ~ c o ( l + r ) -I-'~:I
denotes the operator norm of
d e r i v a t i v e of
Co(r )
in
B(X)
and
C~(r)
~(×).
Let us define a class of s o l u t i o n s for the equation k ~ ~+
and
given.
(L-k2)v = Z with
C ~ F(I,X).
DEFINITION 2.2. such t h a t
is the strong
(radiative
function).
½< 5 ~ min(¼(2+c), ~ i ),
Then an X-valued f u n c t i o n
~mc~ion for
{L, k, il},
I)
v ~ H~'B(I,X)Ioc.
2)
v'
3)
For a l l
Let
Let
v(r)
6
be a f i x e d constant
~ e F(I,X) on
I
and
k ~ ~+
be
is c a l l e d the r,~i~mi~e
i f the f o l l o w i n g three c o n d i t i o n s hold:
ikv E L 2 , ~ _ I ( I , X ). ~# • C~(I,X)
we have
(v, (L-k2)¢)O = <~., ¢>. For the notation used above see the l i s t beginning of ~l. and hence v
The c o n d i t i o n 2) means
of notation given at the
9v _ ikv 3r
is small at i n f i n i t y ,
2) can be regarded as a s o r t of r a d i a t i o n c o n d i t i o n .
This is why
is called the r a d i a t i v e f u n c t i o n . THEOREM 2.3.
(limiting
absorption p r i n c i p l e ) .
Let Assumption 2.1 be
fulfilled. (i) for
Let
{L, k, ~}
(k, ~') c ~+ × F(I,X) is unique.
be given.
Then the r a d i a t i v e f u n c t i o n
23
(ii)
For given (k, ~) c {+ × F~(I,X)
radiative function
(iii)
Let
v = v(., k, Z)
for
{L, k, C} which belongs to
K be a compact set in
the r a d i a t i v e function f o r
{L, k, £}
there e x i s t s a p o s i t i v e constant
there exists a unique
~+.
Let
with
C = C(K)
v = v ( - , k, 2)
k e K and
be
£ ~ F6(I,X ).
depending only on
K
Then
(and
L)
such t h a t
(2.6)
IIVIIB,_~S =< c'l! '!il 6'
(2.7)
IIvll
+ IIv'-ikvll
+ IIB~vll6-i :< c!ll 'ill,,s
'
and
llvll ,_6,(r, )
(2.8) (iv)
The mapping:
is continuous on mapping from
c2r-(2 -1)111 '11!
6+ x FS(I,X )
Therefore i t
into
.
I'B (I,X) (k, 2) , v(., k, r) e H0,-6
6+ x F~(I,X)
6+ x F~(I,X).
(r > I)
is also continuous as a
H~'B(I,X)Ioc.
Before we prove Theorem 2.3, which is our main theorem in t h i s chapter, we prepare several lemmas, some of which w i l l
be shown in the succeeding
two sections. LEMMA 2.4. {L, k, 0}.
Let
Then v
k ~ {+
and l e t
is i d e n t i c a l l y
The proof of t h i s lemma w i l l
For
f e L2,#(I,X )
<~[f]' @>= ( f ' ¢)0 (2.9)
for
with
v
be the r a d i a t i v e function f o r
zero.
be given in ~3.
B 2 0 we define
¢ ~ C~(I,X).
Illi,[f]lllB
Z[f] c F~(I,X)
by
It can be seen easily that =< llfll 3~ .
24 LEMMA 2.5. {L, k, ~'[f]}
Let
with
a compact set in
f e L2,S(I,X)~ ~+
B'2v ~ L 2 , 4 _ I ( I , X ) (2.10)
k ~ K and l e t
and
5
v
be the r a d i a t i v e
and l e t
function
for
where
K
v c L2,_.:~(I,X),
is as in D e f i n i t i o n
and there e x i s t s a constant
2.2.
is
Then
C = C(K)
such that
"v' - ikv"~_ 1 + IfB'~v,5_l =< C{llvll # + ! I f l l ~) ~ . -
'
and (2.11)
ilvll 2-~.,(r,-~)
LEMMA 2.6.
Let
C2r-( 2'-S-I )
=<
Vn(n = 1,2 . . . . )
~}
ilfll
be the r a d i a t i v e
kn c ~+ and 2n e F~(I,X).
where
+
(r
=>
function
I)
.
for
{L, k n, Zn },
Assume that
ik n + k e_ (~+ (2.12) in
!~'n ~ "~' as
n -~ 2
F~(I,X)
and that there e x i s t s a constant I
IlVn il -e~+ IIv'n
CO such that
iknvnll 5-1 <= CO '
(2.13) ilvnll2- ~ , ( r , ~ ) for a l l
n = 1,2 . . . .
which is the r a d i a t i v e (2.14) as
n
Then
{v n}
function vn ~ v
÷
~,
2 -(25-1) ~ Cor
(r ~ l )
has a strong l i m i t for in
{L, k, el.
I'B(I,×)
H0
v
in
L2,_~(I,X)
Further we have loc
25 LEMMA 2.7. with
B > O.
Let
k 0 • ~+
with
Imk 0 > 0
(v, (L-k~)#) 0 = <~, ~>
has a unique s o l u t i o n
v0
in
(2.16)
v0
is the r a d i a t i v e
Let
Then the f o l l o w i n g
Clllzlilm (c
satisfies
= C(k O, 6))
•
{L, k O, £}.
be proved in ~4.
~ ~ F6(I,X)
and
be a unique s o l u t i o n
k 0 e ~+, Imk 0 > O.
Let
(a) and (b) are e q u i v a l e n t :
(a)
v
is the r a d i a t i v e
(b)
v
is represented as
function for v = v 0 + w,
{L, k, ~l(k2-k~)vo ]}
PROOF. Lemma 2.8 d i r e c t l y (2.17)
v0
v 0 • H~'B(I,X) r~ L 2 , ~ ( I , X )
of the equation (2.15) with
function for
and
function f o r
Lemmas 2.5, 2.6 and 2.7 w i l l LEMMA 2.8.
(# • C~(I,X))
H~'B(I,X)
IIvoIIB,B~
k • ~+.
I • FB(I,X)
Then the equation
(2.15)
Therefore
and l e t
fL, k, Z}. where
w
is the r a d i a t i v e
.
follows from the r e l a t i o n
for
v • L2(I,X)Io c
(v - v O, (k-k2)#)O
= (v, (k-k2)¢)
- ~',
2 ~> + ((k2-ko)vo , 6) 0 (l:~ ~ CO(I,X))
and Len~a 2.7. Q.E.D.
28 By the use of these lemmas Theorem 2.3 can be proved in a standard way used in the proof of the l i m i t i n g
absorption p r i n c i p l e
f o r the v a r i o u s
operators. PROOF of THEOREM2.3. (I)
The proof is d i v i d e d i n t o several steps.
The uniqueness of the r a d i a t i v e
(k, C) ~ ~+ x F ( I , X ) (If)
function
f o l l o w s from Lemma 2.4.
Let us show the e s t i m a t e ( 2 . 6 ) ,
replaced by
llflls f o r the r a d i a t i v e
k c K, v e L 2 , _ ~ ( I , X ) suffices
and
function
f (- L 2 , 6 ( I , X ) .
v
l lclllc
from (2.18) and Lemma 2.5. each p o s i t i v e
integer
fL, k n Z [ f n ] }
(2 , 19)
n
we can f i n d
=
k
CO = 2C,
we sheuld note t h a t {v n}
1
{L, k, O} v
where
llf
'
with
which s a t i s f i e s
is identically
C
(2.7)
Then f o r
and the r a d i a t i v e
(n
that
function
=
1,2
'" . . )
kn ~ k E K
as
n .... .
in Lemma 2.6 hold good w i t h
= I.
(2.8) w i t h
as
n ~ ,=.
to the r a d i a t i v e But i t
zero, which c o n t r a d i c t s
we have shown ( 2 . 6 ) ,
is false.
Thus
Cn = ~ [ f n ] ,
i s the same constant as in Lemma 2.5 and
L2,_(~(I,X ) IIvli
easily follow
such t h a t
n [I ,~t =< n1
U~[fn]]l ~ ~ llfnrl 5 -~ 0
converges in
c K
n
fn E L2,~(I,X )
can be seen t h a t (2.12) and (2.13) and
ilfil~
Let us assume t h a t (2.18)
with
llVn II -6
replaced by
We may assume w i t h o u t loss of g e n e r a l i t y
2.6,
fL, k, ~ ' [ f l }
to show
because (2.6) and (2.8) w i t h
= 0
for
!~lll5
In view of Lemma 2.5 i t
Ilvli_.,,S < Cllfll: ,
it
lit
(2.7) and ( 2 . 8 ) w i t h
(2.18)
v n for
f o r given
function
by Lemma
for
f o l l o w s from Lemma 2.4 t h a t
the f a c t t h a t
rli21iI~
Therefore,
livll_~ = I .
replaced by
llfll~
Thus
f o r the
27
radiative function
v E L2,_6(I,X )
{L, k, 2rfl}
for
with
k e K and
f c L 2 , 6 ( I , X ). (Ill) k ~ ~+
The e x i s t e n c e o f the r a d i a t i v e
Imk > 0
and
I c F~(I,X)
with
vn
Imk 0 > 0
the r a d i a t i v e
and l e t
The e x i s t e n c e o f (2.16) with
vn
~ = ~,
v0
and
function
whence e a s i l y
function
follows
{L, k n, ~ . [ ( k 2n - k o2) V o ] } .
for
2 L 2 , 6 ( I ' X ) , ( k2-k n o)Vo
and
as has been shown i n ( I f ) ,
Let
for
k 0 e ~+
{L, k O, ~ } . Then, u s i n g
that
,,
Since
wn = v n - v 0
(r ~ I) wn = v n - v 0
vn
and
belong to
v0
is the r a d i a t i v e
belong to
L2,6(I,X),
and hence,
we have
illWnll_,~ + IIw' - iknWnll .
~ = 6.
c!ll~llt6
On the o t h e r hand, by making use o f Lemma 2 . 8 , function
with
k n = k + i / n ~ ~+
{ L , k n, t'}.
~ C2r-(26-1)!l!Z!l125
~ IIVoii2-..,(r,=,)
(2.21)
for
Set
are assured by Lemma 2.7.
IIvoIIB, 6 ~ C = C(ko),
(k, Z)
we have
(2.20) with
k c R - ~0}.
be the r a d i a t i v e v0
for
has been shown by Lemma 2.7 w i t h
Let us n e x t c o n s i d e r the case t h a t and denote by
function
n
< CIk2-k21
~-1
=
n
O'
IIvoII 6
'
(2.22)
L,,Wn,=,,R,. ( r ,o~ ) :< where
C = C(Ko)
(2.22)
that
{kn}
and and
Ik
K0 = (kn} U{k}. {v n}
satisfy
n4!
It
IIVoIIG
follows
(2.12)
(2.20),
Hence, by Lemma 2 . 6 , t h e r e e x i s t s
L2 _ ~ ( I , X )
which is the r a d i a t i v e
a strong
for
limit
{L, k, Z}.
,
(2.21)
and ( 2 . 1 3 ) w i t h
Lemma 2.6.
function
(r>l)
and
~n = ~ v
in
Thus the
in
28 existence for the r a d i a t i v e f u n c t i o n for (IV)
F i n a l l y l e t us show ( 2 . 6 ) ,
be a compact set in radiative function as
v = v 0 + w,
by Lemma 2.8.
~+. v
Let
for
k0
{L, k, i'}
has been established.
(2.7) and (2.8) completely.
and
,[L, k, t>}
v0
be as in ( I l l ) .
(k ~= K, ~ e F(,,(I,X))
w being the r a d i a t i v e f u n c t i o n for I t follows from ( I I )
Let
K
Then the is represented
{L, k, L'[(k2-k~)vo ]}
that the estimates
l , Cik 2 -ko~; i IIw!I_5+ IIw'-ikwII~_ l + IIB~wII~_ ~ 2 IIVoII, , (2.23) i
~.,wll 2-~,(r,,~) hold with
C = C(K).
< Cr-(2~-l)
=
Ik 2-koI2
2
!tVo"2a
(r > 1)
(2.23) is combined with (2.20) and (2.21) to give
the estimates (2.7) and (2.24)
Ilvll2_~;,(r,~ )
(2.6) and (2.8) d i r e c t l y the mapping: from ( I I I )
~
. .2 C2r-(24-1)i'ltl,i~,... ,
f o l l o w from (2.7) and (2.24).
(k, L') + v ( . ,
k, ~')
in
t41'B~(I,X) um
The c o n t i n u i t y of
is e a s i l y obtained
and Lemma 2.6. Q.E.D.
2g §3.
Uniqueness of the Radiative Function.
This section is devoted to showing Lemma 2.4. the l i n e of J~ger {lJ
and { 2 ] .
It will
But some m o d i f i c a t i o n
be proved along
is necessary, since
we are concerned with a long-range p o t e n t i a l . We shall
now give a p r o p o s i t i o n
of the s o l u t i o n (L - k2)v = 0
v E D(J), with
J = (R, ~)
k E ~ - {0}.
a f t e r the proof of Proposition PROPOSITION 3.1.
(3.I)
R > 0,
of the equation
Here the d e f i n i t i o n
of D(J)
is given
1.3 in 51.
the f o l l o w i n q
Let
v E D(J)
( I ) and (2):
The estimate [(m-km)v(r)[x
holds with
with
Let Assumption 2.1 be s a t i s f i e d .
(J = (R,~) , (R~O) s a t i s f y (I)
r e l a t e d to the asymptotic behavior
~ c(l+r)-26[v(r)[x
k e IR - z0sr ~ and some
c > 0,
(rcJ) is given as in D e f i n i t i o r
where
2.2. (2)
The support of
v is unbounded, i . e . ,
the set
{r e j/Iv(r)Ix>O}
is an unbounded set. Then (3.2)
lim{Iv'(r)]2x
+ k 2 1 v ( r ) I x2 } > 0 .
r-~
The proof w i l l
be given a f t e r proving Lemmas 3.2 and 3.3.
ve D(J) (3.3)
(Kv)(r) = [v'(r) '2 'x
+
k2 Kv(r)i
- (B(r)v(r),
(C0(r)v(r), v(r))
v(r)) x
(rcJ).
Set for
30 Kv(r) is absolutely continuous on every compact interval LEMMA 3.2.
Let
c I >0
R1 >R
and
vED(J) and l e t (3.1) be s a t i s f i e d .
in J.
Then there e x i s t
such that
(3.4)
d-~(Kv)(r) ~ - C l ( l + r ) - 2 ° ( K v ) ( r )
(a.e. r > R1 )
where a.e. means "almost everywhere". PROOF. Setting
(L-k2)v = f
and using
(3.1) and
(QI)
of Assumptior
2.1, we obtain d
dT(Kv) = 2Re{(v",v') x + k2(v,V')x - (CoV'V')x} - d ~ ( { B ( r )
(3.5)
(Bv,v') x
+ Co(r)}x'X)xix = v
= 2Re(ClV-f, v') x + ~(Bv,v) x - (C& v,v) x 2 :> _2c2(i+r)_2~ Ivlxlv'l× + T(Bv,v) x - (C~v,v)x
with
c2 = c + Co,
2.1.
From the estimate
(3.6)
cO being the same constant as in
/-2Ikl 21vlxlv'Ix : Tk-T -2 (v~- vlx)Iv' Ix <=
and (3.5) i t follows that we have with d Kv > -c I ( l + r ) - 2 6 ( I v dr
(QI)
T•
of Assumption
(Iv' I~ +
k2 2) -2-I vl x
c I = c2~I
+ kl 21vi / + (Bv,vl x (c~v,v) x
(3.7)
2 = -Cl (l+r)-2'4(Kv) + ( 7 -
Cl (I+r)-26)
+ l~Ik2cl(l+r)-2~Ivi 2x - ({ci(I+r)-26C0 z -Cl(l+r)-26(Kv ) + J l ( r )
+ J2(r).
(Bv'v)x + Co}v'V)xl
31
Jl(r)
~ 0
for sufficiently
for sufficiently
large
r .
C~ (r)Ir = O(r -2~) (r-~o) too.
large By
r,
because
(O.O)
~ 0
of Assumption 2.1 llCl(l + r ) -2~cO(r) +
holds, and so
Therefore there e x i s t s
2r -I - c i ( I + r ) - 2 6
J2(r) ~ 0
for sufficiently
RI> R such that (3.4)
is v a l i d f o r
large r ~ RI.
Q.E.D. Let us next set Nv = r{K(edv) + (m2-~o9 r) r - 2 ~ l e d v l x 2} ,
(3.8) where m
is a p o s i t i v e
LEMMA 3.3. for fixed
Let
veD(J)
~E(I/3,
< ~ < ~1 , d = d ( r ) = m( l _ u ) - I r 1 -~
integer,
I/2)
(3.9)
satisfy there e x i s t (Nv)(r)
PROOF. Set
w = edv.
Then
( I ) and (2) of Proposition mI ~ 1
~ 0
and
,
(3.10)
m =>
r-2~lwI~ + 2(m 2 - Zo~ r)rl-2~Re(w~
= lw'I~ + {k 2 + (l-2~)r-2~J(m2-1og
w) x
r) - r -2p}
+ 2rl-2~(m 2 - Zo9 r)Re(w',w) x
Now we make use of the r e l a t i o n s
(3.11)
=
edv '
+
as f o l l o w s :
r)lwl~
- (BW,W)x - (CoW,W)x + r d-d~(Kw)
w'
mI )
is c a l c u l a t e d
d--(Nv) : (Kw) + rd-~(Kw ) + (l-21J)r-2~(m2-Zo9 dr -
Then
R2 ~ R such that
(r >: R2
(d/dr)(Nv)(r)
3.1.
mr-~v
w" = edv '' + mr-lJedv ' + mr-~w ' - ~mlr-14-1w = edv ,, + 2mr-~w , _ (imm -,u-I + m2r-2!J)w
2 lWlx
r
32 to obtain
d~(KW)dr = 2Re(w" +k2w - CoW - Bw,w')x -d-~ ({B + Co}X,X)xlx=w = (3.12)
2edRe(Clv
- f , w ' ) + 4mr-~J]w 'Ix
2
2(~mlr -~I-I + m2r-2~)Re(w,W')x d
- d--~-({B +
Co}x,x)xl
I t follows from (3.10)~(3.12)
X=W
that
d~(NV)dr = (4mrl-~ + I) lW'Ix2 + {k 2 + ( l _ 2 ~ ) r - 2 ~ ( m 2
_
log r )
-r-2U}lw]2x
({C O + rC6} w,w) x + 2{redRe(ClV - f,W')x (3.13) -2~mr -~ +
rl-2~log
r)Re(w,W')x
+ (Bw,w) x
By the use of (3.1) and Assumption 2.1 we arrive at d (Nv) = > [ k 2 + m2(l-2~J)r -2)I d-7
(3.14)
{(I-21~) log r + l } r -2!J
- 2Co(l+r)-~:] lwl2x + (4mr IHj + I) lw'I~ - 2{(c + Co) (l+r) - ( 2 6 - I ) + ~mr -~ + rl-2!J~og r}lWlxlW, lx
Thus we can find
R3 > R,
independent of
m ,
such that
d---(NV)dr => {-12-k2 + m2(l-2~)r-2~J}lWlx2 2 {2rl-2!J~Jg r + ~Jmr-~}lwlxlW ' I x
(3.15)
+ (4mr I-I~ + I) [w' x2 -P (r,m) lwl~ - 2Q r,m) lwlxlW'Ix + S(r,m)lw'12x (r=>R3,m>l)
33 This is a quadratic form of
of
lWlx2
lWlx
and
The c o e f f i c i e n t
lW']x
P(r,m)
satisfies
(3.16)
P _>k2/2
( r > O , m>O)
Further we have PS - Q2 >= 4mrl-up _ Q2 = 2mk2rl-~ + 4(l_2~)m3r I-3~ _ {~2m2r -2~ + 4~mrl-3~Zo9 r + 4r2-4!4(~o9 r) 2} : m2{4(l-2!J)m - 1~2r-l+~}rl-3~
(3.17)
+ m(k 2
4~r-21JZo9 r ) r l-~J
+ {mk 2 _ 4rl-31J(Zo9 r ) 2 } r I-~L => k2/2 with s u f f i c i e n t l y
large
(r__>R4,m>__l) R4>R 3 , R4
can be taken independent of
m~l
.
Thus we obtain (3.18)
d (Nv)(r) d-r
On the other hand
Nv(r)
that is, (3.19)
>= 0
(a.e. r => R4, m => I)
is a polynomial of order
2 with respect to
Nv can be r e w r i t t e n as (Nv)(r) : r e 2 d { [ m r - ~ v + v ' l ~ + k2Jv!~ - (Bv,v)x - (CoV,V)x + r-2~(m 2 - Zo~ r)
[v[~}
= re2d{2r-2~Iv 2m2 + 2r-lJRe(v,V')xm+(Kv-r-2~Ivl~lo~ x By
(2)
exists
(3.20)
m ,
in Proposition 3.1 the support of R2 ~ R4
v
such that
Iv(R2) Ix > 0
r)}
is unbounded, and hence there
3,1
Then, since the c o e f f i c i e n t there e x i s t s
mI ~ 1
(3.21)
of
m2
in
(Nv)(r)
is p o s i t i v e when
r = R2 ,
such that
(Nv)(R 2) > 0
(m > mI
(3.9) is obtained from (3.18) and (3.21). ~.E.D. PROOF of PROPOSITION 3.1. a sequence
{tn}CJ, tnf=~(n~),
Then there e x i s t s
F i r s t we consider the case that there e x i s t s such that
r 0 ~ R1 such that
(Kv)(ro) >0,
3.2.
Let us show t h a t ( K v ) ( r ) >0 for a l l fr (3.4) by exp{C 1 j r 0 ( l + t ) - 2 5 d t } , we have
(3.22)
n = I, 2,
being as in Lemma In f a c t , m u l t i p l y i n g
(a.e. r > r o ) ,
follows -ci]
(3.23)
R1
r ~ r0 .
d { clI~o(l+t)-26dt } d-r e .(Kv)(r) __>0
whence d i r e c t l y
for
( K v ) ( t n) >0
( K v ) ( r ) >_ e
rr
(] + r ) - 2 5 d t ro (Kv)(ro)>O
(r ~ ro) .
I t follows from (3.23) that
Iv'(r)I~+
k21v(r)
=
(Kv)(r)+
(B(r)v(r),
+ (Co(r)v(r), (3.24) > e
v(r)) x
v(r)) x
-clIr,~:+ t ) - 2 ~ d t "r 0 (Kv)(r O)
- C o ( l + r ) - C k - 2 ( I v ' ( r ) 1 2 x + k21v(r)12x )
(r =>. r o ) ,
35 which implies that
k21v(r)j2x )
r--~olim( j v , ( r ) i 12 x +
(3.25)
co
> e -fro
(l+t)-2~dt
Next consider the case that R5 > R2, R2
being as in Lemma 3.3.
(Kv)(r O) >0
(Kv) (r) ~ 0
for a l l
r ~ R5 with some
Then i t follows from (3.19) and (3.9)
that 2r-2~"Jv(r)12x m2 + 2 r - ~ R e ( v ( r ) ' v ' ( r ) ) x m
(3.26)
- r - 2 ~ J v ( r ) l#Zog r :> 0
(r => R5, m => mI)
,
which, together with the r e l a t i o n d 2~ d ~ I v ( r ) J x : 2Re(v(r), v ' ( r ) ) x ,
(3.27) implies that
d--rdi v ( r ) ix2 => r - ~ { lml Zo9 r - 2 m l } I v ( r ) I x 2 >= 0
(3.28)
with s u f f i c i e n t l y support of
large
v ( r ) we have
R6 ~ R5 .
Because of the unboundedness of the
Iv(R7)Jx > 0
with some R7 ~ R6 , and hence i t
can be seen from (3.28) that Jv(r)Jx ~ JV(Rl)Jx > 0
(3.29)
(r => R7)
whence follows t h a t (3.30)
(r >= R6)
lim r ~ {Iv , ( r ) I 2x + k 2 j v ( r ) J x2 } > k21v(R7)l~ > 0 O..E.D.
3~ In order to show Lemma 2.4 we need one more p r o p o s i t i o n which is a c o r o l l a r y of Proposition 1.3 ( r e g u l a r i t y PROPOSITION 3.4. RN and l e t
Q(y)
v m L2(I,X) loc
(3.31) with
Let
theorem).
be a r e a l - v a l u e d continuous f u n c t i o n on
be a s o l u t i o n of the equaLion
(v, (m-k2)~)O = (f"~)O k c ~+
and
(3.32)
f ~ L2(I,X~o c .
Iv'(r)-
holds for a l l
Then 2 2 + k2v(r)l~ + kllV(r)[x
ik v ( r ) [ # = i v ' ( r ) +
(~>~Co(I'X))
2 4k~k211vllo, (O,r) + 2kllm(f'V)o,(O,r)
r c I, where
k I = Rek
and
k 2 = Imk
PROOF. As has been shown in the proof of Proposition 1.3, vEUH2(RN) I oc
and there e x i s t s a sequence
(3.33)
as n ~ =~ ,
where
{~n} c C~(R N)
{ '#n ~ v = U-Iv
in H2(]RN) loc"
I (T-k2)~n ~ f = U - I f
in L2(RN)Ioc
T = -4 + ~(y).
such that
Set Cn = UCn
Then, proceeding as in the proof of Proposition 1.3 and using the r e l a t i o r (L-k2)U = U(T-k2),
we have ~n -~ v
in L 2 ( I , X ) I o c ,
#n(r) ÷ v ( r )
in X
(rel),
~(r)
in X
(rcl),
(3.34) -~ v ' ( r )
(L-k2)# n -~ f
in L 2 ( I , X ) I o c
37
as n > ~, from
Set
0 to r
fn = (L-k2)%n '
integrate
and make use of p a r t i a l
((L-k 2) ~i~n, (1)n) x = (fn %n)x
integration.
(d,:'n , '~5'n)O,(O,r) + ({B+C-k2}#n,¢n)O,(O,r)
Then we a r r i v e at r - (gn(),~n (r) )x "
(3.35)
: (fn' #n)O,(O,r) By l e t t i n g
n ~ ~
in (3.35) a f t e r taking the imaginary part of i t
(3.35)
yeilds (3.36)
I m ( v ' ( r ) , v ( r ) ) x = -2klk211Vllo,(O,r ) - I m ( f , V ) o , ( O , r ) .
(3.32) d i r e c t l y (3.37)
follows from (3.36) and Iv'(r)
- i k v ( r ) 12x = i v ' ( r )
2+ 2 2 + k2v(r)l x kliv(r)T x
-2kllm(v'(r),
v(r)) x O~.E.D.
PROOF of with
kE~+.
from (3.32), (3.38)
LEMMA2.4. I t suffices by s e t t i n g
Iv'(r)
Let
v be the r a d i a t i v e
to show that
v = O.
function for {L,k,O} Let us note that we obtain
f=O,
- ikv(r)Ix2 = Iv'(r)
+
2 k2v(r)T2x + k# Iv(r)Vx2 + 4k#k21tv, O,(O,r)
Let us also note the r e l a t i o n (3.39)
lim
'v'(r)
- ikv(r)12x = 0
which is implied by the fact thdL
v' - i k v e L 2 , ~ ; _ l ( l , X )
then i t follows from (3.38) and (3.39) that
If
k 2 > O,
38
(3.40)
which means that
1
i"--~ r~>l Ifv li ~, (O,r)
-#---4k k2
l l v " ~ , ( O , = , ) = O,
k 2 = O, i . e . ,
lira I v ' ( r ) r,-~,o
that is,
k = kle: ~ - {0}
.
- ikv(r)l x
v = 0 .
=
0 ,
Next c o n s i d e r the case
Then from (3.38) and ( 3 . 3 9 ) we
obtain lim 2 + k21v(r)l~) r-~ (Iv'(r)Ix
(3.41)
lim =-~T~Iv'(r)
- ikv(r) l x
= 0
Therefore Proposition
3.1 can be a p p l i e d to show t h a t the s u p p o r t o f
v(r)
I ,
is bounded i n
continuation
and hence
v = 0
by P r o p o s i t i o n
1.4 ( t h e unique
theorem). _o. • E.D
39 ~4. Now we s h a l l
Proof oF the lemmas
show the lemmas in 52 which remain unproved.
some more p r e c i s e p r o p e r t i e s o f the mapping Theorem 2.3
will
v = v n,
k = kn
(4.1)
, v = v(.,k,L)
Applying P r o p o s i t i o n and
~' = ~n'
1.2 (the i n t e r i o r
llVnl:B(O,R) ~ C(llvnlio,(O,R+l ) + III nlll O,(O,R+I))
with
C = C(R),
{L n}
are bounded sequences in
because
(k n}.
i t can be e a s i l y seen t h a t f o r m l y bounded f o r
for
estimate)
we have
(R (- I ,
fore,
in
be shown.
PROOF o f LEMMA 2.6. with
(k,c)
After that,
for fixed
L2,_6(I,X )
l[Vnlio,(O,R+l )
n = 1,2 . . . .
R ~ I,
n = 1,2 . . . . .
is a bounded sequence.
the r i g h t - h a n d
{v n }
of
and
Since
F~(I,X),
side of (4.1)
~v~n}
and
respectively,
l~ILn!ll O,(O,R+I)
with a fixed positive
Thus, P r o p o s i t i o n
e x i s t a subsequence
and
n = 1,2 . . . . )
number
are u n i R.
There-
is u n i f o r m l y bounded
1.5 can be a p p l i e d to show t h a t there
{Vn}
and
v (- L 2 ( I , X ) I o c
such t h a t
vn
m
converges to
v
in
m
L2(l,X)loc.
Moreover, the norm
;Iv n l : _ ~ , ( r ~)
tends
m
to zero u n i f o r m l y f o r (2.13),
(4.2) as
m-~.
(4.3) v
m = 1,2 . . . .
as
r - ~ ~: by the second r e l a t i o n
and hence we a r r i v e a t
Vnm By l e t t i n g
v
in
L2,_~(I,X)
m-~ ~,
in the r e l a t i o n
(v n ,(L - k2 )¢) = <~' ~:;> n n ': m m m
(:~ (- C o ( l , X ) )
i s seen to be a s o l u t i o n o f the equation
(4.4)
( v , ( L = k-2)(I~) = <~,,q.>
(c><_ C~:(I,X)).
'
of
40 Since (4.5)
u = v
nm
satisfies
np
2
(u,(L-I~)'-i,') m
Proposition (4.6) •
- v
=
the equation
2
z
-k ) v I + ', -~n ' +> nm np Up nm p
(,.:, ~ C O ( I , X ) ) ,
1.2 can be a p p l i e d again to show t h a t
ilVnm-v n p'II B , ( O , R )
-~{ C{]lv nm-v n pII O, ( O , R + I )
+
k 2 k 2 I IIVnpll I nm- np O,(O,R+I)
+ Ill n -~n IF O,(O,R4-i)' -> 0 m
as
m,p + =,
p
whence f o l l o w s t h a t
(4.7)
vn
÷ v
in
H~'B(!,X)Ioc
m as
m ÷ ~.
Thus, l e t t i n g
(4.8)
,Iv n~
m
m + ~,~ in the e s t i m a t e
i k n v n il ~-I , rnD~ ~ , , , j ~ C0, m
m
which is obtained from the f i r s t
(4.9)
relation
of ( 2 . 1 3 ) , we see t h a t
#Iv' - i k v l l ~ _ l , ( O , R ) ~ CO,
and hence, because of the a r b i t r a r i n e s s
of
Therefore
for
v
i s the r a d i a t i v e
function
R > O,
v'-ikv
c L2,6_I(I,X).
{L,k,~}.
As is e a s i l y seen from the d i s c u s s i o n above, any subsequence of {v n} contains a subsequence which converges in the r a d i a t i v e {v n}
itself
function
v
converges to
for v
{L,k,Ll. in
L2 _ ~ ( I , X ) n H ~ ' B ( I , X ) I o c Therefore,
v e s t i g a t e some p r o p e r t i e s o f the f u n c t i o n
(4.10)
follows that
L2 _,~(I,X) " H ~ ' B ( I , X ) I o c .
Let us turn to the proof of Lemma 2.7.
Hilbert
it
~.E.D.
To t h i s end, we have to i n space
space obtained by the completion o f II~ll 2B,-~ = ~I ( l + r ) - 2 ~ { l ' >
to
H~IB~(I,X).
CO(I,X)
It is a
by the norm
ix2 + IBm2<>Ix2 + i~!
}dr.
41 HI ' B (I ,X) c H0I,B( , I , X ) l o c 0,-~' represented as Obviously
and the inner product and norm are
V')x + (B2u-B2V)x + ( u , V ) x } d r ,
(u,v)B,_6 = [ l ( l + r ) - 2 B { ( u '
(411) IlUllB,_6 = [ ( u , u ) B _ B I ~. Let
t
be a p o s i t i v e number.
(4.12)
Then the norm
11 IIB,_B,t
" U l l ~ , - 6 , t = ~I ( t + r ) - 2 B { ! u ' j 2 + x
is e q u i v a l e n t to the norm
II lIB _6
l i n e a r continuous f u n c t i o n a l e a s i l y show that the norm
on
iB1~Ul2x + lUl2x }dr
(= II I!B,_B,I ).
H~[B_B(I,X)
fllLlll B of
defined by
is
The set of a l l a n t i -
FB(I,X),
~ e F6(I,X)
because we can
is e q u i v a l e n t to the
norm (4.13)
tit ~ = sup ~ ,<,.~,~>l/¢ ~ C~ ( I,X) , -
11¢11B ~ = 1}.
'
~
'IIZ lli
I t can be e a s i l y seen, too, t h a t the norm
-
L1
is e q u i v a l e n t to the
norm defined by (4.14) for a l l
I!1~,Ill 6 , t = sup{l<£'(t+r)({'i~>l/° ~ C o ( I ' X ) ' t > O.
PROOF of LEMMA 2.7. on
I,B t"1,x) NO,_B
x H _:B(I .
(4.15)
Consider a b i l i n e a r continuous f u n c t i o n a l ,X)
At
defined by
A t [ u , v ] = (u,v)B _ n , t - 2 3 f l ( t + r ) - 2 6 - 1 ( u , v ' ) d r + fl(t+r)-2~({C
where
I1~,11B -- 1}
t > 0
and
positive definite
~ • ~
+
with
for some t > O,
- k~ - l } u , v ) x d r ,
Im k 0 > O. i.e.,
Let us show t h a t
there e x i s t s
tO > 0
At and
is
/_? C = C(to,k O) > 0
lAto{U,ul
(4.16) It suffices (~J t O)
such that
and
~:=C l'uil~_ Z , t o
to show (4.16) for
u = ';; ( C " ( l , X ) .
C(r) = C(r) - 1 - ;,.
(4.17)
(u e H~'B ( I , X ) ) . Set
k 02 =
>,+
i;-
Then we have
2 ,~ 4. ,.,'.:;)x dr (:i ~NIB,_~, t + j, i (t+ r) -2 I~(C~
i At [i~,<)] i 2 =
2~f I (t+r)-2"~-l Re (,,~,~' )xdr) 2 + (~j}i (t+r) -2 [~"i ~ ; 2dr x 2 ;~i ( t + r ) - 2 ~Im(..!~, :~' )xdr) 2 . By the use of a simple i n e q u a l i t y ("~ > O,
cL,b c IR)
(4 18)
'.At[',>,~>]i 2 ~½1(ii~,.i '2
•
(a z b) 2 ~ (l - ,.~)a2 + (l - c~-l)b 2
we have from (4.17)
"B,-.,,t
e
+ ~I(t+r)-2~(C,~,m)xdr)2 -
-
+ ~j2 (IT ( t + r ) - 2 ~ i ¢,i 2xdr)2 } 4 ~2 { (f i ( t + r ) - 2 {~'-I Re (¢.,~ ' )xdr) 2 + (fi(t+r)-2~-I 12 l l ( t )
Im(<~,e..,' )xdr) 2}
- 4(~212(t)"-
can be estimated as follows:
ll(t) (4.19)
Ii(t)
~
(I
-
cz)ii~ll 4 , - ~ , t B
+ f 2 _ (I~, _ l ) l l ~ l l i ( f l ( t + r ) - 2 # i ~ i ~ d r ) 2 (0 <,'~ < I ) .
Set
~ = 21rCIi(2[iCii + ~,2)-I
in (4.19).
Then we a r r i v e at
43 2 iJ 4 I 1 ( t ) 7-'-. . . . . . 2 ilC !1+I~2 !1l~;llB , - 3 , t "
(4.20)
On the other hand, we have
(4.21)
12(t)/ll~ll 4B , - # , t ~<-
- ~,
2(fi(t+r
xdr)2/t211 ~i, , - # , t ÷ o
x
( t ÷ ~). I t follows from (4.20) and (4.21) t h a t
Ato
is p o s i t i v e d e f i n i t e
with some
t o• Let
~ • FS(I,X).
By the Riesz Theorem, there e x i s t s a unique
f • HL'B(5(I'X)u,- such that
I <~, ~ "~,>
=
(f'#)B,-[~,
to
(4.22) II f II[B, - .~, t o ~ C l[IilliS
where
C = C(@,t O)
does not depend on
Theorem (see, e . g . , Yosida [ I ] , exists
w • HI ' B ( I , X ) 0,-6
f
or
~,.
Now the Lax-Milgram
p. 92) can be applied to show t h a t there
such that
IAto[W,@ l = ( f , 6 ) B , _ # , t 0
(@ ~ C~(I,X))
llwllB,_6,t ° ~< Cllfll B _ B , t o
(C = C ( ~ , t o ) ) .
(4.23)
By p a r t i a l (4.24)
integration,
we obtain, from (4.22) and (4.23),
( ( t O + r)-26w,
(L - k-~o)#)0 = <~,~>
(4.22) and (4.23) can be used again to see t h a t f i e s the estimate (2.16). tion for
{L,ko,~}.
Thus,
(~ G C ~ ( l , X ) ) .
v = (t O +r)-2Bw
v = (t O +r)-2Bw
satis-
is the r a d i a t i v e func-
The uniqueness of the r a d i a t i v e f u n c t i o n has been
already proved by Lemma 2.4.
Q.E.D.
44
In order to show Lemma 2.5, we have to prepare two lemmas. LEMMA 4 . 1 . a positive
Let
constant
(4.25)
set in (~+.
K be a compact C = C(K)
Then tilere e x i s t s
such that
k211Vlll_,3 -~ C{llvli_( S + IIv' - ikvil~.:_ 1 + Ilfll S,
holds f o r any r a d i a t i v e k2 > 0
and
Proof.
function
v
for
{L,k,2[f]}
Proposition all
by
r c I.
k = k I + ik 2 ~ K,
f e L2,~(I,X). Let
k ~ K with
Im k = k 2 > 0
and l e t
Then, by Lemma 2.7, there e x i s t s a unique r a d i a t i v e such that
with
v , v ' • L 2 , ~ ( I , X ). 1.3 and
v
Moreover,
satisfies
imaginary part.
i n t e g r a t e from
the equation
function
v = v(.,k,SIf])
(L - k2)v = f
by for almost
((L-k2)v(r),v(r)) x = (f(r),v(r)) x
T
(0 < R < T < ,~)
and take the
Then
T (2-25)Im # ( l + r ) l - 2 6 ( v ' , v ) x d r
(4.26)
R to
L 2 , ~ ( I , X ).
v • D(1) rq H ~ ' B ( I , X ) I o c
M u l t i p l y the both sides of
(l+r) 2-23,
fc
- (l+T)2-251m(v'(T),v(T)) T - 2klk 2 ~ ( l + r ) 2 - 2 6 1 v l ~ d r
+ (l+R)2-2~Im(v'(R),v(R))x T = Im / ( l + r ) 2 - 2 d ( f , v ) x d r . R
Since
v,v' • L2,a(I,X),
lim(l+r)2-2a(v'(r),v(r))x = 0 r ~: follows from (3.36). Therefore,
lim I m ( v ' ( r ) , v ( r ) ) = 0 r-TO along a s u i t a b l e sequence k211vll~_ ~ ~ 1 21kll (4.27) 1
~T ] n
Jl
R -~ O,
{(2-2~)fl(l+r)l-2~'iv'
{(2-2C)J 1 + J2 }.
2{k I
Let us estimate
and
and
J2"
holds, and letting
T +
we have
IxlVlxdr + fl(l+r)2-2'SrflxlVlxdr}
45 J1 ~'~ IIv'!l ,~!lvll :~: { F v ' - i k v i l -{S -, 1-{';
(a = m a x l k } ) , k~K
all vll _6}11 vii 1 -~,
+
]
(4.28)
)
I
~d 2 ~, II fll 1 _~.~1!vl! 1 -,~' where we used the Schwarz i n e q u a l i t y .
Set
b = min ) k l l kcK i
i °
Then i t f o l l o w s
from (4.27) and (4.28) t h a t 1
+ cdl vii
k211vii I - 6 ~ 2-~ (II v ' - i kvll
+ I[ f!l
)
(4.29) 1
<% 2-~ (II v -i kvll ~:-1 + all vii _~ + II fll 4), where i t
should be noted t h a t
directly
from ( 4 . 2 9 ) .
LEMMA4.2. C2
function
on
Let I
and
I-~
< 5.
(4.25) f o l l o w s
Q.E.D. v ~ D(T) such t h a t
I° (4.30)
_x < .~-I
5(r) =
and
k ~ 6 +.
0 ~ ~ # I,
C(r)
Let
be a r e a l - v a l u e d
and
(r ~ R)
[~1 (r ~ R+I)
with
R > 0. ~
2 R
R T ~ Re f ~ ; ( l + r ) C J ' ( f , v ' - i k v ) x d r R
+
t..
T - Re f~(l÷r)CS(C, ' I V ' V ' - i k v ) x d r R
T "~( c~: T f~(l+r) C~v,v)xdr + ~ S#:(l+r)CZ'-l(CoV,V) R
(4.31)
m
R "
dr x
T - k 2 f,L(l~r)"~'(CoV,V)xdr R l T + ~ f~'(l+r)IZ!IiB'~v"2 + (CoV,V)x}dr R ~x + ½ ( l + T ) ~ q i v ' ( T ) - i k v ( T ) l : X2 - ( C o ( T ) v ( T ) , v ( T ) ) x ; ,
46
where
f = (L-k2)v,
o, e IR
PROOF. The r e l a t i o n (4.32)
-(v'-ikv)'
and
T
R+I
TM
(L-k2)v = f
can be r e w r i t t e n as
- i k ( v - i k v ) + (B(r) + Co(r ) + C l ( r ) ) v
: f
whence follows t h a t (4.33)
-((v'-ikv),v'-ikv
i k l v ' - i k v l x 2 + ((B + C0 + C l ) V , v ' - i k v ) x
x
= (f,v'-ikv)x. Take the real part of (4.33) and note that
de~'lv'-ikvlx:
2
(4.34)
k 2 ~ O.
Then we obtain
x'
d .(Cov,V)x } - ½ (C~v,V)x + k2(Cov'V)x - 2l d~ + R e ( C l V , V ' - i k v ) x ~ R e ( f , v ' - i k v ) x.
M u l t i p l y i n g both sides of (4.34) by and making use of p a r t i a l
integration,
PROOF of (2.10) of LEMMA 2.5. f u n c t i o n for
{L,k,~,Ifl}
v • D(1) r~ H~,B(I,X)Ioc for almost a l l
r • I.
~ ( l + r ) m,
with
i n t e g r a t i n g from
T
we a r r i v e at (4.31). Q.E.D.
Let
v ~ L2,_~(I,X)
k ~ K and
m = 2~-I and
be the r a d i a t i v e
f • L2,~(I,X).~
by Proposition 1.3 and we have Let
R to
R= 1
Then
(L-k2)v(r)
in Lemma 4.2.
f(r)
Then, i t
follows from (4.31) t h a t (6
-
½ )ll~i2(v'-ikv)!l i -23 2 i.I. 2 cS-I,(I,T) + ( - ~)II~'2B'2vlI~-I(I,T)
% TfL](l+r) 2~"-'i J f! v ' - i k v l x d r 1 ' 'x (4.35)
+ c o Tf ~ ( l + r ) _ 2 ~ i V l x [ V , _ i k v Ix dr 1
+ TCoTf~(l+r)-2~S vi 2xdr + .2. ~. .I c o Tf ~ ( l +r) -2~ r v l x2d r 1 1 T 2 + Cok2 f~L(l+r)l-2"~IVl Ix2d r + ½ ( : ~ ' ! ( I ÷ r ) 2 ~ - l ~ ' tB~-~v'x + 2 Co!V!2x}dr
47 +½~(l+T)25-11v'(T)-ikv(T)12
+Co(l+T)l-261v(T)l~} , X
where c O is the constant given in Assumption 2.1 and the following estimates have been used: (l+r)26-111Cl(r)l I ~...~co(l+r )
2~,-I
-~':I ~. c o (l+r)-I
=Co(l+r)-~+(6-1)
(l+r)26-111C~(r)II ~ c O(l+r) 26-2-c # co(l+r)-2~, (4.36) (l+r)2~-211CO(r)ll ~ c0(I+r)2~-2-~ ~ c0(I+r)-26, I
(I+ r)26-111Co(r)ll ~ Co(l+r)2~-l-c ~ c0(I+r)I-26 By Lemma 4.1 and the Schwarz inequality, we have the right-hand side of (4.35) 11
IIfll611~'2(v'-ikv)ll6-I + Co IIv II_611~m(v '-ikv)II6-I + 6CoIIVII26 + CoCIIvll ~{17vI',~ + Hv'-ikvII~_1 + IIfll6} (4.37) + ½(I+c0)cr326-I, vI',2 ,° ' B,(l
,2)
+ ½(l+m){(l+T)26-21v'(T)-ikv(m)12 with
C£ = maxlE'(r)l.
Since
v ' - i k v c L2,6_I(I,X)
X
+ Co(l+T)-2~Slv(T)12x } J and v ~ L2,_~(I,X),
r
the last term of (4.37) vanishes as {Tn},
T ~ ~ along a suitable sequence
and hence we obtain from (4.35) and (4.37) I.
!I v'-ikvll~-I, (2,~,) + ii B"~vI!,~-I, (2,~,~) (4.38) .~ C{IIvlI_L< + llfll~ + , v ' - i k v l l o , ( l , 2 ) + llVllB,(l,2)}. (2.10) follows from (4.38) and Proposition 1.2 (the i n t e r i o r estimate). O.E.d.
48 PROOF of (2.11) of LEMMA 2.5.
From (3.32)
Iv(t) 12x-~- ki2' kv(t), v' (t)-i
(4.39) Multiply
(I + t ) -25
(4.39) by
in P r o p o s i t i o n
3.4, we have
12x + 2!k l ,I-lil f!i ,,.',l!vl; -o<"
and i n t e g r a t e from
r
to
~,~. Then, i t
fol-
lows t h a t 2 1 - 2..t; II vli -.x~ (y,,'J "~'~-7 f (m+t) Iv'-ikv'2dtx '
k1 r
+
...... .2 .....
(l~r)
-(2:~-1
)1' fll ,~11vii
':k 1 (2"-1) (4.40)
1 ( l + r ) -2(2 :,-1 )1! v ' - i k v l 2~- l , ( r ¢,:.) "-; -~ kI
+ (2.11)
"
kl ! (2;3-I)
is obtained from (4.40) and ( 2 . 1 0 ) .
Now t h a t a l l
(l+r)
(2..~-1)11f!i .~,iivl! _~.
Q.E.D.
the lemmas in ~2 and Theorem 2.3 have been proved c o m p l e t e l y ,
we can show more p r e c i s e p r o p e r t i e s of the mapping (4.41)
¢+ x L2 ' ( I , X ) )
(k,£)
~ v( . , k , L [ f ] )
c L2,_6 ( I , X )
by reexamining the proof of the lemmas in ~2. ]=~4MA 4~3.
~r f n'
Let
converges weakly to in
~+
such t h a t
radiative {v,n 1 m
of
{v n}
(4.42) as
m-~ c~,
n
as
+ k c ¢ ~ with
for
{ L , k n , ~ , [ f n]~.
L2,,~(I,X)
n + ,~,. k c ~+
as
Let
such t h a t
fn
~k "," be a sequence
n ~ ~-'. Let
vn
be the
Then, there e x i s t s a subsequence
such t h a t v
where
f ~ L2,5(I,X) k
function
be a sequence in
n
-~ v nl v
in
L 2 , _ { ( I , X ) ," H ~ , B ( I , X ) l o c
denotes the r a d i a t i v e
function
for
{L,k,2,[f]}.
,ig PROOF. Since in
L2,C(I,X ).
{fn )
is weakly convergent,
{fn ~ is a bounded sequence
Therefore, i t follows from (2.7) in Theorem 2.3 t h a t
is a bounded sequence in L 2 ( l , X ) l o c , bounded sequence in
L2(RN)Ioc.
(4.43)
-"gn +
we can see that
{v n
n
that i s ,
Noting that
knVn =
fn
{~n}(~ n = U-Ivn ) Vn
satisfies
is a
the equation
(?n = u - l f n )'
is r e a l l y a bounded sequence in
"
{Vni
H2( ]RN)I oc"
Then,
by the repeated use of the R e l l i c h Theorem, i t can be shown that there e x i s t s a subsequence
(Vn }
of
,[~YnI
which is a Cauchy sequence in
Hl(IRN)lo c.
m
Therefore,
{v m }
is a Cauchy seuqence in
H~'B(I,X)loc
with the l i m i t
n
v e H~'B(I,X)Ioc"
Moreover, i t follows from (2.8) in Theorem 2.3 that
{v n I" converges to v in L2,_~,(I,X ), too. In q u i t e a s i m i l a r way to m the one used in the proof of Lemma 2.6, we can e a s i l y show that v is the r a d i a t i v e f u n c t i o n for
(L,k,~,[f].
Q.E.d.
From the above lemma, we obtain ZH_y._Q~E~I_4._4. Let Assumption 2.1 be s a t i s f i e d . (i)
Then the mapping
(4.44) is a
~+ -~ k-~ v ( . , k , ; ! . [ . l ) ~(L2,6(I,X),L2,_~(I,X))-valued
-e L2,_4(I,X ) -F
continuous f u n c t i o n on C ,
that is,
i f we set v(-,k,L[f])
(4.45) then in
(L - k2) -I c m ( L 2 , a ( I , X ) , L 2 k ~ 6+ (ii)
from
= (L - k2) -l f
(f ~ L2,~:I(I,X)),
_4(l,X))
and
in the sense of the operator norm of For each
L2,~(I,X )
into
k ~ ~,
(L - k2) - l ,
L2,_~(I,X).
(L - k2) -I
is continuous
~ (L2,~(I,X),L2
_cS(I,X)).
defined above, is a compact operato~
50 PROOF. Suppose that the assertion Then, there e x i s t a p o s i t i v e nurnber {kn} c 6 +
(i)
is false at a point
i~ ~ 0
and sequences
{llfnlE 6 = 1
(n = 1,2 . . . . ),
i kn ÷ k
(n .... ),
tl IIv(.,k,L IfnJ ) - v ( . , k n , L I f nl)ll ~ 13 With no loss of generality, to
f
with
we may assume that
f c L2,4(I,X ).
show that there exists a subsequence
[
v(.,knm,~[fnm])
(4.47)
Lv(.,k, in
L2 _~(I,X )
as
(4.48)
as
of ( i )
%[fnm~
m ÷ ~.
(ii)
).
converges weakly in
Then, Lelm~a 4.3 can be applied to {n m}
of p o s i t i v e integers such that
÷ v(-,k,L[f]),
) ~ v(.,k,~[f])
]) - v ( " k n '~[fn I)" -3 + 0 m m m
which contradicts
is complete.
fn
(n = 1,2 ....
Therefore, we obtain
llv("k'#[fn
m ÷ ~,
{fn } ~ L2,~(I,X)
such that
(4.46)
L2,5(I,X)
k e ~+.
the t h i r d r e l a t i o n of (4.46).
follows d i r e c t l y
from Lemma 4.3.
Thus, the proof Q.E.P.
F i n a l l y , we shall prove a theorem which shows continuous dependence of the r a d i a t i v e function on the operator THEOREM 4.5. (4.49)
Let kn -
Ln,
n = 1,2 . . . . .
C(r).
This w i l l
be useful in if6.
be the operators of the form
d2 d ~ + B(r) + Cn(r),
Cn(r ) = Con(r) + Col(r) (r c I)
with
Cjn(r) = O~jn(rm)x
for
j = 0,I.
Here
O.On(Y) and
Qln(Y)
are
51
assumed to be real-valued functions on ~N of Assumption 2.1 with tively.
QO and
The constants c,~ and
independent of
n = 1,2 . . . . .
(4.50)
lim ~ n ( y ) n-~o
with
Q1
Qo(y )
and
Ql(y)
which s a t i s f y
replaced by
c O in
(0.0)
QOn and
(QO) and
(QI)
Qln'
and
(QI)
respec-
are assumed to be
Further assume that : ~(y)
(j = 0 , I ,
y ~ IR N)
which s a t i s f y Assumption 2.1.
{Ln,kn,g n} such that
be the r a d i a t i v e function f o r
Let
v n,
kn ~ ~+'
n = 1,2 . . . . .
Ln E Fg(I,X)
and kn ÷ k
in
{+
Cn ÷ L
in
FB(I,X )
(4.51)
as
n ÷ ~ with
(4.52)
k ~ ~+
and
vn
in
where
v
stant
C such that
-~
v
L c F6(I,X ).
Then we have
H I,B ( I , X ) , 0,-6
is the r a d i a t i v e function for
{L,k,L}.
And there exists a con-
ilvnllB,_6 ~ C III~.IIi 6. (4,53)
11Vn-iknVnt]~-I
+ ilB'~Vn"6 1 ~ C tll~l,
llVnll~,_~,(r,~) ~ C2r - ( 2 ~ - I ) ,l]Clll ~ for all in
n = 1,2 . . . . .
is bounded when
k
moves in a compact set
~+. PROOF. Let
wn
C = C(k)
(r~l)
gn
be the r a d i a t i v e
be the r a d i a t i v e function for
Im k 0 > O.
We have
Function for
{Ln,ko,~n}
{Ln'kn~[( k 2n- k2~ o~gn l } '
v n = gn + Wn by Lemma 2.8.
Lemmas 2.5 and 2.7, we can find a constant
C,
where
and l e t k0
~+,
Reexamining the proof of which is independent of
52 n = 1,2 . . . . .
such that {l!gnU,,< 4-i!Bt:ign,=l ,: + IIgnll,. ~.~ C ~llLn!l' ~, I
! (4.54)
I',wn - iknWnlld_ 1 + IrB~Wnll6_ 1 -~ C(llgnll 5 + ,Wnll_6),
ll wn, for a l l
n = 1,2 . . . . .
CZr-(2
S-l/(,gn,
Thus, we obtain,
for all
+ ,Wn, 2 /
(r ->.-1)
n
I,
ri!v,n - ik n vn 115_I + I!B2Vnfl~,_l -~ C( ill~Slll 6 + llWn![ - ~~)'
I
(4.55) 2 llVn:l_~,(r,,~ ) ~ C2r - ( 2 5 - I ) ( !l; LIII ~ + llWnl! 2_6,)
(r ~ l)
l
with a constant
C > O.
(4.56)
IlWn!l_5 < Cllgnll6'
The estimate (n = 1,2 . . . . )
can be shown in q u i t e a s i m i l a r way to the one used in the proof of Theorem 2.3 and Lemma 2.6. subsequence for
{L,k,O},
{h m}
In f a c t , of
{Wn/IIWnfl_~}
We have
uniqueness of the r a d i a t i v e follows
llhll_5 = I. function
from (4.54) and (4.56).
2.6, we can show the convergence of which, together with (4.53), H~'B~(I,X).
~.E.D.
then there is a
which converges to the r a d i a t i v e
where we have used the i n t e r i o r
(4.54) and (4.50).
(4.53)
i f we assume to the c o n t r a r y ,
estimate
(Proposition
On the other hand,
h = 0
function 1.3), by the
(Lemma 2 . 4 ) , which is a c o n t r a d i c t i o n . Proceeding as in the proof of Lemma {v n}
implies t h a t
to {v n}
v
in
L2,_6(I,X)
converges to
r~ H~,B(I,X)Ioc , v
in
Chapter I I .
Asymptotic Behavior of the Radiative Function
§5. Construction of a Stationary Modifier Throughout t h i s chapter, the potential the following conditions,
Q(y)
is assumed to s a t i s f y
which are stronger than in the previous chapter.
ASSUMPTION 5.1. (Q)
~(y) and with
can be decomposed as
Q(y) : ~o(y) + ~l(y)
Q1 are real-valued function on
~N
such that
~0
N being an integer
N > 3.
(0~0) There e x i s t constants c O > O, m0 C function and
0 < c ~ 1
!DJOo(y) l < Co(l + ly!) - j - c
(5.1) where
Dj
such that
(y c IR N,
denotes a r b i t r a r y derivatives of
jth
_~l(Y)
is a
j = 0 , I , 2 . . . . mO) order and
m0
is
the least integer s a t i s f y i n g (5.2)
mo > Let
mI
(5.3)
(mI - l ) c < I.
Qo(y) ~ 0 Ql(y)
and
m0 => 3o
be the least integer such that
we assume that
(~i)
2_~ 1
Further
(5.4)
with the same constant (I)
is assumed to s a t i s f y
N and there exists a constant
such that
'QI(Y) I ~ Co(l + 'Yl)
REMARK 5.2.
~o(y)
When m0 > mI ,
(IYl ~ l ) .
is a continuous function on
c I > max(2 - ~,3/2)
mlc > I.
-~l
c O as in
(y • IR N) (o~0).
Here and in the sequel, the constant
tion 2.1 is assumed to s a t i s f y the additional
conditions
in Defini-
54 It
~',.._< m i n ( ~ , . , l - l )
when
'~ < L .<_ I ,
(5.5) :i: (
From the c o n d i t i o n s contradict
(5.2) and ( 5 . 4 ) ,
the c o n d i t i o n
(2)
If
(3)
The c o n d i t i o n
an i n t e g e r ,
c > I~,
replaced by C~
(4)
it
then
m0 = 3 and mI = 2 .
(ml-l)c r.
< 1
function
and
on
is t r i v i a l .
for a little
(5.3) i s also t r i v i a l , ~(y)QO(y)
is easy to see t h a t (5.5) does not
~ > -~.
we can exchange
The c o n d i t i o n
valued
~ 1 -I +~"!
IR N
such t h a t
1 ~.
when
s m a l l e r and i r r a t i o n a l
because
(l-¢(y))0.0(y)
In f a c t ,
Qo(y)
and
+ ~l(y),
¢(y) = 0
is ~'.
.~l(y)
can be
5
is a real-
where
(!y! < I),
= I(IYl
.> 2)
A general s h o r t - r a n g e p o t e n t i a l -I -c0)
(5.6)
Ql(y) = 0(jy I
does not s a t i s f y
(5.4)
("0 > 0,
in Assumption 5.1.
;Yl . . . . )
In !}12, we s h a l l discuss the
Schr~dinger o p e r a t o r w i t h a general s h o r t - r a n g e p o t e n t i a l . The f o l l o w i n g
i s the main r e s u l t
THEOREM 5.3.
( a s y m p t o t i c behavior of the r a d i a t i v e
Assumption 5.1 be s a t i s f i e d . Z(y) = Z ( y , k ) of
y
on
of t h i s chapter. function).
Then, there e x i s t s a r e a l - v a l u e d f u n c t i o n
IRN × (IR - { 0 } )
such t h a t
Z ~ C3(1R N)
as a f u n c t i o n
and there e x i s t s the l i m i t
(5.7)
F(k,~) = s - l i m e - i i ~ ( r ' ' k ) v ( r )
in
X
r-~)
f o r any r a d i a t i v e ~ FI+~(I,X),
function
where
Let
v
~J(y,k)
for
{L,k,~..l
is defined by
with
k (: IR - {0~, and
55 ~J(y,k) = rk - >,(y,k)
l
(5.8)
r
~ (y,k) with
r : lYl,
= y/lyl
if
0 < c. __<=½
(
if
1 ~-< c <
0
This theorem w i l l THEOREM 5.4.
Let
function for
being as in ( 5 . 9 ) . u = ei~v - {0}.
and
~
and
= I~-c
(5.9)
ative
f Z(tc~,k)dt 0
1.
be proved in ~7 by making use o f the next. Assumption 5.1 be s a t i s f i e d .
{L,k,~}
with
Then we have
k • IR - {0} u' - i k u ,
i s given by ( 5 . 8 ) .
Then, there e x i s t s
and
let
be the r a d i -
L • FI+S(I,X),
K
B
where
be a compact set in
such t h a t
i. 3 _< cl!l~]lll÷~ llu , -ikull B + llB'2ull
(5.10)
v
B½u ~ L 2 , B ( I , X ) ,
Further,
C = C(K)
Let
(u : e i ~ v )
and
Iv(r) Ix £ CII[~!IIi+ ~
(5.11) f o r any r a d i a t i v e where
~
function
v
for
{L,k,~}
with
k • K and
L • FI+~(I,X),
i s as in ( 5 . 9 ) .
As was proved in J~ger [31, Saito [3]
(r c T)
we may take
Z(y) z 0
when
Qo(y ) m O.
(and Ikebe [ 2 ] ) showed t h a t we may take
(5.12) in the case t h a t approximation ~ of
½ < s < I. Z(y)
It will
be shown t h a t (5.12)
in the general case.
The f u n c t i o n
i s the " f i r s t ~(y,k)
is
56 called a s t a t i o n a r y
modifier.
In the remainder of t h i s section, of the s t a t i o n a r y lowing problem:
modifier
,~(y,k).
Find a r e a l - v a l u e d
we shall c o n s t r u c t
the kernel
Z(y,k)
To t h i s end, l e t us consider the f o l function
>,(y,k)
on
(~ > l ,
r
IR N x (m
{o})
such that (5.13)
l(L-k 2) ( e i ~ X ) ! x = O(r -~)
f o r any
x ~ D,
X(y) T 0
where
is a s o l u t i o n
we have to i n v e s t i g a t e ~'N
on SN-I
.
as in (0.22),
~
is given by ( 5 . 8 ) .
of t h i s problem. some properties
Let us introduce
If
Qo(y ) z O,
then
In order to solve t h i s problem,
of the Laplace-Beltrami
polar coordinates
operator
( r , g l , 8 2 . . . . . ~N_I )
i.e., Yl = rc°s(~l '
i
(5.14)
Yj
rsinOlsint2...sinO.
(YN =
rsinglsinO2""sinS:N-2sinON-l'
I
where
r > O,
0.<_ 0 I,~)2 . . . . . (~N-2 -<- ~'
j-I ,cose.3
(j = 2,3 . . . . . N - l ) ,
0.< ON_1 < 2",~. We set
!b I = I, I
(5.15)
i
bj = bj(e) = s i n O l s i n O 2 . . . s i n S j _ l
Mj
Mj(e)
bj(@) -I ~@ ~ 3
(j = 2,3 . . . . . N - l ) ,
(j = 1,2 . . . . . N - l ) .
Then we obtain from (0.23) (5.16)
ANX-
and hence, s e t t i n g
N-I >i bj(O)j=l
2 (sinOj )-N+j+I
A = -A N + ~ ( N - I ) ( N - 3 ) ,
,~ { ( s i n o j ) N - j - I ~(,~j
we have
}
~_x , ,~r,j
57
N-I
IA"~xl 2 = ~ [MjX!2x + ICNXl2
(5.17)
x
where
x(m)
(5.17) on
j=l
is a sufficiently
t h a t the o p e r a t o r
½
D ,
i.e.,
for
xe
( 2N :
x
Mj
smooth f u n c t i o n on can be n a t u r a l l y
D½,
we d e f i n e
MjX
~,~(N-I)(N-3))
sN-I .
'
I t f o l l o w s from
extended to the o p e r a t o r by
M j X = s - l i m MjX n,
where
n-><x}
{x n} and
is a sequence such t h a t A~Xn ÷ A~x
in
X as
xn ~ C I ( s N - I ) ,
n ~ ~,.
as the extended o p e r a t o r on
D~-~.
Xn ~ x
In the sequel Let
~(y)
be a
in
X as
n .....
Mj
will
be considered
C2
f u n c t i o n on
IR N
and l e t us s e t "¢ : ~(y) : 9(y;,\) = r -2
(5.18)
)~ (MjX) 2, j=l
p = P(y) = P(y;~,) = r'2(AN ~) M=
LEMMA 5.5. with
N-I
Let
Z(y) e C2(IRN).
N-l ~ j=l
(MjZ)Mj.
xc
D and set
r >,(y) = ~ Z ( t w ) d t 0
(r = I Y l ,
~ = Y/[Y[)
Then we have
(e±i~B(r) - B(r)e±i~)x = eti~(-¢
± 2ir-2M ± i P } x
(5.19) = (~ ± 2ir-2M + i P ) ( e ± i ~ x ) ,
and (L-k2)(eiUx) = ei'~i{(B(r) + 2 i r - 2 M ) x + (iP + i Z ' + Cl)X (5.20) - (2kZ - CO - Z2 - ¢ ) x } , where with
~,
P,
M are given in (5 18) "
'
Z' = _~_Z ~r'
and
iJ(y) = rk - ~(y)
k e IR - { 0 } . PROOF .
(5.19) and (5.20) can be shown by easy c a l c u l a t i o n
i f we note
58 AN(e~ikx) = e-:i> (ANX _, 2iMx z i(AN;~.)x).
(5.21)
Q.E.D. In order to estimate the term LD~,IA 5.6.
Let
P(y),
h(y) ~- C2(IRN).
we need
Then
N
(5.22)
(Mjh)(y) = p=j ~ bj(O)-Iyp,j
(5.23)
(ANN)(Y) = ~. 2 j=l
N-I
N
bj (~_))-2
~, p,q=j
- (N - l)
~(y),
Z(y)
P
N ~
yp
"
H
3h
~gp '
be as in Lemma 5.5 and let
r--
(Mj~)(y) = }
~2h
Yp,jYq,j Sy~y~
p=l where yp,j = .~,q, . Let J as in (5.18). Then
"~p,
be
-~,
(:" 11
0 p
P(y)
at
3--y-_
y=t~
(5.24) P(Y)
:
~
~I! > h(~12 O j 1 p,q=l
- (,-1)
">: F~
P'JYq'J ~Yp~Yq ly=tm
.~z~
p l L(~ ~Y~ y:t, i
Idt,
where r = IYl, ~): Y/IYl. PROOF. Let us start with (5.25)
~h =
~@j
N ~ yp~J
P=J
where i t should be noted that from (5.25). (5.26)
~h , Oyp ~
yp,j = 0 for
p < j.
(5.22) d i r e c t l y follows
I t follows from (5.25) that N ~h N ~2 h ~2h = - ! ~yp p,q=3 ~ypayq ~@~ P jYP . . . . + ~ .Yp,jYq,j , ~ •
59
Here we have used the relation (5.27)
~yp,j = .~2yp ;Oj ~G~ = - yp J
(j <__p < N, l < j
Thus we obtain N-I cos O. ~. Z bj(O) -2 -~2-h~ + (N - j - I) ~-i-~--iJ~?..~ j=l ~c.a J J N-I N N-I N m 2 ~h = Z_ 7,. oj ,-2 yp,jyq,j ,~2h Z ., bj Yp~-~p j=l p,q=j ~Yp~'Yq j=l p=j
(ANh)(Y) :
(5.28)
N-I Njbj2(N -jj=l p! -
cos 0. ~h I ) s i n 01 Yp ' j ~Yp '
and, hence, we have only to show N-1 N <~' _(j_ ~ ~h ~ bj(0) -2 N j I) .~_.~_0 j : l p=j YP'J - YP~-Yp (5.29)
N,
~h
(N - l) p!lyp ,~yp. The order of summation in the left-hand side of (5.29) can be changed as N-I
(5.30)
N
Z Z:Z
j=l p=j
N-I
p
N
Z +Z.
p=l j : l
j=l
Therefore, i t is sufficient to show
(5.31)
I bj(~)_21 cos 0. } Jp = -= (N-j-I) sin 0-~.Yp,j - Yp = - (N-l)yp j 1 L (p : 1,2 . . . . . N-I)
Nl
{
cosO
1
JN = jZlbj(O)-2.= (N-j-l) e---~. sin Yp,j - YN = - (N-I)YN"
To this end, let us note that
OO f
i{ l II '~--2- . . . . . I Iyp = I tsln Oj
cos 0. (5.32)
(j < p < N),
sin e~. Yp,j J (J : P L N - I ) ,
i-Yp whence follows that P~Ibj((,,)-2 (N_j_I) I 1 JP = j=l Isin 2 0j (5.33)
: [i i l l { b j + l ( O ) - 2 ( N - j - l )
= - (N - 1)yp The relation
- BJ(0)-2(N-J)} - bp(0I-2(N-p)IYPi
(I < p < N-I).
JN = -(N - l ) y N can be proved in a quite similar way, which
completes the proof of (5.23) Let
1 - 1 yp - 6p(0)-2(N-p)yp J
X(y)
(5.24) follows from (5.22) and (5.23).
be as in Lemma 5.5 and let "DJZ(Y) =O(IYl - j - c )
Z(y)
~.E.P.
satisfy the estimate
(lYl -~ ~,
j : 0,1,2),
(5.34) 2kZ(y) - Qo(y ) - Z(y) 2 - ¢ ( y )
: O ( l y l -~)
(I < ~ ~ l+c lYi ÷ ~)-
Then the estimates
(MjX)(Y),(ANX)(Y) =O(lYl l-e)
from (5.24), where we should note that
(!Y! .... )
16j(8)-lyp,jl
~ lYl
are obtained for
j ~ p.
Therefore, i t can be seen from (5.20) in Lemma 5.5, that O / r -~)
holds good under the condition (5.34).
which satisfies {Xn(Y,k)}
i(L-k2)(eiUx) I'x = In order to obtain ~(y),
(5.34), let us consider three sequences
(y ~ ~N,
~ sN-I,
k ~ IR - {0})
{bn(Y,k)},
defined by
{~n(o~,k)},
61
~'I (y) = o r
(r = lyl,
~,~ : Y / I y l ) ,
#n+l(y ) = 2-!{ {Qo(y ) + (~n(y)) 2 + ~ ( y ; X n ( y ) ) }
(n : 1,2 . . . . .
(n = 1,2 . . . . . m0-1),
(5.35)
0
(m = 2,3 . . . . . m l - l ) ,
~n ((o) : -~]~n (tw) - #n_l(tW)}d t + #n_l(W)
(n = ml,ml+l..,m0 ),
r
Xn(y ) = ~ #n (tu))dt + ((r)~n(m) (r = I Y l , where
~( )
is a r e a l - v a l u e d
C~-function
n = 2,3 . . . . . m0 ),
w = Y/IYl,
on
T
such t h a t ~0
(5.36)
0 < ((r)
< I,
~'(r)
> O,
and
~(r)
II and
m0,m I
for a l l
are as in Assumption 5.1.
n = 1,2 . . . . . m0-1.
If
m0 ~ ml,
In the f o l l o w i n g
(r ~ l ) ,
J (r ~ 2),
then we set
~n(~) = 0
lemma, i t can be seen t h a t
these sequences are well defined by (5.35). LEM~ 5.7. pair
(i)
( y , k ) E ~N x ( ~
@n(Y,k) = 0 (ii)
~n(Y,-k) {0})
= -C~n(Y,k), and any
~n(~,-k)
n = 1,2 . . . . . m0.
[Yl < 1 and n = 1,2 . . . . . m0. m0~l-n m0 + l - n @n(y ) E C (IR N) and ~n(~) ~ C (S n - l )
(5.37)
There e x i s t s a constant
C > 0
IDJ~n(y)l ~ C(l + l y l ) - j - c
such t h a t
f o r any
Further,
for
n = 1,2 . . . . . m0 • (iii)
= -~n(~,k)
for
62 and (5.38)
IDJ(#n(y) - ~ n _ l ( y ) } !
hold f o r any Dj
y c IR N,
n = 1,2 . . . . . m0
denotes an a r b i t r a r y (iv)
continuous functions on
and
j = 0,1,2 . . . . . mo+l-n,
d e r i v a t i v e of the
As functions of
k,
~
{
n (0}.
lyl) -a-n~
~ C(I +
and
jth
order. mo+l-n are C (IRN)-valued
~
n The constant
side of (5.37) and (5.38) is bounded when
where
k
C
in the right-hand
moves in a compact set in
IR - {0}. PROOF.
(i),
(ii)
and ( 5 . 3 7 ) ,
i t should be noted t h a t
O~(Y) = 0
(Mjh)(y) =
(5.39)
(5,38) can be shown by i n d u c t i o n . for
O(lyl I-=)
lY[ ~ 1 if
which follows from (5.22). n = 1,2 . . . . . mo-I that
mI < mO,
because
If
m0 ~ mI ,
and we may simply set any l o g a r i t h m i c
(ml-l)c < I.
(iv)
and
Dh(y) =
(= > o,
O(]y[ -T)
lyi + ~ ) , then
~n(m) a 0
for all
Xn(y) = f~#n(tm)dt.
In the case
term does not appear in the estimation
is c l e a r from the d e f i n i t i o n s
of
~n
Q.E.d. DEFINITION 5.8.
We set
Z(y) : Z(y,k) = #mo_2(y) + ~ ' ( r ) # m _2(m), r 0
(5.40)
i>.(y)
A(y,k)
~mo-2(Y) = 0f z ( t ~ ) d t ,
IY(y) = Y(y,k) = 2kZ(y) -. Qo(y) - Z(y) 2 with
r = iYl, REMARK 5.9.
Z(y) E C3(IRN),
~(y;~)
~ = Y/]Y](I)
Here
From Lemma 5.7, i t can be e a s i l y seen t h a t
Z(y,-N) = - Z ( y , k ) ,
Z(y) = 0
for
]y] # 1
and
and
~n"
63
i
I(DJZ)(y)l £ C(l + lyl) -j-~
(5.41)
'
r(DJY)(Y)I L C(I + l y l )
-J<mo-l)s
(y • ~ N,
(y •
~ N
,
j = 0,I,2,3),
j = 0,I).
Further, taking account of ( 5 . 5 ) , we have (5.42)
where
!(DJY)(y)! < C(l + iyL) -j-2%-2:S
# (2)
on
As a function of The constant
in a compact set in
(5.43)
k,
Z
is a
q
C3(IRN)-valued continuous function
C in (5.41) and (5.42) i s bounded when
k
IR - { 0 } .
Let us consider the case t h a t
Z(y,k)
= Zl(X,,)
=
I~ < ~ ~ I .
Then,
m0 = 3 and
%(y).
This case was treated in Saito [3] and Ikebe [21 ( c f . Remarks).
j = O,l)
is given by ( 5 . 9 ) .
IR = { 0 } .
(3)
(y • ~N
(12) of Concluding
moves
64
~6. An e s t i m a t e f o r the r a d i a t i v e Let
Z(y)
and
prove the f i r s t
k(y)
Now we are i n a p o s i t i o n
be as i n ~5.
to
h a l f o f Theorem 5.4.
Let us set f o r a f u n c t i o n
(6.1)
co(G) = sup
proposition
G(y)
IR N
on
(I + iy!)~"!G(.y)
y~IR The f o l l o w i n g
function
.
N
is an i m p o r t a n t step to the p r o o f o f ( 5 . 1 0 )
in
Theorem 5.4. PROPOSITION 6.1. that ]R
Qo(y ) {0}.
Let
Assumption 5.1 be s a t i s f i e d .
has compact s u p p o r t in Let
3
IR N
be as in ( 5 . 9 ) .
Let
K
F u r t h e r assume
be a compact set i n
Then t h e r e e x i s t s
C = C(K,O)
such
that i,
(6.2)
Ilu'-ikuil[~, + IIB:'ul:,~ ," C"[I,f l I I + F + ilvl; _o}
holds f o r any r a d i a t i v e f c L2,1+B(I,X), constant
function
where
C = C(K,Q) Here
tive of
o r d e r and
m0
(L-k2)v = f ~(y)
Let
~,(.y)
~,,c I
i s given in ( 5 . 4 0 ) .
Dj
and The
j = 0 , I , 2 . . . . . mO,
means an a r b i t r a r y
deriva-
are given i n Assumption 5.1.
proposition,
- {0}
k c K
c. I ( Q I ) , { : j + c ( D J Q o ) ,
we need several
v e D(1) ~ H ~ ' B ( I , X )
k ~ ~
with
and
lemmas.
be a s o l u t i o n
f ~ L 2 ( I , X ) I o c.
Set
o f the e q u a t i o n u = ei)'v
with
d e f i n e d by ( 5 . 4 0 ) . (i)
(6.3)
with
and
{L,k,~Jfl~
is given by ( 5 . 1 ) ,
In o r d e r to show t h i s LEMMA 6 . 2 .
for
is bounded when
are bounded. jth
u = ei~v
v
Then we have -(u'-iku)'
- ik(u'-iku)
+ Bu
= ei)'f - 2iZ(u'-iku)
+ (Y-C1-iZ'-iP)u
- 2ir-2Mu,
65 where
Y is as in (5.40) and P,M are given by (5.18). P (ii) Let V(y) = =~igjG j with C ] functions gj on IR N - {0} J and operators Gj in X such that Gju ~ Cac(I~X) with (Gju)' = Gju'
in
k2(I,X)loc.
Then T T _.[(Vu,u'-iku)xdr. = ~-~-k1 t~[ ( V u , u ' - i k u ) x IT - J(V{u , , - i k u } , u , -iku)xd~ R
R
T f(V'u,u'-iku)xdr
T + 2i f ( Z V u , u ' - i k u ) x d r
R
R
T + f(Vu,{Y-Cl-iZ'-iP}u
(6.4)
+ ei~f)xdr
R
T 2 - }(Vu,Bu)xdr + 2i f ( V u , r - MU)xdr} R
R
(0 < R < T) P ~gJ Gj. ~ Dr j=l PROOF. ( i ) is obtained by an easy computation i f we take note of
with
V' =
Lemma 5.5 and the r e l a t i o n
(L-k2)v = f.
both sides of (6.3), multiply i t by Then, by the use of p a r t i a l LEMMA6.3.
Let
Take the complex conjugate of
Vu and integrate on
integration,
x,x' c D and l e t
SN-I x (R,T).
we a r r i v e at (6.4). S(y)
be a
Q.E.D.
C1 -function on
IR N.
Then (6.5)
(SMx,X')x + (Sx,MX')x = -r2(SPx,x ') - ((MS)x,x') x,
PROOF .
By p a r t i a l
(Sx,(Mj~)MjX')x : (6.6)
integration,
we have
~N_ 1S(r~)x((*~) (Mj~)(rw)bj(O) -I ~Oj ~ ' -dm (d~, = (sin O l ) N - 2 . . . ( s i n
@ j ) N - j - I . . .sin ON_2)
66
= _ [
~,~T-{S(rc~)x(,*O(Mj>,)Cr',O(sin L : j ) N - j - l } b j ( o ) - l £ ' ( w ) -
sN_ 1 ",,j
ds.)
•
= - S
{(Mj>.)Mj(Sx)
+ Sxbj(9)-2(sin
ej)-N-j-I
sN-I a
.
i},~), J
((sin uj )
,N-.i-1 ~
~
J
)~'~d~ ,
whence we obtain (S ,MX')x = -(M(Sx) + SX(ANA),X')x (6.7)
= _ (SMx,x')x
This is what we wanted to show. LEMHA 6.4.
Let
Qo(y)
r a d i a t i v e function for where
B
Q.E.D.
be as in Proposition 6.1.
{L,k,~[f]}
with
i.e.,
B = ~ -~:
is as in (5.9),
((MS)x,x') x = r 2 ( S P x . x ' ) x .
k • IR - {0}
Let
v
and
f • L2,1+~(I,X),
(0 < c ~ ) ,
= 0
be the
(~ < E ~ I ) .
I.
Then we have
v ' - i kv,B 2v c- L2,[~( l,X)
PROOF. Let function for
k n = k + i_ n
{L,k,L[fl}.
(n = 1 , 2 , 3 , .
''
)
and l e t
v
n
be the r a d i a t i v e
From Lemma 2.7, i t follows that
v'-ikvn n'B'2Vn • L2,1+B ( I ' X ) "
M u l t i p l y both sides of
-((Vn-ikvn)"V'n-iknVn)x-
iknlVn-iknVn 12 +x
(BVn,V'-ik~v_)xu . .
(6.8) + (CVn,Vn-iknvnlx = (f,Vn-ikvn) x by
~(r)(l
+ r) 2[~+I,
integrate from
0
to
~(r) T.
being given by (5.36), take the real part and Then, proceeding as in the proof of Lemma 2.5
and using (5.5), we a r r i v e at
67
(g + ½ ) ~ - 2i~... iv n,_iknvnl~dr + (l__ 1!)}(r23!B'~2v. nl2xdr 0 0 2
! ~-(I+T)2~+I I V n ( T ) - i knVn(T) I x
(6.9)
+
I TIE'(l+ 0
r)23+1 ~,I i B]~Vn 12 -]Vn-iknVn] 2xTdr : x
T + C [ ~ ( l + r ) - ~ + g ! v ' - i k n n n x VI Ivnlx dr 0 T + I~j(l+r) 21~+I Iflxlv'-ik n 0 whence, by the use of P r o p o s i t i o n 2.3 (the l i m i t i n g
absorption
n
v n Ixdr
'
1.2 (the i n t e r i o r
principle),
it
estimate)
and Theorem
follows that
i,,
(6 • I0)
livn-iknvn!l ~ + llB2v n iI.~ |,=-< Cllfll 1+!~
w i t h a constant
C which is independent of
(n = 1,2, "'" )
n = 1,2 . . . . .
Let
n + ~
in
the estimate I
(6 .I 1 )
ii
Vn-i kVnll #, (O,R) + II B'~Vnll ~, (O,R) -<- Cli fil l+f~"
Then i t f o l l o w s from Theorem 2.3 t h a t )
(6.12)
i ] v ' - i k v , g,(O,R) + [i B'~vii ;5,(O,R) < C"flll+i5
holds f o r any a r b i t r a r y
i.
R > O,
which implies t h a t
v ' - i kv,B2v E L2,~(I , X) •
CL•[.~• LEMMA 6.5. pact set in with
T>R+I
IR - {0}.
k(z K and
2,(r-R+l),
Let
where >R>I
~(y) Let
be as in P r o p o s i t i o n v
be the
f e L2,1+I~(I,X), ~(r)
radiative
6.1 and l e t
K be a com-
function
for
{L,k,~[fJ}
Let
~.(r) = (~R(r) =
{~ being as in ( 5 . g ) .
is given by (5.36)•
Then we have f o r
68 T
,,
R+I
#,r2m(lu'-ikul2+~B~ul2)dr R
x '
(6.13)
x
+ Cl
~
'
::r2~;-'~'(iu'-iku ,2x + iB."~,ui )dr + llfl~l+ B + llv II
~-mI - 1 k-n- 1
u = e
v,
n(T)
x
li
+ C2 In:O where
'- 2 ' - i k u i x 2+IB~u ! x )dr
~ ~,(T) + Cm ~ (lul2+lu
dr-' T 27-I( Re j,,r znMu,Bu) R x .] ,
is a f u n c t i o n
of
T
satisfying
lim n(T) = O, T-~×)
mI
is as in Assumption 5.1,
K,
and
Cp = Cp(K,Q),
j = 0,1,2 . . . . . mO,
p = 1,2,
are bounded when
R and
pcl(Ql),Pj+c(DJQo ),
are bounded.
PROOF. M u l t i p l y over the region
CR is a constant depending only on
both sides of (6.3) by
sN-I x (R,T)
,zr23+l(o~~O-),
and take the real p a r t .
K : Re } .,~r 2~+I"~ _ ( ( u ' _ i k u ) ' , u ' - i k u )
integrate
Then we have
x * (Bu,u'-iku)x~dr
(6.14) : Re } <~r 29+I { ( e l ~""f , u ' - i k u ) x
+ I m } ~2ri ~ + I : R
+ ((Y - Cl)U,U - i k u ) x } d r
(Z' + P ) u , u ' - i k u ) x d r
+ 21m ~or23-I ( M u , u ' - i k u ) x d r The l e f t - h a n d
side
K of (6.14)
T 2'~ K > (~ + . 1 ) f ' x r ~ I u ' - i k u R
= Kl + K2 + K3.
is estimated from below as f o l l o w s : 2 T 2"' i. xdr + ( ~ - - f i ) . I ~ r ~']B2Ul2xdr R
(6.15) + K1
~- R}Ic~'r2[~'+l{ u _ikui2x_iB1~uI~}dr _ ½T2(~'+IIu'(T)-iku(T)I 2 R x"
can be estimated as
(6.16)
K1 £ C{llfIll+i~
ar2L'; I
'~ '> R+I -iku]2xdr +llf!l~+~+(R+l) 1-25 i(z'lu[2xdr + llvl12 I
@9
+ TI-2+Iu(T)I#}, where we i n t e g r a t e d by parts and used (5.42) in estimating the T ~r26+i term Re f , (Yu,u'-iku)xdr. Here and in the sequel the constant R depending only on K and Q(y) w i l l be denoted the same symbol C. I t follows from
(6.14) - (6.16) t h a t
T e 2 ~" 2)d r c I ~ a r 2 ' ~ ( l u ' - i k u ! x + IB2ul x R (6.17)
< CTT26+llu t" ' , - i k u ! ~ + T l - 2 6 [ u ( T ) ! 2 + (I + R) 2~+I Ril o,lB+ul~dr X
+ ,v, 2-~ In order to estimate 6.3 ( i i ) .
Set
+ K2
R+I ~R c~'lul2xdr + Ilfll 2l+#~~ + K2 + K3 and
K3
(Cl = ,~_(,2-f~)) "
in (6.17), we have to make use of Lemma
V = mr2B+l(z' + P)
in ( 6 . 4 ) .
Then
K2 ~ - ~ R e { T 2 ~ q + I ( ( z ' ( T ) + P ( T ) ) u ( T ) , u ' ( T ) - i k u ( T ) ) x } + F(T) + G(R)
(6.18)
+
Im
R +
where
F(T)
and
G(R)
I,,
((z'+P)u,Mu)xdr,
R
are the terms of the form T
F(T) = C{ILfU#+ 3 ÷ (6.19)
, 'zr2~ ~(lu'-ikur2'x + lB~u[2)drx + llvll 2- 5)
R}l(lul ~ IS(R)
CR
+
lu'
Iku! 2 + X
R CR being a constant depending only on sary to estimate the term in ( 6 . 4 ) .
R and
((Z' + P)u,Bu) x,
Then i t follows t h a t
K.
'
IB'2ul2)dr}, X
Here, (5.17) is neces-
Let us next set
V = ~r21~-IM
70 1 Re~T2~'~-1 ( M u ( T ) , u ' ( m ) - i k u ( T ) ) x : m3 L - -k
+2 (6.20)
+ F(T) + G(R)
T
Im f..,r 2"-1 (ZMu, u' -i ku) xdr R
+ -~ Ini ~r,,r 2 3 '- l ( M u , ( Z I + p ) u ) x dr R + ~- Re ~.~r2i~-l(Mu,Bu)xdr R Re(M(u'-iku),]'-iku) x = - (r212)(P(u'-iku),u'-iku)x
Here we used the r e l a t i o n
which follows from (6.5) with
x = x' = u ' - i k u
was also used in order to estimate the term from (6.17),
and
S(y) = I.
Re(Mu,Yu) x.
Lemma 6.3
Thus, we obtain
(6.18) and (6.20)
m
f.~r
2'-
R
i,
r)
:~(lu'-ikul 2 + IB~-uiL)dr < T..,I(T) + F(T) + G(R) x ;~ ~+ C{
(6.21)
~_
T, 2"~+I " (Z(Z'+P)u,u'-iku)xdr
Im j~r R
+ 2 Im ):~r ~ R
(ZMu,u'-iku)xdr
Re
"
(Mu,Bu)xdr }
-- ql(T) + F(T) + G(R) + C{J 2 + J3 + J4 }' where we have used the fact that (6.22)
v ' - i k u , B 2 v • L2,~(I,X)
pact in
and
hi(T) = C{T2~+Iiu'(T)-iku(T)12'x + TI-25!u(T) Ix2 + T28+I B.~u(T) ix I,
Since
Im{((Z'+P)u,Mu) x + (Mu ( Z ' ~ ) u ) x} : O,
by Lemma 6.4 and the support of
IR N by the compactness of
O.o(y),
Z(y)
Is com-
i t can be e a s i l y seen that
11
u'-iku,B'2u c L2,[5(I,X),
which implies that
lim q(T) = O.
J2
and
J3
in
m-~<~
(6.21) can be estimated in quite the same way as in the estimation of and
K3
(6.23)
respectively.
By repeating these estimations, we a r r i v e at
T o !2 + [B+u i x2)dr ]mr2~+(Ju'-iku R
x
K2
71 q(T) + F(T) + G(R) [mI -I + Cln!O k-n-I Re }"~r2~-I (znMu'Bu)xdrR + k -ml Im ~ar2B+l . (Zml(z'+P)u,u'-iku)xdr R
+ 2k where
TI(T)
Im
c~ R
(zmlMu,u'-iku)xdr
is the term s a t i s f y i n g
lim q(T) = O.
Z(y)
By noting that
ml
:
m-~o~
O(;yl-l),
(6.13)
Q.E.D.
is obtained from (6.23).
We have only to estimate the terms sition
1 ,
in order to show Propo
Re(ZnMu,Bu)x
6.1 completely.
LEMMA 6.6.
Let
S(y)
be a real-valued
S(y)l Lc
Cl
function
such that
(y e mN),
(6.24)
,DS(Y)I 5 c r - l with a constant
c > O,
(6.25)
I Re(SMx,Ax~• x .... I < Cr l - c ( iA~xl2x '~ +Ix'2,x )
holds with
where
(IYl ~ l)
~(N
PROOF.
(I)
(j = 1,2 . . . . . N).
C = C(c,K,~ O) which is bounded when
j = 0 , I , 2 . . . . . mO, are bounded. A = -A N +
D = --~--.3yj
Then the estimate
(r _> l ,
c and
ke
K,
pj+c(Qi]),
K is a compact set in
~
- {0}
and
- 1)(N - 3).
We shall
d i v i d e the proof i n t o several steps.
By the use of (5.17)
J = Re(SMx,Ax) x
can be r e w r i t t e n
rN-I 1 J = Re(SMx'/~X)x = Rel I ! 1 (Mn(SMx)'MnX)x n (6.26) + c 2N Re(SMx,X)x = Jl + J2
as
x e D)
72 K an O,K. term when K is
Throughout this proof we shall call a term dominated by Cr l-E(IA~x;x,2 + ix!~ ) _
IMXlx ~ Cr I ~IA~xlx ,
r ::> 1 with
we can easily show that
we have only to consider the term (II)
for
L.
Let us calculate
J2
Since
is an O,K. term.
Thus,
Jl"
Jl"
i
Jl = Re ni l((Hns)Mx~Mnx)x
}
IN 1 N-I + ReLn=l(Si j=l>~ (MnMj~')(Mjx)'MnX)x
(6.27)
C = C(c,K,Q).
}
rN-I N-I } I "' (S ~ (Mj~)(MnMjX),MnX) x = Jll + J12 + J13" + Re n)--'l j&l By noting that an O.K. term.
MnS(Y )
Before calculating
J13'
tbo; (6.28)
{y e IRN/Iyl __>1 ~,, Jll
is bounded on
MnMj - MjMn =
we mention
cos Oj ~6-;,-i]-
Mn
l I
is seen to be
(n>J)' (n = j ) ,
cos o
i_b-I n Mj n sin On which is obtained d i r e c t l y from the definition of
(n < j ) , Mj
(see (5.15)).
Using
(6.28), we have FN-I~ n-I b- 1 cos 0. >i (Mj>,) J (Mnx)'Mnx)x J13 = Re ILn=l~. ~(S ~ j=l j si-n-#i-jN-I cos On ~I - (S ~ (MjA)bn I ~-~-n---cT n- (Mjx),MnX)x j =n+l J~
(6.29)
Here,
J132
MnX ,
we have
is an O.K. term, because, making use of (6.5) with
x = x' =
73
1 J132 : - 2- Nil{r2(SP(MnX)'(MnX))x + ((MS)MnX'MnX)x}" n=l
(6.30)
Therefore, let us consider (III)
J12 + J131"
Set
r F~z-I tdt Zp(r,~) = I imypl y:t,~ 0 :_ _~
(p : 1,2 . . . . . N),
(6.31) Z~(rw) : (bjbn)-I jfl
p=j 0 l~Y-~q_ YP'JYq' _j y=t',~ q=n (j,n : 1,2 . . . . . N-I),
where
= Yp,j
obtain (6.32)
~Yp ~0. • J
Then, setting
cos ON ~ 1 and starting with (5.24), we
Mj~. = b]l J {-bj+IZ j
+
N cos 0. 3 cos 0 p }, ~ bpZp sin----~-~ p=j+l j
and Zjn (6.33)
[,'IM.k
nj
=
Znn
(j > n), _ b- 2
N
n p!nZpbpCOS Op
cos @. Zjn + b] 1 sin c~ (MnX) J Then J12
(J < n).
is rewritten as J12
I i n-lbj I cos gj N I(S j ~'>l _ sin O[ (Mnk)(Mjx)'MnX)x 'j
= Re ~ n~l
N
N-I
(6.34)
(j = n),
"1
~, (Sbn 2 p!nZpbp COS Op(MnX),MnX)xl n=l
{"ii S nj=l~l Zjn(Mjx~
+ Re n
i
MnX)x = J12.1 + Jl 22" J
74 Here,
J122
is an O.K, term because
estimate of (5.34).
Zjn(Y) = O(iy! l-c)
Hence, in place of
by the f i r s t
J12 + J131' i t is sufficient
to
consider J' = J121 + J131 (6.35) : Re Ln=l(SFn(MX ),MnX )x
IY
} ii
= n-1](Mj b] ] cos Oj j! ~) sin Oj
NnZpbpb~2cos
+ Re
(n--I
(SGn,Mnx)x != Ji + J2 I
with f
Fn
~ P
Op
= Fnl
Fn2 ,
d
(6.36)
I i
L (IV)
n-I -1 cos ej N-I cos ^ = (MnA x) ~ ::~n (Mj~)(Mjx) Gn j~ibj sin Oj )(Mj j=n+16n I ~ On
Now let us calculate
O'.
Using (6.32) and interchanging the
order of summation, we arrive at (6.37)
N
N
Fnl = -p~lZpbp cos
Op + ~ bn2bpZpCOS ep. p=n
and hence (6.38)
N
Fn = - p~l':ZpbpCOS Op,
which implies that
J~ is an O.K. term.
As for
J~,
we can interchange
the order of summation to obtain #I-I N-I cos ,~. : {(Sb] 1 J~ Re 19! I j ! l , sin Oj3 (Mn>,)(Mjx )'MnX)x (6.39)
.-I
_ (SbjI cos ej (MnX)(MnX)'MjX)x~ sin Oj
=0.
75 Thus, we have shown t h a t each term of
J = Re(SMx,Ax) x
is an O.K. term,
Q.E.Do
which completes the proof.
Proof of Proposition 6.1.
By Lemma 6.6, the l a s t term of the r i g h t F(T),
hand side of (6.13) is dominated by the term of the form F(T)
is given in (6.19).
sequence
{T n}
Therefore, by l e t t i n g
T
where
along a s u i t a b l e
->- oo
in (6.13), we obtain oo
[j6(lu'-iku[2
X
R
+ IB½uI2x)dr
(6.40) [~ < G(R) + Ci£c~r2B-~(
u'-ikulx2
+
IB~ul 2x )dr + ,,f ,, #+ B+,,v ,, !
!, )
where
C = C(K,Q)
is i n d e p e n d e n t
large in (6.40).
of
R > O.
Take
R = R0
sufficiently
Then i t fol 1ows t h a t
co
(6.4t) Since
~ r 2 B ( l u , _ i k u i 2 x + iB½U{2x)dr < C{llfll~+~B+llvll2} + G(Ro ) (C = C(K,Q)). RO+I = G(Ro) is dominated by the term of the form C{llfll~+B+llvll!6} by
using Proposition 1.2,
(6.2) e a s i l y follows from (6.41).
the proof of Lemmas 6.2 - 6.6, we can e a s i l y see t h a t remains bounded are bounded.
when
P~I(QI)'°J+c(DJQo)"
Re-examining C = C(K,Q)
J = 0,1,2
m0
Q.E.D.
Now t h a t Proposition 6.1 has been shown, we can prove (5.10)
in
Theorem 5.4. PROOF for where
of (5.10) in THEOREM 5.4.
{L,k,~}
with
k c K,
6 - 1 ~ B ~ 1 - ~.
can be decomposed as {L,ko,~}
and w
Let
v
be the r a d i a t i v e f u n c t i o n
the compact set of Let
k 0 ~ 6÷
v = v0 + w ,
where
~
- {0},
such t h a t v0
and
Im k 0 > O.
L c FI+ ~, Then
v
is the r a d i a t i v e f u n c t i o n f o r
is the r a d i a t i v e f u n c t i o n f o r
{L,k,~[f]},
f = (k2-k~)vo •
76 It follows
from Lenlma 2.7 t h a t
Set
Col]l'~llll+ 6 (c o = Co(ko,[~))
IIv~lll+ 6 + IIB'~vOIIl+[~ + IIvoIll+c~ <
(6.42) u0 = e
iX
v O.
Noting t h a t u~ - iku 0 = eiX(v~ - ikv O) + iZei>'Vo ,
(6.43) and
( { i P + 2ir-2M-~!Vo,Vo) x,
0 ~ (B(r)Uo,Uo) x = (B(r)Vo,Vo) x (6.44)
iB2vo~x' 12 + C{ivoi2x + IB~vo 12} with
C = C(K),
which f o l l o w s
from (5.19)
in Lemma 5.5, we o b t a i n from
(6.42) 1,
u, 1 Clll~rl~F+
. • , + IIB'2Uoll £ :< IlUo-lkUo,i#
(6.45) Therefore,
it
suffices
t h i s end, we shall
to show the estimate
approximate
Qo(y)
(C = C ( k o , K , # ) ) -
(~10) w i t h
by a sequence
u = ei~w. {QOn(Y)},
To where we
set
f~vli (6.46) and
o(r)
(r ~ I ) ,
QOn (y) : °'--~"Q~(Y)t n )~u is a r e a l - v a l u e d , = 0
(r ~ 2).
bounded u n i f o r m l y
smooth f u n c t i o n
Then i t n = 1,2 . . . .
on
can be e a s i l y with
T
us denote by w n
f = (k 2 - k~)v 0
the r a d i a t i v e
(n = 1,2 . . . . ).
such t h a t
seen t h a t
function
For each
L n,
p(r)
= 1
pj+~(DJQo n)
j = 0 , 1 , 2 . . . . . m.
d2 L n . . . . . . .dr 2 + B(r) + Con(r) + C l ( r )
(6.47) and l e t
for
(n = 1,2 . . . . )
Let us set
(Con(r)
= O.on(rm)x),
for {Ln,k,~Ifl~ the f u n c t i o n
is
with
z(n)(y)
77 can be constructed QOn(Y ),
u n = ei~(n)w n
Now, Proposition (6.49)
n
Since
formly on
IRN
as
uniformly
n ÷ ~
6
replaced by
r = fz(n)(tm)dt) • 0
+ llB2Unll
f o r each
j = 0,I . . . . . mO,
in
DJQo(y)
i t follows
on every compact set in un + u
(n = 1,2,. -" ,) '
< C {llfltl+B+llWnlr_6}
B =
IR N.
as
that
n ÷ ~
~ ( n ) ( y ) ÷ ~(y)
Therefore,
'I'611,X) UO_6~
as
uni-
by the use
n + ~o
Let
in the r e l a t i o n
(6.50)
Ilun-ikunll~,(O,R) R > O,
(6.51) Since
(~(n)(rm)
DJO~on(Y) converges to
of Theorem 4.5, we obtain
with
O.o(y)
6.1 can be applied to show llu'-ikunVE
C = C(K).
n + ~
5.8 with
and we set
(6.48)
with
according to D e f i n i t i o n
which is a d i r e c t llu,_ikullB,(O,R)
R > 0
(6.52) (5.10) follows
is a r b i t r a r y ,
+ IIB2UnllB,(O,R ) < C{llflll+ 6 + IlWnll_6} consequence o f ( 6 . 4 9 )
.
Then we have
+ llB½ull~,(O,R)< C{Hflll+ B + llvlJ 6 }.
we have obtained
IIu'-ikuII~ + IIB½BII~ ~ C{lifIIl+ ~ + llvIl_~}. from (6.42),
(6.45),
(6.52) and ( 2 . 7 ) .
Q.E.D.
78 ~7.
Proof of the main theorem
This section is devoted to showing (5.11) in Theorem 5.4 and Theorem 5.3 by the use of (5.10) in Theorem 5.4 which has been proved in the preceding section. PROOF of (5.11) in THEOREM 5.4. v is the radiative
Let us f i r s t
function for { L , k , L [ f ] }
consider the case that
with keK and feL2,1+B(l,X ).
Using
i) of Lemma 6.2, we have for u=ei~v
(7.1)
I d---~e2irk(u'(r)-iku(r),u(r))x}=e2ikrg(r I dr ~ g(r)=lu'-ikui2-1B~2uI2-(eiXf,u) +2i(Z(u - i k u ) , u ) x
i
'X
X
X
-((Y-C,-iZ'-ip)u,U)x+2ir-2(Mu,u)x.
I t follows from (5.10) that g(r) is integrable over I with the estimate I g(r) I xdr
2+11 2ull2+:1fll ~II vii 0
0
,,
-6
1,
(7.2)
+C{II u ' - i kull J vii -a°+ll vii v2 "+11B l l -~u!l o 311 -~,~}
Integrate the
f i r s t r e l a t i o n of (7.1) from r to R (O
(7.3)
k I v(r)l~=kl u(r)12 x =Im(U' (r) ,u ( r ) ) x - l m { e 2 i k ( R - r ) (u' (R)-i ku(R) ,u(R))x } R .
+Imfre21k(t-r)g(t)dt. By l e t t i n g
R-~o along a sequence {Rn} such that (u'(Rn)-iku(Rn),u(Rn))x+O
(n-~o) (7.3) gives
<
u r) x +f r I g ( t ) ' d t } .
Let us estimate I m ( u ' ( r ) , u ( r ) ) x.
(7.5)
d~d
,. ~ I m (,u ' ( r ,
Since
~,u(r) )x)= Im{~-~d (u ' (r) , u ( r ) ) x }
=Im(e -2ikr-~d r e 2 i r k ( u ' ( r ) - i k u ( r ) , u ( r ) ) x } dr '
1 = ling(r)
79 and (7.6)
Im(u'(r),u(r))x=(Z(r.)v(r),v(r))x+Im(v'(r),v(r))x-~O (r÷O)
by (3.36), i t follows that (7.7)
llm(u'(r),u(r))xl ~
Ig(t) Idt. 0 Thus we obtain from (7.4), (7.7) and (7.2) (7.8)
I v(r) rx~Cll fli i+ ~
with C=C(K).
(rEl)
Let us next consider the general case.
v is now assumed
to be the radiative function for {L,k,~} with kEK and ~EFI+B(I,X). Then, according to Lemma 2.8, we decompose v as V=Vo+W, where v 0 is the radiative function for {L,ko,R,} with koe£ +, Imko>O and w is the radiative function for {L,k,LE(k2-k~)vo]}.
I t follows from Lemma 2.7
and (1.8) in Proposition 1.1 that
I vo(r)l~v~i Voll~<_CIII~IIII+ ~ (tEl),
(7.9)
llVoll+~CIi~llll+ B with C=C(ko). (7.10)
On the other hand we obtain from (7.8)
Iw(r) I x<=Cllv 0111+B
with C=C(K,ko).
(5.11)
(r~l)
d i r e c t l y follows from (7.9) and (7.10). Q.E.D.
The proof of Theorem 5.3 w i l l be divided into the following four steps.
Lemmas 7.1, 7.2, 7.4 and Corollary 7.3 w i l l be concerned with
the asymptotic behavior of the radiative function v for { L , k , ~ I f J } witF ke ~-{0} and fCL2,1+B(l,X). LEMMA 7.1.
Let v be the radiative function for { L , k , L l f ] } with
keA-{O} and fEL2,1+B(I,X).
Then I v ( r ) I x tends to a l i m i t as r-~x,.
PROOF. Let Ro>O be fixed.
(7.3) are combined with (7.5) to give
IS()
iF
(7.11)
Iv( r)l#=~ I m(u'(RO),u(RO))x +Im] Rog(t)dt ,<~
.
+ I ~.21 k(t-r) g(t)dt}, Jr whence follows that (7.12)
lim]v(r) x=k 2 ,~im(U '(Ro),u(Ro))x+Im g(t)dt}, r-~ R0 which completes the proof. Q.E.D.
LEMMA7.2. Let v be as in Lemma 7.1. (7.13)
Then there exists the weak limit
F(k,f)=w-lim e - i ~ ( r ' ' k ) v ( r ) ~×'
in X, where #(y,k)=rk-~(y) and X(y) is given by (5.40).
PROOF. Let us set I C~k(r)=21-~e-i/1(r" 'k)( v' (r)+ikv(r)) (7.14) I
=2~keirk(u' (r)+i ku(r)-iZ(r. )v(r))
[ ~k(r)=eirk(u' (r)-i ku(r) ) =eirk+ix(r" ) (v' ( r ) - i k v ( r ) + i Z ( r . ) v ( r ) ) with u=e i~v. (7.15)
Let XC-D. Then, by (i) of Lemma 6.2,
~T~_(m k d (r),X)x=-~l.e-irk(-(u'-iku)LiK '-ik(u'-iku)+iZ'u+iZ(u'-iku),x) 1 o-irk{ (ei >,f =2ik (u'B X)x'x )x+i (Z(u'-iku)'x)x
X
+( (Cl-i P-Y )u, x)x-2i r -2 (u ,M X)x }=2]-~-i rkg (r ,x), where we have used the relation (6.5) in Lemma 6.3 with x=u, x'=x and S(y)zl.
Similarly we have
(7.16)
d ,~~k ~ 'r'j ,x )=eirk[ (u,B X)x- (ei~f, x)+2i (Z(u '-i ku) ,x )x ~-~-t
-((Y-Cl-iZ'+iP)u,X)x-2ir-2(u,MX)x}=eirkh(r,x Therefore
'
)
by (5.10) in Theorem 5.4 and the fact that IMxl X :
g(r,x) and h(r,x) are integrable over (1,~), which implies that there exist limits
81
I im (,~k(r) ,X)x=m(k,x), r-~ (7.17) I im(~k(r ) ,X)x=~ (K,x). t->co
Here ~(k,x)=O.
In fact, since u'-ikuEL2,[~(l,X), there exists a sequence
{ r n} such that l~k(rn)Ix tends to zero as n÷oo. Therefore, by taking account of the fact that (Z(r-)v(r),X)x>O as r-~o , which is obtained from the boundedness of l v ( r ) I x ( r : l ) ,
i t follows from the second relation
of (7.17) that l i m e-i'~ ( r " ' k ) ( v ' ( r ) - i r~
kv ( r ) ,X)x
(7.18) lim { e - 2 i r k ( ~ k ( r ) ' X ) x - i e - i u ( r ' ' k ) ( z ( r ' ) v ( r ) ' X ) x } = O r÷~ Thus we o b t a i n from ( 7 . 1 8 ) and the f i r s t r e l a t i o n of ( 7 . 1 7 ) :
Ilim
{(e-i~(r''k)v'(r),X)x+ik(e-iU(r''k)v(r),X)x}=2ik
~(k,x),
(7.19)Ir÷~ |lim
{(e -i~(r
Whence f o l l o w s ~(k,X)
'k)v'(r),X)x-ik(e-iu(r''k)v(r),X)x}=O
the e x i s t e n c e
of the l i m i t s
= lim ( e - i n ( r ' ' k ) v ( r ) , X ) x r÷~
(7.20) -
for of
x~D.
1 lim(e-in(r',k~,(r),x) x i k r÷~ Because of the denseness of D in X and the boundedness
[ v ( r ) l x on I ,
which completes
the p r o o f . Q.E.D.
COROLLARY 7 . 3 . Iv'(Rn)-ikV(Rn)l÷O (7.21)
F(k,f)
PROOF. {Iv(Rn)lx},
r n} be a sequence such t h a t Let :R as n+~.
: w-lim~k(Rn)
Since it
Then t h e r e
exists
Rn +~ and
the weak l i m i t
in X.
{ I v ' ( R n ) I x ~ i s a bounded sequence as well
follows
from
(7.20)
that
the weak l i m i t
as
82
w-~
e-i~(Rn''k)v'(Rn)
Therefore,
(7.21)
exists
is e a s i l y
in X and is equal
obtained
to i k F ( k , f ) .
from the d e f i n i t i o n
of
mk(r). Q.E.D. LEMMA 7 . 4 . { r n} such t h a t
rn+~ (n÷~)
im ( r )nI ~÷~Iv
(7.22)
PROOF.
I
There e x i s t s
Let v be as above. and
x = [F(k,f)l
x"
Let us take a sequence
{ r n} which
L~i~Iv'(rn)-ikv(rn)l all
n=l,2 .....
above. 2.3. (7.24)
satisfies
( l + r n ) ~ + ~ I B 1 ~ ( r n ) U ( r n) i x ~ CO
~i~ ( +rn)3+½[B1~(rn)U( ' (7"23)llV(rn)iix ~ CO , rn)'x
for
a sequence
= 0
x = O,
where Co>O is a c o n s t a n t ,
Such a { r n} s u r e l y
exists
u=eiXv,
and ~ is as
by Theorem 5.4 and Theorem
Let us s e t ~_k(r)
-
2 i1k e i ~ ( r ' ' k ) ( v
(r)-ikv(r)).
Then we have (7.25)
v(r)
= ei~(r''k)~k(r
the d e f i n i t i o n
of mk(r)
) + e-i~(r''k)c~ being
given
in
-k
(r)
(7.13),
whence f o l l o w s
that ,
(7.26)
iV(rn)Ix
2
= (~k(rn), + ((~_k(rn),
e-
i~
(rn
•
'
k)v
(rn)) x
e+i~(rn "'k)v(rn))x
= a n + bn
Here b n ÷0 as n÷~ because of the t h i r d and f o u r t h r e l a t i o n s of (7.23). T h e r e f o r e , i t s u f f i c e s to show t h a t a n ÷ ] F ( k , f ) [ x2 as n÷~. (7.27)
Setting Fn(k,f ) = e-iU(rn''k)v(rn
we o b t a i n
from
(7.15)
),
83
- IR ~ (dm k ( t ) , F n (k ' f ) ) x dt rn
an = ( m k ( R ) ' F n ( k ' f ) ) x (7.28)
: (~k(R),Fn(k,f))
I IRr - i k ( t - r 2ik e n
x
where (7.29)
g(r,x)
= (Bu,x)x
+ ((Cl-iP-Y)u-e
+ i(Z(u'-iku),X)x Now l e t
us e s t i m a t e
and the t h i r d (7.30)
Ig2(r,U(rn)i[
As f o r g 3 ( r , u ( r n ) ) (7.31)
f'X)x
4
+ {-2ir-2(u,Mx)x}
g ( r , U ( r n ) ).
relation
) n g(t,u(r n )dt,
of (7.23) :< C o ( ! f ( r ) l
!t
: ~ig j r , x ) . J
follows
from ( 5 . 5 ) ,
5.42)
that x + C(l+r)-26[v(r)Ix)
= go2(r) .
we have s i m i l a r l y
Ig3(r,u(rn))!
~ CoC(Z+r)-C-B!(Z+r)B(u'-iku)l
x
=< C o C ( l + r ) - ~
= go3(r),
(l+r)3(u ' _ikU)Ix
because (7.32)
-~-B = i -s =< -6
6=0),
-~-(6-~)
g4(r,U(rn))
= -6
can be estimated
Ig4(r,U(rn))J
~=6-E).
f o r r ~ r n (~I)
~ C(1+r)-2(
(7.33)
as
l+r)Z-ElUlx[A½u(rn)!x
= C(Z+r)-Z-C(l+rn)½-BlUlxl(l+rn)B+½B½(rn)U(rn)l> CCo(l+r)-E-~-~IVlx
where ~=~ ( 0 < ~ ½ ) , Theorem 5.4,
(7.32)
= c (½<~1).
Here we have used (5.11)
and the f i r s t
enter into the estimation of
=< C ' ( l + r ) -½-~ = go4 ( r) ,
relation
g(r,U(rn)).
of ( 7 . 2 3 ) .
Let us
We have from the
i n e q u a l i t y aB ~ ½(a2+B2)
Igl(r,U(rn))l (7.34)
Set g o ( r )
2 < ½1 B2Ulx+½r ~ ~ 2 = - 2 {A2u(rn)Ix
~ ½1(Z+r)BB~ul 4 : j=~ l g o j ( r ).
in
+½r-2[A½u(rn)Ix
gol(r)+½r-21A½u(r
)I 2 n x Then go is i n t e g r a b l e and i t
follows
84 from
(7.28)
that
1 IR 4--~ I~ 2J R dr < 2 - ~ rn go(r)dr+ IA'2u(rn)Ix r n --2r (7.35) {an-(C~k(R)' Fn(k,f))xl = 1
IR 1 i r g o ( r ) d r + 4 j - - ~ , r n l B ~ (rn )u (rn )I x2 n
~
Let R÷~ in as n ~ .
(7.35)
Then,
(7.36)
along
a sequence such t h a t
using C o r o l l a r y
fan - ( F ( k , f ) , F n ( k , f ) ) x l
By the second r e l a t i o n s i d e of
(7.36)
tends
of
7.3,
(rn>Z) = "
Iv ' ( R n ) - i k v ( R n ) I x
we a r r i v e
÷0
at
- - I ~ r go ( r ) d r + ~ ] ~ T r n I B ~ ( r n ) U ( r n ~ 121kl~ n (7.23)
and Lemma 7.2 the r i g h t - h a n d
to zero and ( F ( k , f ) , F n ( k , f ) )
x coverages
to
I F ( k , f ) [ x2 as n÷~, which completes the proof. Q.E.D. PROOF of THEOREM 5.3. {L,k,~}
with
ke IR-{O} and ~ c F ~ ( I , X ) .
v is rewritten for
{L,ko,~}
for
{L,k,~[f])
easily
In f a c t ,
along (7.39)
as V=Vo+W, where v 0 i s with
koCC+,
with
According
function
to Lemma 2.8
the r a d i a t i v e
function
Imko>O and w i s the r a d i a t i v e
f = ( k 2 - k ~~) v o .
for
Since V o ~ H)~ ' ,B ( I~, X
function it
can be
seen t h a t
(7 • 37) s - lri m ~
(7.38)
Let v be the r a d i a t i v e
v 0 (r)
letting
= 0
R+~ in the r e l a t i o n
I V O ( r ) I x2 = [ V o ( R ) I x2 - 2 I R r R e ( v o, ( t ) , v o ( t ) ) d t a sequence {R n} which s a t i s f i e s ] v o ( r ) l x2 =< 2 [,~r [ V o '( t )
whence ( 7 . 3 7 )
follows.
IVo(Rn)IX÷O
x ] V o ( t ) I x dr < llvol12B , ( r , ~ > ) ,
As f o r w we can a p p l y
and 7.4 to show i w-~m . e-i~(r-,k)w (7"4°)~lw(r)I
(n .... ), we have
(r
x = IF(k,f
= F(k,f) Ix ,
in X,
Lemmas 7 . 1 ,
7.2
i
85 which implies
the strong convergence of e - i ~ ( r ' ' k ) w ( r )
Therefore the existence of the strong l i m i t F(k,~) (7.41)
= s-limr÷~ e - i ~ ( r ' ' k ) ( v = F(k,f)
O(r)+w(r)) ( f = (k 2 -ko)v 2 o)
has been proved completely. Q.E.D.
86
;8.
Some p r o p e r t i e s of
lira e - l P v ( r ) r - ><>~
In t h i s section we s h a l l i n v e s t i g a t e some p r o p e r t i e s of s - lim e - i l J ( r ' k ) v ( r ) ,
F(k,&) =
whose existence has been proved f o r the r a d i a t i v e
r-~
function
v
for
{L, k, &}
with
k•
r e s u l t s obtained in t h i s section w i l l
A - {0} ,
Sc FI+~(I ,X).
The
be useful when we develop a
spectral representation theory in Chapter I I I . LEMMA 8.1.
Let Assumption 5.1 be s a t i s f i e d and l e t
F(k,~,)
be as in
Theorem 5.3, i . e . ,
(8.1)
F(k,~) = s - lim e _ i : ~) r , [ k ,I v ( r r+~
where
v
is the r a d i a t i v e function f o r
where
6
is given by ( 5 . 9 ) .
{L, k, ~
with
in X,
k~
Then there e x i s t s a constant
FI+3(I,X),
C = C(k)
such
that
(8.2) C(k)
JF(k,Z)[ x _-< clll#lU~ i s bounded when PROOF.
k
moves in a compact set in
]R - {0}
As in the proof of Theorem 5.3 given at the end of ~7,
i s decomposed as
v = Vo+W-
v
Then, as can be e a s i l y seen from the proof
of Theorem 5.3,
(8.3) Set
JF(k,~)lx = limJw(r)Jx
go =
(k 2
2 - ko) Vo'
follows from (3.32) with
k0
being as in the proof of Theorem 5.3.
v=w '
f=go'
kl=k'
k2=O t h a t
It
87 i w ( r ) [ 2x <=k21 { l w ' ( r ) - i k w ( r ) I x2 - 2k Im (gO,W)o,(O,r) }
~1
(8.4)
lw'(r)
= ~l ] w ' ( r )
<
with
C = C(k)
have been used . iv'(rn)
and Let
- ikv(rn)Ix
-
ikw(r) I x2 + ~_~_PCII goll62
- i k w ( r ) I x2 + T~T c'
C' = C ' ( k ) .
Here (2.7) in Theorem 2.3 and Lemma 2.8
r~,> along a sequence + 0
as
lll~Ill~
n- ~
{ r n}
r n +~
such that
and
Then we obtain (8.2). Q.E.D.
Since the r a d i a t i v e f u n c t i o n respect to
~ ,
v(-,k,&)
for
{L,k,~}
is l i n e a r with
which follows from the uniqueness of the r a d i a t i v e
f u n c t i o n , a l i n e a r operator
F(k)
from
FI+B(I,X)
into
X is w e l l - d e f i n e d
by 1
(8.5)
F(k)L : F(k,~)
(LE F I + B ( I , X ) ,
In the f o l l o w i n g lemma the denseness of
FI+B(I,X)
in
LEMMA 8.2.
Let
0 <= ¢ < ~ .
FG(I,X) Then
into FB(I,X)
: O(~<e~l))
F6(I,X)
shown, which, together with Lemma 8.1, enables us to extend to a bounded l i n e a r operator from
l
B:~-e(O
will
g(k)
be
uniquely
X. is dense in
F¢(I,X)
PROOF. As has been shown before the proof of Lemma 2.7 in §4, c F (I,X) hl ,B , O , _ ~ i l , x) (8.6)
can be regarded as a bounded a n t i - l i n e a r which is defined as the completion of II~II~,_¢ = Z l O + r ) - 2 ~ { l ~ ' ( r ) I ~
By the Riesz theorem there e x i s t s
f u n c t i o n a l on
C~(I,X)
+ IB~(r)~(r)12x
by the norm
+ Im(r)I~]dr
w = w~ E UO,-¢k'l'B 'I,X)
such that
88
(8.7)
<~,v)
where
( ' )B,-~
l]l~J[l~
is e q u i v a l e n t to
llwllB,_~
such that
(8.8)
<~'n' v> Wn(r)
=
as
'~J(t)
(t~O),
=0
Obviously
.
The norm smooth
, and set
(n=l,2,--) R'n
e
and we have
F#( I , X )
which implies that
II1~" - ~nlll~ + 0
Thus the denseness of
,
be a r e a l - v a l u e d , (t]l)
= (wn, V)B,_ ¢
n-~o ,
(8.9)
HI'BO,_~(I,X)
Let
~(t)=l
~(r-n)w(r)
llW-WnllB,_~ ÷ 0
(v e HI'B O, ~ ( I , X ) )
is the inner product of
f u n c t i o n on R
with
= (w,v)B,_ ¢
F~(I,X)
in
(n~)
F (l,X)
(0~
#)
has been proved. ~.E.D.
From Lemmas 8.1 and 8.2 we see that the operator
F(k), k • ~ - {0} ,
can be extended uniquely to a bounded l i n e a r operator from X.
F6(I,X)
into
Thus we give DEFINITION 8.3.
extension of
We denote again by
F(k) . When ~=~lf]
(8.10)
F(k)~Ifl
Now we shall show t h a t radiative function c F~(I,X) HI,B , . O,_6~i,X)
'
(8.11)
can be represented by for
{L,k,~,}
.
On the other hand any r a d i a t i v e
(k e ~ ,~, • F6(I X))
belongs to
HI,B ( l , X ) 0,-~
"
= lim<2,,Vn
>
n-><x~
and the
functional
Therefore
,
[
As we have seen above,
function
and we have
<~,v>
we shall simply w r i t e
= F(k)f
F(k)~
v = v(-,k,~)
f c L2,5(I,X),
can be regarded as a bounded a n t i - l i n e a r
+
well-defined,
with
F(k) the above bounded l i n e a r
v
for <~,v>
on {L,k,~} is
8g
where
vn • H 'B(I,X) THEOREM 8.4.
the r a d i a t i v e (j=l,2)
satisfies
v n -~ v
.HO,_6L l,B 'l,X).
in
Let Assumption 5.1 be s a t i s f i e d .
function
for
{L,k,~,j}
with
vj(j=l,2)
Let
k ~ ~ - {0} ,
~j
be
• F . ( I X) o '
Then
(8.12)
(F(k)~l'
where the r i g h t - h a n d
_
F(k)~2)x
1
2ik {<~2' Vl > - }
side is w e l l - d e f i n e d
means the complex conjugate. (8.13)
'
as stated above and the bar
In p a r t i c u l a r
jF(k)~12 : _I__ Im < L , v > x k
f o r the r a d i a t i v e • F6(I,X)
function
When
v
for
Lj = ~,[fjl
{L,k,~}
with
with
k •~
- {0} and
fj • L2,~(I,X),j=I,2,
(8.12)
takes the form (8.14)
(F(k)fl'
_ 2ik 1 ~(Vl " 'f2)o _ (fl,V2)o }
F(k)f2)x
F u r t h e r we have 1 Ira(v, f) 0 I T ( k ) f l x 2 = "~
(8.15) f o r the r a d i a t i v e
function
v
for
{L,k,~, I f ] }
with
k e R - {0}
and
f • L 2 6 ( I , x) PROOF. Let us f i r s t where
6
consider a special
is given by ( 5 . 9 ) .
case t h a t
S t a r t i n g w i t h the r e l a t i o n
~j.c F I + 6 ( I , X ) , (v I ,
(L-k2)@)0 =
co
< ~ I ' ~> (~ • C 0 ( I ' X ) ) ' (8.16)
i {(v~,~')x I
we obtain :.... + (B'~vI'B:'~)
x
j=l,2,
+ ( (C_k2)Vl ,q~)x}dr =
90
Let
p(r)
O_
be a r e a l - v a l u e d
p(r)=l
(8.17)
(r
,
C1
function
on
I
such t h a t
=O(r__>2) and l e t us set
Pn(r ) = p(.r.)
(n=l,2 .... )
Then i t can be e a s i l y seen t h a t
{,
(8.18)
On(r) = 0 iPn(r)l< c
Substitute
(8.19/
f
= pn v2
if
r
i
in
(8.16).
or
r>2n
1
(r • l , ~ c : m a x l p ' ( r ) [ rel
, ).
Then we have
+ pn{(V~,V~) x + (B~-v I ,B1~V2)x + ((C-k2)Vl
I
V2)x}]dr
= < L l , PnV2 > Quite s i m i l a r l y ,
by s t a r t i n g
with
((L-k2)6'
v2) = < ~ 2 ' ' > '
it
follows
that (8.20)
r
-
Jl
~
V
Pn ( l , V 2' ) x +Pn{(Vl,V2) ' ' x + ( B. Vl ,B'~V2)x + (( C-k2)v I v 2)x~i dr = < ~2,PnVl >
(8.19) and (8.20) are combined to give 2n (8.21
S
n
p 1 { ( v l , v ~ ) x - ( v ~ , V ~ ) x } d r = <~2,PnVl > - <~l,PnV2 >. " '
Since (vl(r),v~(r)) (8.22
= (vl(r),
x - (v~(r),
v2(r)) x
v~(r) - ikv2(r)) x - (vi(r) 2ik(vl(r),v2(r))
x
- ikvl(r),
v2(r)) x
91 and (8.23) with
(vl(r),v2(r)) h(r)÷O
as
x = (e-l~Jvl(r),
r~
e-ll'v2(r))x
by Theorem 5.3, 2n
(8.24)
= (l--(k)~, 1, F(k)~,2) x + h(r)
we obtain,
noting that (8.18) and
]2n
In P'n(r)dr
=
[Pn(r ) ,n
= -I
,
2n
(8.25)
I
Pn{(VlV2)x - ( V l ' V 2 ) x } d r
- 2ik(F(k)Ll'F(k)C2)x
n
< = c
2n{ lh(r)l
+ (r-6lVl[x)(r~-llv ~ - ikv21x) + (r~-llv ] - ikvll x
n
X(r-61V21x) } dr max =n
!h(r)l
+ llvlli_~,(n,~)llv ~ - ikv211~_l,(n,~ )
=
+ IIv] - i k v l l l s _ l , ( n , ~ ) l l v 2 ! l _ 6 , ( n , ~ ) } (n-x~)
÷0
On the other hand,
PnVj
tends to
vj
in
HI'B~ (I'X)O,-,
as
n÷~o, and
hence (8.26)
- <~l'PnV2 > ÷ <%2'vI > -
as
(8.12) follows
n ~o.
general case that {C2n }
~jEF~(I,X)
.
Let us next consider the
Then there e x i s t sequences
such that Ljn
(8.27)
from (8.25) and (8.26).
c
FI~(I'X)
~jn ÷ ~j
in
(n=l,2 . . . . F S(I,X)
(j=l,2)
j=l,2)
,
~rZln~
and
92
Let
Vjn ( j = l , 2 ,
be the r a d i a t i v e
n=l,2 . . . . )
function for
{L,k,Lnj}.
Then we have from Theorem 2.3 (8•28)
as
n~ °
letting
in
v. -~v. jn j for j=l,2.
.UO,_6~i I,B ",X)
(8.12) in the general case is e a s i l y obtained by
n-~o in the r e l a t i o n
(8.29)
(F(k)Lin,
F(k)&2n) x = <~,2n,Vln > - ,
which has been proved already.
(8.13) 4"(8.15) d i r e c t l y
follows from
(8.12).
~.E.D. Now i t w i l l D,
be shown that the range
and hence i t
is dense in
defined by (5.36),
i e. •
X .
O<E(r)
~
=
,
=
Let
{F(k)f/f
for
x•D
definition w0
fo(r)
and of
k•
R - {0}.
p(y,k)
is the r a d i a t i v e
and
~(r)=O
(8.31) PROOF. Let (O,n), n=l,2 . . . . . for
{L,k,~IXnfo]}
--
5
= l(r<2) =
•
Set
,
Then, as can be e a s i l y seen from the
function for Let
(r
I
= (L-k2)wo
and (5.20) in Lemma 5.5,
PROPOSITION 8.5.
contains
~(r) be a smooth f u n c t i o n on
WO(r ) = ~ ( r ) e i ~ 1 ( r ' , k ) x
(8.3o)
• L2,~(I,X)}
xeD
fo e L 2 , 6 ( I ' X )
and
fL,k,~ffol}. and l e t
fo
be as above•
Then
F(k)fo=X . Xn(r )
be the c h a r a c t e r i s t i c
and l e t and
wn
and
~n
{L,k,~[(l-Xn)fO]}
f u n c t i o n of the i n t e r v a l
be the r a d i a t i v e , respectively.
functions Note t h a t
93 ×nfO~ L2,1+S(I,X),
w0 = wn + q n
the r e l a t i o n
Xnf 0 is compact.
X .
Since
(8.33)
Then, by
and Theorem 5 3, we have
F(k)(Xnf) = s-lim { e - i p ( r " k ) w o ( r )
8.32) n
because the support of
e-i~wo(r) = x
for
- e-i#(r"k)qn(r)}
r~2, there exists the l i m i t
x n = s - lim e - i ~ ] ( r " k ) q n ( r )
in X ,
r-~oo
and hence the r e l a t i o n (8.34)
F(k)(×nf) : x-x n
is v a l i d for each
n=l,2 . . . .
On the other hand, proceeding as in the
derivation of (8.4), we obtain from (3.32), r,lr ~ Iqn(r)Vx2 <= ~1 r''n~ ' - i k q n ( r ) i x~2 + i -C~ ll(l-Xn)foll ~
(8.35)
(n=l,2 . . . . ) with
C=C(k),
whence follows by l e t t i n g
{ r m}
in (8.32) that
~
[Xn]~ ~ T C~ iJ(l_Xn)fOil2a
(8.36)
along a suitable sequence
( n - l , 2 , "'" )
Thus we a r r i v e at <
(8.37)
!F(k)(XnfO) - x tx = Cll(l-Xn)follcs (C = C(k),
When
n
tends to i n f i n i t y ,
Xnfo+f 0 , i.e.,
n=l,2 . . . . )
( l - X n ) f 0 *0
in
L2,,3(I,X),
and (8.31) is obtained from (8.37).
Q.E.D.
94 In the remainder of t h i s section we shall of
F(k)
to
L2,6(I,X)
,
consider the r e s t r i c t i o n
which is denoted by
F(k)
again.
Let us
consider the mapping (8.38)
~+]k
By the use of Lemma 4.3
~ F(k)(F(L2,6(I,X),X)
F(k)
continuous f u n c t i o n on ~ - { 0 } compact operator from THEOREM 8.6. (I) on
R-{O}
(II)
Then
can be seen a E ( L 2 , ~ ( I , X ) , X ) .
Further we can see that
L2,6(I,X )
into
F(k) is a
X
Let Assumption 5.1 be s a t i s f i e d .
F(k)
is a B ( L 2 , 5 ( I , X ) , X )
-valued continuous f u n c t i o n
.
For each
L2,5(I,X)
into
X
k e R-{O} F(k)
(i)
Then there e x i s t a p o s i t i v e number {kn} c ~ - { O }
is a compact operator from
p
PROOF. Let us assume that
is not true at a point c > 0
and sequences
(8.39)
{fn} c L2,6(I,X),
kn ÷ k
(n=l,2 . . . . ) (n-~O (n=l,2 . . . . )
I g ( k n ) f n - ~:(k)fnl x ~ c
With no loss of g e n e r a l i t y L2,~(I,X)
k e ~-{0].
such that llfnll 6 = 1
to some
{fn }
f ~L2,~(I,X)
may be assumed to converge weakly in We shall show that there e x i s t s
subsequence
{n m}
{F(knm)fnm}
converge s t r o n g l y to the same l i m i t
which w i l l
valued
of p o s i t i v e integers such t h a t
contradicts
{F(k)f n } m F(k)f
the t h i r d r e l a t i o n of (8.39).
as
We shall
and ~
, consider
95
the sequence
{F(kn)f n}
only, because the sequence
be treated in quite a s i m i l a r way. k=kn '
f=fn '
V=Vn
where
that
vn is the radiative function for
a subsequence {v n }
of
{v n}
(n=l,2" "" )
{L,kn,~Ifnl} .
can be chosen to satisfy
By Lemma 4.3 {v
m
converges in
can
I t follows from (8.13) with
1 im(Vn, fn)O IF(kn)fnlx2 = knn
(8.40)
{F(k)f n}
} nm
L2,_5(I,X)
to the radiative function
v
for
{L,k,L[fl},
whence we obtain
(8.41)
limlF(kn ) fnm 12 x : II~
Let
xeD
IF(k)ffx2
Im(v ' f) :
m
and set Wnm(r) : E~(r)eiU(r"knm)X'
(8.42) gnm(r ) = ( L - k ~ ) w n (r) m
where
~(r)
is defined by (5.36).
,
m
Then i t follows from Proposition 8.5
that (8.43)
l:(kn )gn = x m m
By taking note of Remark 5.9, (2) i t can be easily seen that w -* w0 nm
in
L2 _.3(I,X)._
gnm-" fo
in
L2,6(I,X)
(8.44)
96 as
n~x~ with
w0
and
fo
defined by (8.30).
Therefore we have,
using (8.13) in Theorem 8.4, lim (F(knm)fnm,X)x = lim (F(k n )fn ' (knm)gnm)X ~ m+~, m m = lim
~211knm . - - - - {(Vnm gnm)0
m-><x..~
(fnm'Wnm)O >
(8.45) - 2ikl { ( V , f o )0 - (f'Wo)o} = (F(k)f,
F(k)fo) x = ( F ( k ) f ,
x) x ,
which, together with (8.38) and the denseness of that
{F(kn )fn } m
shown ( i ) . { F ( k ) f n} in
converges to
F(k)f
D in
s t r o n g l y in
X .
X
, implies
Thus we have
m
In order to prove ( i i ) is r e l a t i v e l y
L2,6(I,X)
.
i t is s u f f i c i e n t
compact when
This can
{fn }
to show t h a t
is a bounded sequence
be shown in q u i t e a s i m i l a r way used in
the
proof of ( i ) . Q.E.D. THEOREM 8.7. let
k~R-{O}.
Let Assumption 5.1 be s a t i s f i e d .
F ( k ) f = s - lim e - i ; J ( r n " k ) v ( r n
v
feL2,~(l,X
) and
Then
(8.46)
where
Let
is the r a d i a t i v e f u n c t i o n for
sequence such t h a t
rn+~
and
{L,k,~[f]}
v ' ( r n) - i k v ( r n ) ÷ O
)
in
and in
X,
{r }
is an
n
X as
n-~°.
Before showing t h i s theorem we need the f o l l o w i n g the Green formula.
97 LEMMA 8.8.
Let
(8.47) with
v j e L 2 ( l , X ) l o c satisfy the equation
(vj (L - k-2) ~)0 = ( f j ' # ) 0 f j ~ L 2 ( l , X ) l o c and kcR-{0}.
(# E C~(I,X), j=l,2)
Then we have
r 0 { ( v l ' f 2 ) x - (fl'V2)x } dr
(8.48) = (v~(r) - i k v l ( r ) ,
v2(r)) x
(vl(r),v~(r)
- ikv2(r)) x
+ 2ik ( v l ( r ) , v2(r)) x for
re I.
PROOF. The idea of the proof resembles the one of the proof of Proposition 3.4.
As has be shown in the proof of Proposition 1.3,
vj=U-IvjeH2(RN)Ioc
(j=l,2).
Therefore we can proceed as in the
proof of Proposition 3.4 to show that there exist sequences {#2n }
such that I
(8.49)
¢ In, #2n eCo(l,X ) and ~.jn ÷ vj @. ( r ) + jn vj(r) ~jn(r)~vi(r)
in
L2(I,X)Io c
in
X (roT)
in
X (rEl)
(L-k2)~jn, = fnj ÷ f j as n-~
(j=l,2
in
L2(l,X)loc
Integrate the relations ((L-k2)#in,#2n)x = (fln,#2n)x
(8.5o)
(#in,(L-k2)#2n)x = (~in,f2n)x
~ i n } and
g8 from
0
to
r
and make use of p a r t i a l
integration.
Then i t f o l l o w s
that
(#~n(r),
¢2n(r))x + (#in(r),#'2n(r))
x = (fln,@2n)O,(O,r)
(8.51) (@In'f2n)O,(O,r) where we should note t h a t
(#~n(O),@2n(O)) x = (#in(O),@~n(O))
(8.48) is e a s i l y obtained by l e t t i n g
= 0
n-~o in (8.51). Q.E.D.
PROOF of THEOREM 8.7.
Set in (3.32)
k]=k,
k2=O
and
r = rn
Then we have (8.52)
which implies t h a t (8.48)
1
JV(rn)l~
~ k2 I v ' ( r n ) {JV(rn)Ix}
vI = v2 = v ,
- i V ( r n )12X - ~ I m ( f ' V ) o , ( O , r n )
is a bounded sequence.
f l = f2 = f
and
r = rn
Next set in
Then i t
follows
that
le-i~L(rn" , k ) v ( r n ) Ix = ~1 I m ( v ' f ) o , ( O , r n
(8.53) + ~1 Im(v( r n ) , v' (r n) - i k v ( r n ) ) x The second term of the r i g h t - h a n d n-~
because
J v ' ( r n) - i k v ( r n ) I x + O
side of (8.53) tends to zero as as
n~>
and
[V(rn)Jx
bounded, and hence we have (8.54)
lira le-i!4(rn"k)v(rn)12x n-><)o
= ~ Im(v,f)o
= JF(k)f[2x ,
is
99 where (8.15) has been used. (8.30).
Let
Then, setting in (8.48)
xeD
and let
Vl=V,
w0 and
v2=Wo, f l = f ,
fo
f2=fo
as in and
r=r n,
we obtain (e-ip(rn',k)v(rn),
(8.55)
-
e-i~(rn',k)wo(rn))x
2ikl {(V(rn),W~(rn)
_ ikwo(rn))x _ ( v , ( r n ) _ i k v ( r n ) ,
+ (V,fo)o,(O,rn) Since
e-i~(rn"k)wo(rn)
= -iZ(rn.)Wo(rn)÷O
(8.56)
in
= x
(r n ~ 2)
X (rn÷~) ,
(f'Wo)o,(O,rn)}
and w~(r n) - ikwo(r n) i t follows from (8.55) that
l i m ( e - l P ( r n " k ) v ( r n ), x) x ~ {(V,fo) 0 n-.oo = (F(k)f, F(k)fo) x = (F(k)f,X)x ,
where (8.14) and Proposition 8.5 have been used. and the denseness of
Wo(rn))x
(f,Wo) O}
From (8.54),
(8.56)
D (8.46) is seen to be valid. Q.E.D.
CHAPTER I I I SPECTRAL REPRESENTATION §9.
The Green kernel
In t h i s chapter the r e s u l t s obtained in the previous chapters w i l l be combined to develop a spectral representation theory f o r the operator d2 L = - d 7 + B(r) + C(r)
(9.1) given by Q(y)
(0.20)
and
(0.21) .
Throughout t h i s section the p o t e n t i a l
i s assumed to s a t i s f y Assumption 2.1. Now we shall define the Green kernel
( r , s • Y = [0,~) , k • {+) s •
T,
x • x and l e t
G(r,s,k) Let
and i n v e s t i g a t e i t s p r o p e r t i e s .
Z[s,x]
be an a n t i - l i n e a r
function on
HI~B(I,X)
defined by (9.2)
(9 e HI~B(I,X))
= (x,¢(s)) x
Then i t follows from the estimate
(9.3)
I (S)lx ~
(¢ e HI~B(I,X))
~~11~11 g
which is shown in Proposition I . I ,
that
# I s , x ] e F~(I,X)
,
f o r any
f~ > 0
and the estimate
(9.4) is valid. {L,k,£[s,x]}
III ~[s,x] III~ ~ ~- (I + s)~alxlx Denote by .
v = v(-,k,s,x)
the r a d i a t i v e function f o r
Then, by the use of (1.8) in Proposition 1.3 and (2.7) in
Theorem 2.3, we can e a s i l y show that
!v(r)i x < C(l + (9.5)
s)':ixlx
(C = C(R,k), r c: [O,R], s c T , x E X)
101
f o r any
R~ I .
T h e r e f o r e a bounded o p e r a t o r
G(r,s,k)
on
X
is w e l l
d e f i n e d by (9.6)
G(r,s,k)x
DEFINITION 9.1. G(r,s,k)(r,s
linearity
( t h e Green k e r n e ] ) .
e T , k ~ {+)
The l i n e a r i t y of
= v(r,k,s,x)
will
Z[s,x]
The bounded o p e r a t o r
be c a l l e d
o f the o p e r a t o r
.
the Green kernel
G(r,s,k)
w i t h r e s p e c t to
directly
x .
for
follows
L . from the
Roughly speaking,
G(r,s,k)
satisfies (L - k 2) G ( r , s , k )
(9.7) the r i g h t - h a n d
side denoting
the Green kernel
G(r,s,k)
PROPOSITION 9.2. (i) on
Then {+
T x
G-function.
will
The f o l l o w i n g
be made use o f f u r t h e r
properties
of
on.
Let Assumption 2.1 be s a t i s f i e d .
G(.,s~k)x
× X .
= 6 ( r - s) ,
is an
Further,
L2 _ 6 ( l , X ) - v a l u e d
G(r,s,k)x
continuous function
is an X - v a l u e d f u n c t i o n
on
I x I x (~+ × X , too. (ii)
G(0,r,k)
(iii)
Let ( s , k , x )
interval have
= G(r,0,k)
e T x 6+ x X
such t h a t the c l o s u r e o f
G(.,s,k)x
exists
R> 0
C = C(R,K)
(9.8)
where
Let
J
(r,k)
e I x 6+
be an a r b i t r a r y
c o n t a i n e d in of
I - {s} D(J)
.
open Then we
is given a f t e r
1.3.
and l e t
K
be a compact set o f
6+ .
such t h a t
ilG(r,s,k)ll
II II
J
and l e t
~ D(J) , where the d e f i n i t i o n
the p r o o f o f P r o p o s i t i o n (iv)
f o r any p a i r
= 0
< C
means the o p e r a t o r norm.
(0 < r ,
s < R, k e K) ,
Then t h e r e
102
(v)
We have f o r any t r i p l e
(9.9)
G(r,s,k)*
G(r,s,k)*
Let us f i r s t
l<~[s,x]
G(r,s,k)
.
s ~ T
of
and
2[s,x]
in
x c X .
In f a c t we o b t a i n
I.I.
- l[s',x'],
l(x - x',
,
show the c o n t i n u i t y
(!+r)B¢>l o
<=
of
, B => 0 , w i t h r e s p e c t to
from P r o p o s i t i o n
c I x I × ~+
= G(s,r,-k)
d e n o t i n g the a d j o i n t
PROOF. F~(I,X)
(r,s,k)
(]+s)%(S))xl
=
l(x, (]+s)~t(S)x
(x', (l+s)~(#(s)
+
l+s')~(s'))
(x',
- ¢(s'))) x
(9.]0)
+ J(x', ((l+s) B - (l+s
)~)~(s'))xl
< {~/Tix- ×'ix(l+s)~+ l×'IxU+S)~Is-s'~2+#TlX'Ixl(l÷s)~-(]+s')~l}il4i~ whence f o l l o w s V
for
{L,k,%}
the c o n t i n u i t y .
R C $+
Since
c Hl~B(l,X)loc
(s,k,x)
and
C c F(~ ( I , X )
hand,
= (x,~(O)) X = 0
is the r a d i a t i v e
G(r,O,k)x m 0
function
and i n follows
G(O,s,k)
for
function
such t h a t
,$,(r)
on
I
small neighborhood o f
v(r)
= ~J(r)G(r,s,k)x
g(r)
= -,~"(r)G(r,s,k)x
>(r)
On the o t h e r , and hence
Therefore function.
Thus ( i i )
Take a r e a l - v a l u e d , = 1
on
r = s .
is the r a d i a t i v e
for all
= 0 .
.
by the uniqueness o f the r a d i a t i v e Let us show ( i i i ) .
immediately.
~ ~ HI~B(I,X) {L,k,O}
- 2~'(r)~ r G(r,s,k)x
J
and
Then i t
function .
function
HI~B(I,X)Ioc
G(O,s,k)x = 0
for all
is c o m p l e t e l y proved.
sufficiently
, (i)
, we have
T x ~+ × X , which means t h a t
G(.,O,k)x
t h a t the r a d i a t i v e
is c o n t i n u o u s both in L2 _ ~ ( I , X )
w i t h r e s p e c t to G(.,s,k)x
By r e c a l l i n g
for
5(r)
smooth = 0 in a
is easy to see t h a t {L,k,l[g]}
Proposition
with
1.3 can be
A
made use o f to show t h a t
v c D(1) , which means t h a t
G(',s,k)x
c D(J)
103 (iv)
is obvious from ( 9 . 5 ) .
,~(t) = G ( t , s , k ) x I
=
~ W and 'n
and
Let us e n t e r i n t o the proof o f ( v ) .
w(t) = G(t,r,-k)x
with
x,x' ~ X .
Set
Then, s e t t i n g
g V in the r e l a t i o n s n
I
"
i
(v',~;')
o + (B'~v,
B~) o + (v,(C
(®',w')
0 + ( B'~,,B~) ~ . . . 0. + ((C
- F)~) o = (x,~(s))
x ,
(9.ll)
i
respectively, (9.12)
k2)'~ , w) 0 = ( ~ ( r ) ,
x') x
and combining them, we a r r i v e a t
(x,w(s)) X- (v(r),X')x
= ~0 g ~ ( t ) { ( v ' ( t ) , w ( t ) ) x - ( v ( t ) , w ' ( t ) ) x } d t (n : 1,2 . . . . ) ,
where R
~n(t)
= ~(n(t-to))
such t h a t
satisfy fact that
~(t)
is a r e a l - v a l u e d ,
: 1 (t ~ 0), = 0 (t > I)
t O > max ( r , s )
n ¢ n(t)
, ~(t)
.
÷ - d ( t - t O)
(x,G(s,r,-k)x')x
n ....
Let as
, and
smooth f u n c t i o n
on
t O is taken to
in (9.12) and take note o f the
n -~
Then we have
- (G(r,s,k)x,x')
X
(9.13) iV(to),
w ' ( t O) + i k w ( t o ) ) x - ( v ' ( t o )
- ikv(to),
W(to)) X
(t O > max(r-s)) By making
t O ÷ => along a s u i t a b l e sequence
{ t n} , the r i g h t - h a n d
. side
o f (9.13) tends to zero, whence f o l l o w s t h a t (9.14) f o r any
(x,{G(s,r,-k) x,x' e X .
- G(r,s,k)*}x')
X = 0
Thus we have [)roved (v) • Q.E.D.
104
~I0.
The eigenoperators
The purpose of t h i s section is to construct the eigenoperator
(reT, k ~R-{O})
n(r,k)
was defined in ~9.
by the use of the Green kernel
In t h i s and the f o l l o w i n g
sections
G(r,s,k) O(y)
which
will
be assumed to s a t i s f y Assumption 5.1 which enables us to apply the r e s u l t s of Chapter I I . We shall f i r s t
show some more p r o p e r t i e s of the Green kernel in
a d d i t i o n to Proposition
9.2.
PROPOSITION I 0 . I .
Let Assumption 5.1 be s a t i s f i e d .
Then the f o l l o w i n g
estimate f o r the Green kernel t
(10.1)
II G(r,s,k)li
<
(k.'- ~+, Irlk > 0 , r , s : - T )
I'
Cl(k)
j,
C2(k)min{ ( l + r ) l + '3
, (l+s)l+
6
(k,'-R - .'/01.,r,s ( - T )
holds, where and
C2(k)
and R -
~Q= r~ - a
are bounded when
{0},
respectively.
(10.2)
and PROOF.
fcL2,5(l,X
and
)
(9.4) with
B=O
mows in a compact set in
Cl(k)
{ke¢+/Imk>O}
= ~[G(r,s,k)f(s)ds
(reT)
v ( . , k , ~ I f J)
{L,k,'>',IfJ}
for
with
o
v = G( , s , k ) x
estimate of ( I 0 . I )
and the constants
Further
function
( i ) Assume that
Lemma 2.7 that first
k
v(r,k,L If])
holds f o r any r a d i a t i v e ke~ +
1 (~:<~I)
(0<E<=.), = 0
k(=~ + (s
with
T, x
X)
Imk>O .
belongs to
is obtained from ( 9 . 3 ) , Next assume that
Then i t
follows frorn
II~'B(I,X)
.
The
(2.16) in Lemma 2.7
kc}R - {0"." .
Applying (5.11)
105
in Theorem (5.4) and using (9.4), we have
!G(r,s,k)Xlx
~ C(k)(l+s)l+i~IXlx
which implies that (10.3)
llG(r,s,k)il
The second estimate of ( I 0 . I ) G(r,s,k)*
= G(s,r,-k)
C(l+s) l+i~
I
(r,s~T)
.
follows from (10.3) and the r e l a t i o n
((v) of Proposition 9.2).
with compact support in u = G(',r,-k)x I,B to H0 ( I , X )
<
and l e t
and the r a d i a t i v e
k(=~ + function
(ii)
Let
Imk>O v
for
fcL2(l,X )
Then (L,k,~,l f I}
belong
and s a t i s f y
( u ' , ¢ ' ) 0 + (B':~u,B:'¢), + ( ( C - ~ ) u , ~ ) o : ( x , ~ ( r ) )
(I0.4)
X
(¢"V')o + (B':¢'B'~v)o + ((C-k2)4!"V)o = (#'f)o for
¢CHo,B(I,X)I
.
Set
Then, using the r e l a t i o n
(i) = v
in the f i r s t
G(s,r,-k)*=
r e l a t i o n of ( I 0 . 4 )
G(r,s,k),
(x,v(r)) x = (u',v') 0 + (B:'u,B'v)o
.
we have
+ ((C - k2)u,V)o
(I0.5) = (m(-,r,-k)x,f)o
whence (10.2) f o l l o w s .
Let
can approximate
rk + i~
k
by
f
be as above and
n
(n=l,? . . . . )
where we have made use of the c o n t i n u i t y v(.,k,9~t f ] ) (I0.2)
k
has been established f o r
support in by
with respect to
{fn}
I.
In the case that
' where
= (x,/iG(r,s,k)f(s)ds)x
function
and (9.8) in Proposition 9.2. k c ~+
and
f c- L 2 , ~)( I~, X
f.'; L2(I,X)
Thus
with comoact
we can approximate
fnC L2(I,X) with cornpact support in
f
Then we
to obtain ( I 0 . 2 )
of the r a d i a t i v e
taking note of the c o n t i n u i t y of the r a d i a t i v e with respect to
k c IR - .i0}.
function
T
Then v ( - , k , L l f I)
and the estintate (10.1), we a r r i v e at ( I 0 . 2 ) . Q.E.D.
f
1 9G
Let
v
k c
R
- ~ . '0r "
is the r a d i a t i v e
,
s
[- I
function
x
for
{L,k,%l s , x l
t lll~ts,xj~l!l+r~ :_.'.~ ! 2 - ( l + s )
with
1+31
(II X
set
~I ,
and
xlx ,
v(r)
= G(r,s,k)x
~" and
where
.
'~[s,xl c FI+~,(I,X ).,
{~ is qiven by ( 5 . 9 ) .
Therefore Theorem 5.3 can be applied to show the existence of the limit
(10.6)
F(k)~[s,xl
= s - lira e - i : : ( r " k ) v ( r )
in X .
r-x~.,
It follows
from Lemma 8.1 t h a t
(10.7) with
Ir(k)~l s,xll C = C(k)
bounded l i n e a r (10.8)
,
__
and hence f o r each p a i r
operator
q(r,k)
on
X
(r,k)c:l
x (R - fO})
is w e l l - d e f i n e d
s - lim e-i"(t"k)G(t,r,k)x
= r(r,k)x
a
by (x ~: X)
r+oo
DEFINITION 10.2. (I0.8)
will
The bounded l i n e a r
be c a l l e d
n(r,k)
of this
section
naminq w i l l
(especially
be j u s t i f i e d
L. i n the
i n Theorem I 0 . 4 ) .
Let Assumption 5.1 be s a t i s f i e d .
PROPOSITION I 0 . 3 .
d e f i n e d by
the ei~enop.erQtor a s s o c i a t e d w i t h
The a p p r o p r i a t e n e s s remainder o f t h i s
onerator
Then we have
•
s - lim G(r,s,-k)elP(s"k)x
(I0.9)
= rl ( r , k ) x
S -><x~
f o r any t r i p l e ( 5 . 8 ) and PROOF. 30>0
(r,k,x)
q*(r,k)
c T x (R - {O})×X,
i s the a d j o i n t
Suppose t h a t t h e r e e x i s t and a sequence
fs n}
> 30 I V n ( r O) - rl ( r 0 , k o ) x 0 1 x = Vn(r) = G(r,s,-ko)ei~(Sn"k)x
of
n(r,k)
r > 0 ,
such t h a t
~(y,k)
i s d e f i n e d by
.
k 0 ~ R - {0}
, x0 ~ X ,
Sn#,X, and
holds f o r a l l 0 .
where
n=l,2 ....
where we set
By the use o f the i n t e r i o r
estimate
107 (Proposition
1.2) and ( I 0 . I )
{!I Vn!I B,(O,R) }
it
can be seen t h a t the sequence
is bounded f o r each
R >0 .
R e l l i c h theorem) there e x i s t s a subsequence {Wp}
i s a Cauchy sequence of
e s t i m a t e again. Hol'B(l'X)loc
"
Then
{Wp}
Therefore i t
Cauchy sequence in
X .
L 2 ( I , X ) I o c.
r1*(ro,ko)x 0
weakly in
to
~*(ro,ko)X 0
s t r o n g l y in
{Wp}
of
{Vn}
such t h a t
Make use of the i n t e r i o r
is seen to be a Caucy sequence in f o l l o w s from (1.8) t h a t
Since
to
By Theorem 1.5 (the
X
{wD(ro)}. is a
G(ro,s,-ko)ei~J(s"k)x 0 by ( 1 0 . 8 ) ,
Wp(ro)
converges
converges
X , which is a c o n t r a d i c t i o n . ~E.D.
Let us summarize these r e s u l t s THEOREM 10.4. (i)
in the f o l l o w i n g
Let Assumption 5.1 be s a t i s f i e d .
Then
(I0.I0)
r(r,k)x
= s - lim e-i~(s"k)G(s,r,k)x
inX,
S ~×~
and (I0.II)
q*(r,k)x
= s - lim G(r,s,-k)ei:J(s"k)x
inX
S-~x>
f o r any t r i p l e (10.12) where k
(r,k,x)
c T × (R - (O})×X .
l]n(r,k): " II
!I
(ii) (I0.13)
= l ! , l * ( r , k ) [ I _< C ( l + r ) ~ ,
means the o p e r a t o r norm and
moves in a compact set in
We have
C = C(k)
is bounded when
R - {0} .
The r e l a t i o n 2ik(q(s,k)x,
= ({G(r,s,k)
- G(r,s,-k)~x,x')
x
108
holds for any
(iii)
x , x ' (=X , k e R - [0]
and
p * ( . , k ) x e l1~'B(I,X)loc qD(1)
(10.14) where
and s a t i f i e s
(L - k 2 ) v ( r ) roT,
k(-R-
(i)
follows
PROOF. in (8.12)
[0} , x e X
Then, notin(I that
= 0
Proposition
, L2 = ~ [ r , x ' l
F(k)Lls,xl
the equation
.
from (10.8),
~,I = L l s , x l
r,scT.
,
11.3 and (10.7).
Vl=G(-,s,k)x
= r!(s,k)x
and
and
F(k)L[r,xl
Set
v2 = G(.,r,k)x
.
= r(r,k)x',
we obtain, f
(r;(s,k)x
, r'(r,k)X)x
(lO.15)
proof of ( i i ) .
Let us show ( i i i ) .
}
(x,G(s,r,k)x')x
G(s,r,k)* Set
}
= G(r,s,-k),
completes the
Vs(r) = G ( r , s , - k ) e i ~ 1 ( s " k ) x .
satisfies
(10.16) Let
G(-,r,k)x'>
1 ( (G(r,s,k)x,X,)x 2~-k
together with the r e l a t i o n
Vs(r)
2-]-k i < - ~ - - ~ - ( - ' - ~ ' - ~ )
-<~,[s,xl, =
which,
=
~s n} , Sn(-I
, be an a r b i t r a r y
a subsequence .
This means
L2([,X)Io c . (10.17) Proposition
{tp}
Letting
of
{s n} Iv s} s~
(r!*(.,k)x,
such that itself
(~ (- CO ( l , X ) )
sequence such that
Then, as we have seen in the proof of Proposition
L2(I,X)
oo
(v s, (L - km),~b)O = ( e i ~ J ( s " k ) x , ~ ( S ) ) x
{Vtp}
converges to
in (10.16),
s n +=~ as
. n-~°° .
10.3, there e x i s t s is a Cauchy sequence in n*(-,k)x
in
we a r r i v e at
(L - k2)#,)O = 0
1.3 can be applied to show that
(×1
(¢,~Co(I,X))
.
"!*(" , k)x c H#'B(I,X) loc ,q D(1) O.E.D.
109
The following
two theorems w i l l
eigenoperator and the operator
show some relations
F(k) .
THEOREM 10.5.
Let Assumption 5.1 be s a t i s f i e d .
be as in
Then we have
~8.
(10.18)
fCL2,6(I,X)
(lO.19)
.
Let F(k) ,
rR i q(r,k)f(r)dr
F ( k ) f = s-lim
k e ]R - {0} ,
in X
J0
R- ~ '
for any
between the
In p a r t i c u l a r
F (k)f = I ,~(r,k)f(r)dr JI
holds for
fcL2,~(I,X )
with
l I~ > -~-+ ~,
where the integral
is absolutely
convergent. PROOF.
Let
f c L2,~(I,X )
and define
~f(r) fR(r) = ~0
(10.20)
fR(r)
by
(0 ~ r ~ R), (r > R).
Then i t follows from Theorem 5.3 and (10.2) that F(k)f R = s - lim e - i p ( r " k ) v ( r , k , ~ [ f R l ) = s - lim e - i ! J ( r " k ) f l
G(r,s,k)fR(S)ds
r-~oo
(10.21) =
JrR s - lira { e - i ' ] ( r " ' k ) G ( r ' s ' k ) f ( s ) }
: I0
r1(s,k)f(s)ds
ds
,
where we should note that we obtain from (10.3) (10.22)
=< C(k) ( l + s ) l + ~ I,f ( S ) I x
IG(r,s,k)f(S)Ix
and hence the dominated convergence theorem can be applied. converges to
f
in
L2,,S(I.X)
as
R~
and
F(k)
, Since
fR
is a bounded l i n e a r
110
o p e r a t o r from n-~
L2,~(I,X )
in ( I 0 . 2 1 )
from (lO.12) obtain
.
that
If
into
X ,
(lO.18)
f c L2,~(I,X )
!.l(s,k)f(s)l
(lO.19) from ( l O . 1 8 )
x
with
i s obtained by l e t t i n g 1 ~ > ~ + 5 , then i t
i s i n t e g r a b l e over
I .
follows
Therefore we
. ~.E.D.
THEOREM 10.6. (i)
Let
Let Assumption 5.1 be s a t i s f i e d . ke~
- {0} .
Then
r : * ( . , k ) x c L2 _G(I,X )
f o r any
x • X w i t h the e s t i m a t e
llrl*(',k)xil_,4
(10.23) where
C = C(k) (ii)
< ClXlx
is bounded when
k
( x c X)
,
moves in a compact set in
R - ,[Ol
We have
(10.24)
(~:*(.,k)x,f)o
f o r any t r i p l e
(k,x,f)
PROOF. ( i )
g c L2(I,X )
Let
= (x,F(k)f)x
m (~ - ( 0 } ) x
X x L2(I,X)
.
w i t h compact support in
T .
Then we o b t a i n
from Theorem I 0 . 5 (10.25)
(F(k)g,x) x =
~lq(r,k)g(r)dr,
= { (g(r), JI
Set in ( I 0 . 2 5 )
g(r)
f u n c t i o n o f (O,R) . L2,6(I,X)
into
X ,
T~*(Y, k ) X ) x d r
= ×R(r)(l+r)-26q*(r,k)x, Then, noting t h a t we a r r i v e a t
X) x
F(k)
XR(r)
being the c h a r a c t e r i s t i c
is a bounded o p e r a t o r from
111
IR -25 j (l+r) l~.l*(r,k)x[#dr = ( F ( k ) { ) < R ( l + r ) - 2 ~ n * ( - , k ) x } , X ) x 0 (10.26)
=< C(k)
XR(l+r)-2~n*(',k)x",~,,
IXlx
= C(k) llr,*(-,k)x"_,,S,(O,R)lX[x ,
which implies that (I0.27)
Since (ii)
llq*(.,k)xll_5,(O,R)
R >0 Let
R-~= .
is a r b i t r a r y ,
fcL2,6(l,X ) .
Then, since ( f ( r ) ,
~ C(k)Ixl
(10.23) d i r e c t l y Set in (10.25) ~*(r,k)x) x
x
follows from (10.26). and l e t
g(r) = × R ( r ) f ( r )
is integrable over
I
by
( I 0 . 2 3 ) , we obtain (10.24), which completes the proof. Q.E.D. In order to show the c o n t i n u i t y of Ti(r,k) and respect to
(10.28)
Let
x e D .
rff(r,k)x -
Then we have
1 2ik ~ i ( r ) e i l J ( r " k ) x (r c I ,
~(r)
- h(r,k,x) k c R - {0}),
is a real-valued smooth function on
(r < I) , = 1 (r ~ 2) ,
;1(y,k)
radiative
{L, -k, hi f l }
(10.29)
with
k we shall show
PROPOSITION 10.7.
where
ri*(r, k)
function for f(r)
= ~ (1L
I
is as in (5.8) and with
_k 2 )(~e i ~ ( r " k ) x )
such that ~(r) = 0 h(.,k,x)
is the
112
PROOF. Let Then
WO(r) = ~ ( r ) e i ~ ( r " k ) x
fo e L 2 , ~ ( I ' X )
{L,k,~If01}
.
Let
and
w0
and
fo=(L-k2)wo
is the r a d i a t i v e f u n c t i o n for
g c C~(I,X) .
Then, s e t t i n g in (8.11)
and making use of the f a c t that
T(k)fo=X
(10.30)
2ik {(WO'g)o
fl=fO , f2=g
by Proposition 8.5, we have
l
(x,F(k)g) x =
with the r a d i a t i v e f u n c t i o n for By ( i i )
as in (8.30).
(fo'V)o }
{L,k,~Igl } .
of Theorem 10.6 the l e f t - h a n d side of (10.30) can be r e w r i t t e n
as
(1o.31)
(x,F(k)g) x = ( n * ( . , k ) x ,
g)o
As f o r the second term of the right-hand side we have, by exchanging the order of i n t e g r a t i o n , (fo,V)o = . o ( f O ( r ) , (10.32)
i o g ( r , s , k ) g ( s ) d s ) x dr
= I~ (Jr=~G(s,r,_k) f o ( r ) dr, g(S))xdS 0 = (2ikh(.),g)o
,
where we have used (10.2) repeatedly.
(10.30) ~ (10.32) are combined
to give (10.33)
1 (n*(.,k)x,g) 0 = ( ~w
0 - h, g)o
(10.29) follows from (10.33) because of the a r b i t r a r i n e s s
of
g ~ c~(z,x) Q.E.D.
113
THEOREM 10.8. r!*(r,k)x
Let Assumption 5.1 be s t a i s f i e d .
are s t r o n g l y continuous
PROOF. Let us f i r s t it
is s u f f i c i e n t
in
X , where
Let
e > 0
(10.12),
show
X-valued f u n c t i o n s
the c o n t i n u i t y
of
to show t h a t q * ( r n , k n ) X n r n÷r
in
given .
T ,
Then
kn÷k
in
IR-
- n*(r,k)Xol
.
Ix
and (R - {0} ) xX.
To t h i s end
tends to n * ( r , k ) x
x 0 c D and a p o s i t i v e
In*(r,k)x
on
n*(r,k)x
{ 0 } , Xn+X
Then, by the denseness o f
there e x i s t s
n(r,k)x
in
strongly X
as
n~=
D and the e s t i m a t e integer
n O such t h a t
< m ,
(10.34) lq*(rn,kn)X n - q*(rn,kn)xol
for
n => n O .
<
Therefore we have o n l y to show t h a t
(I0.35)
inX
s - lim q*(rn,kn)X = q*(r,k)x n+oo
for
x c D .
In f a c t i t
q*(r,k)x
f o l l o w s from P r o p o s i t i o n
- ~*(rn,kn)X 1 " r-,k) = 2 i ~ {~(r)e1~J(
(10.36)
+ {h(rn,k,x) + {h(rn,kn,X) Since
H(y,k)
definition
of
i s continuous lJ(y,k)
10.7 t h a t
in
(see ( 5 . 8 ) ) ,
side o f (10.36) tends to zero as
(y,k)
rn)eiU(rn-,kn)}X
- h(r,k,x)} - h ( r n , k , x )} by ( I ) o f Remark 5.9 and the
the f i r s t n~,,~
- £(
term o f the r i g h t - h a n d
The second term of the r i g h t - h a n d
side o f (10.36) tends to zero because of the c o n t i n u i t y function
h(r,k,x)
(~eip(r"kn)x)
.
o f the r a d i a t i v e
I t f o l l o w s from ( I ) o f Remark 5.9 t h a t
converges to
(L-k 2) ( L e i l ' ( r " k ) x )
in
(L-k~)
L2,~(I,X)
,
114 which implies t h a t
h(.,kn,X)
L2,_4(I,X ) n H ~ ' B ( I , X ) l o c .
converges to
(10.37)
T ,
in
Since the convergence in
means the uniform convergence in LO,R~ on
h(.,k,x)
X on every f i n i t e
H~'B(I,X)Ioc interval
we have s - lim
{h(rn,knX) - h ( r n , k , x ) }
= 0
in
X
i
n-~,
Thus (10.35) has be proved.
The proof of the c o n t i n u i t y of
n(r,k)x
is much e a s i e r .
In f a c t , the c o n t i n u i t y follows from the f a c t s t h a t
(10.38)
~(r,k)x = F(k)~Ir,xl
(r~T,
k~
- {0}, xeX)
and t h a t as has been seen in the proof of Proposition 9.2, ~Ir,xJ an
F~-valued, continuous function on
B(F~(l,X),X)-valued,
I x X and t h a t
F(k)
is a
continuous function on ~ - {0~. ~.E.D.
, is
115
§II.
Expansion Theorem
Now we are in a p o s i t i o n to show a s p e c t r a l
representation
theorem
or an expansion theorem f o r the Schrodinger o p e r a t o r w i t h a long-range potential.
To t h i s end the s e l f - a d j o i n t
should be defined and some of i t s
realization
spectral
of
properties
T - -A + Q(y)
should be mentioned.
Here l e t us note t h a t these p r o p e r t i e s were never used in the previous sections Let
(§I - § I 0 ) . ,2(Y)
be a r e a l - v a l u e d ,
Then two symmetric operators
h0
bounded and continuous and
11 in
L2(]RN)
f u n c t i o n on
]R
N
are defined by
p ( h o) : £,,(h) = do(m N) ,
(ll. I)
hof = Tof = -,',f hf = Tf = - A f + Q ( y ) f .
As is well-known,
h0
unique s e l f - a d j o i n t
is e s s e n t i a l l y
extension
self-adjoint
in
L2(]RN )
with a
H0 defined by
D(Ho) = H2(IRN ) , ( I I .2) Hof = Tof = -,_If . where the d i f f e r e n t i a l t i o n sense.
(11.3)
Further,
operator it
TO should be considered in the d i s t r i b u -
is also well-known t h a t we have
o-(HO)
=
Cac(H O)
=
CO,::,}) .
116 Here
o(H O)
is the spectrum of
H0
and
~ac(Ho)
is the a b s o l u t e l y
continuous spectrum. THEOREM I I . I . ~N
with
Let
Q(y)-* 0
essentially extension
Q(y)
as
IYl .....
self-adjoint H .
be a r e a l - v a l u e d ,
in
Let
L2 ( ~ N )
h
continuous f u n c t i o n on
be as in ( I I . I ) .
Then
h
is
with i t s unique s e l f - a d j o i n t
We have C(H) = D ( H O) = H2(~N)
(l1.4) Hf = Tf = -A +
q(y)f (in the distribution sense).
H is bounded below with the lower bound tial
spectrum of
H , is
continuous spectrum of they e x i s t ,
equal to
ko
LO,=~) .
H is absent, and the negative eigenvalues,
are of f i n i t e
0 .
multiplicity
V
V = Q(y)x
(11.5) Since
L2 ( ~ N )
h0
is e s s e n t i a l l y s e l f - a d j o i n t ,
domain ~(H)
{Hof n}
and
h
H .
operator
Q(y) .
Then
is r e w r i t t e n as
h = h0 + V .
a d j o i n t with a unique s e l f - a d j o i n t
4.4).
point other than the
be a m u l t i p l i c a t i o n
is a bounded operator on
if
and are d i s c r e t e in the sense
There e x i s t s no p o s i t i v e eigenvalue of
PROOF. Let
~e(H), the essen-
Therefore on [ko,O) the
t h a t they form an i s o l a t e d set having no l i m i t origin
and
V
is equal to
p(H O)
can be seen to be
the i n t e r i o r
compact in
is also e s s e n t i a l l y
extenstion (see, e . g . ,
Ho-compact, i . e . ,
are bounded sequence in
relatively
h
L2 ( ~ N )
L2 ( ~ N ) .
self-
H = H0 + V and the Kato [ I ] , if
p. 288, Theorem
{ f n } c D(Ho)
, then the sequence
and
{Vf n}
is
In f a c t t h i s can be e a s i l y shown by
estimate and the R e l l i c h theorem.
Therefore the essential
117
spectrum {e(Ho)
Oe(H) of
H0
of
H = H0 + V
(see Kato [ I ] ,
is equal to the e s s e n t i a l spectrum
p. 244, Theorem 5.35).
F i n a l l y we
shall show the non-existence of the p o s i t i v e eigenvalues of Suppose t h a t
~> 0
i s an eigenvalue o f
the eigenfunction associated with function f o r
{L,yzT, O}
radiative function.
~ .
and hence
H and l e t
be
is the r a d i a t i v e
by the uniqueness of the
This is a c o n t r a d i c t i o n .
of p o s i t i v e eigenvalues o f
~ ~ H2(~N)
Then v = U~
v = 0
H .
Thus the non-existence
H has been proved. Q.E.D.
Set
M = UHoU* , (11.6) M = UHU* ,
where in
U i s defined by (0.19).
L2(I,X)
a l e n t to
H0
with the same domain and
and
M .
s t r u c t u r e of the spectrum of
M are s e l f - a d j o i n t
UH2(~ N)
H , respectively.
measure associated with
(-,~,0)
M0
Let
operators
and are u n i t a r i l y E(.;M)
equiv-
be the spectral
We s h a l l be mainly i n t e r e s t e d in the M on
(0, ~,) , because the spectrum on
is discrete.
PROPOSITION 11.2. compact i n t e r v a l in
Let Assumption 5.1 be s a t i s f i e d . (0,~)
and l e t
f,g c L2,5(I,X ) .
Let Then
J
be a
1i8
-~T ( F ( k ) f , ( E ( J ; M ) f , g ) 0 = Sv'-j- -2k2
F(k)g)xdk
(11.7) = IV- j - -2k23T (F(-k)f,
where
¢~J- = {k > Olk 2 • J} PROOF.
z • ~ - R
and
I v~}-> 0 m
{L,d'z-,l[f]} function
F(k)
We denote the r e s o l v e n t
R(z;M) = (M - Z) -I
with
and
.
f • L2, ( I , X ) and
.
Here
v(',V-z-,Z[f])
In f a c t t h i s
and the f a c t t h a t
M by
8.2.
R(z;M) , i . e . ,
R(z;M)f = v ( . , V - z - , Z [ f ] ) ~
for
is the square r o o t o f
is the r a d i a t i v e
follows
,
is given by D e f i n i t i o n
of
Let us note t h a t
F(-k)g)xdk
function
z
for
from the uniqueness o f the r a d i a t i v e
R ( z ; M ) f c UH2(~N)
.
Moreover l e t
us note
that
(]1.8)
lim R(k 2 ± ib; M)f = v ( - , ± ;k I, 2 [ f ] ) b+O
(k ~ m - { 0 } , which f o l l o w s
from the c o n t i n u i t y
to
Then from the well-known
k c ~+ .
in L 2 , _ 6 ( I , X )
f ~ L2,6(I,X))
o f the r a d i a t i v e
,
function
with respect
formula
(E(J;M)f,g) 0
(11 9 )
it
fo]lows
_
1 lim I {(R(A 2~i b+O J
_
1 lim { { ( f , R ( a 2rri b+O JJ
+ ib;M)f,g) 0 - (f,R(a
i
+ ib;M)g)o}da
ib;M)g) 0 - (R(a - i b ; M ) f , g ) o } d a
that (E(J;M)f,g) 0
(11.1o)
( 1 2 fri J j [ ( v ( " / - ~ - , ~ [ f l ) ' g ) o 1 2~i I j { ( f ' v ( " - d - ~ - ' l [ g J ) ) O
- (f,v(i,fra-,~'[g]))o}da - (v("-vr~-'~[f])'g)o}da
"
119 By the use of (8.14) in Theorem 8.4 (11.7) follows from ( I I . I 0 ) . Q.E.D. Let us set 2 F± (k) = ~ V ~
(II.II)
COROLLARY 11.3. operator,
(i)
ikF(~k)
E((O,~);M)M
(k > O) . is an a b s o l u t e l y continuous
i.e., aac(M ) D (0,~<,) .
(ii)
F+ ( ' ) f e
(11.13)
i
L2(O,~),X,dk)
for
f(:
L2,..~(I,X)
with the estimate
oo
2 2 2 IF ( k ) f I dk = II E((O,'~O;M)fll < ]I fll 0 x O0
PROOF. Obvious from (11.7) and the denseness of
L2,6(I,X)
in
L2, ( I , X ) . Q.E.D. The ,3enc~rat]z<~:i P(~umiE~r ;zv:nsfor.~.s
DEFINITION II .4. L2(I,X) = L 2 ( I , X , d r ) (II.14) for istic
[F f ) ( k )
f ~ L2(I,X)
into
Lm((O,<,,),X,dk)
: l.i.m.
F+(k)f R
in
L2(I,X)
operators from t i o n between
L2((O,,=),X,dk) and ×R(r)
into
L2((O,,~),X,dk)
L2(O,~),X,dk) F(k)
and
~(r,k)
into .
are bounded l i n e a r oper-
F± .
F±
L2(I,X)
Let us set
2 n ± ( r , k ) = ~ / ~ ikn (r,~k) (11.15)
is the character-
(O,R) .
I t follows from C o r o l l a r y 11.3 that ators from
from
by
, where fR(r) = ~ R ( r ) f ( r )
f u n c t i o n of the i n t e r v a l
F±
T q+ ( r , k ) = TV ~ i k n * ( r , L k )
denote the a d j o i n t By the use of the r e l a -
120 for
r~-T, k>O. PROPOSITION 11.5.
Let
F±
Let Assumption 5.1, (5.4) and (5.5) be s a t i s f i e d .
be as above.
(i)
Then F±
and
(II.16)
(F+f)(k) = l . i . m ,
(ll.17)
(F+F)(k) = l . i . m ,
F±
~i'~(r,k)f(r)
.
rR
-
where
Let
in
in
and
F c L2((O,,~O,X,dk ) .
f e L2,~(I,X ) .
R
if
Then for each
of
~,:;0
with
in
X .
1 ~ > ~ + # , the integral
of the
is absolutely convergent.
fR(r) = X R ( r ) f ( r )
(O,R) .
k > 0
'
f e L2,3(I,X)
right-hand side of ( l l . 1 8 ) PROOF. Let
Lm(I,X) ,
"
-
In p a r t i c u l a r ,
Lm((O,.~O,X,dk)
.
rR (F+f)(k) = s - lim . r , ~ ( r , k ) f ( r ) d r
(ll.18)
xR(r)
dr
jIR-1%(r,k)F(k)dk
m÷,~
f E L2(I,X) (ii)
are represented as follows:
with the c h a r a c t e r i s t i c
function
Then i t follows from (10.19) in Theorem 10.5 that
F±(k)f R = ~V~2 ikF ( ~ i k ) f = ~ 7 ~2i k
I ~i(r,~k)fR(r)dr Jl
(ll.19) =
~i+(r,k)f(r)dr O -
,
which, together with the d e f i n i t i o n In order to prove ( l l . 1 7 ) (ll.20)
i t suffices
(F+F)(r) = --
for
FC L2((O,~,,),X,dk )
of
F±f ( ( I f . 1 4 ) ) ,
proves ( l l . 1 6 )
to show
n+(r,k)F(k)dk 0
--
with compact support in (0, ~) , because
are bounded l i n e a r operators from Denoting the inner product of
.
L2((O,~),X,dk)
L2((O,,,,), X,dk)
into by
(
L2(I,X) ,
)0
F± .
again, we
121
oo
have for any
f c CO ( l , X )
.
(F±F,f) 0 = (F, +-f)0 =
=
I~ 0 01
(F(k)
,
I"£n+ ( r ,
k)f(
q +(r,k)F(k)dk 0 -
= '.JO - n + ( " k ) F ( k ) d k which implies (11.17). 0 n_.(r,k)F(k)dk respectively,
r)dr)xdk , f ( r ) xdr
, f 0 '
Here i t should be noted that the i n t e g r a l s
and
~0 q ± ( r , k ) f ( r ) d r
because of Theorem 10.8.
are continuous in
r
and in
k ,
( i i ) d i r e c t l y follows from
Theorem 10.5 and D e f i n i t i o n 11.4. Q.E.D. Thus we a r r i v e at THEOREM 11.6 5.1 be s a t i s f i e d .
(Spectral Representation Theorem). Let
(ll.21) where
B be an a r b i t r a r y
Borel set in
Let Assumption (0,,~) .
Then
E(B;M) = ~ Xv~~_ F+ , X
is the c h a r a c t e r i s t i c
f u n c t i o n of
~-B-= {k > O/k 2 E B} .
In p a r t i c u l a r we have (II.22)
E(O,~);M) = F+F+ .
PROOF. J
in
I t s u f f i c e s to show ( I I . 2 1 ) when
(0,~) .
From (11.7) i t follows t h a t (E(a;M)f,g) 0 = I ~ j - ( ( F + / ) ( k )
(II.23)
f,g e L2,6(I,X)
, (F+_g)(k))xdk
= (Xvrj_(F+f)(')__ , ( FO+ g )_( ' ) ) = (F± X_o_FLf,g) 0
for
B is a compact i n t e r v a l
, which implies that
122 #¢
(11.24) Since
E(J;M)f = F+* ×~/_j_ I-±f E(J;M)
L2(I,X)
and
and
(f e L2,6(I,X)
r operators X / are_bounded j _l i n e aF +
F±
L2,6(I,X )
is dense in
L2(I,X)
on
, (11.21) follows from
( l l .24). Q.E.D. In order to discuss the o r t h o g o n a l i t y of the generalized Fourier transforms
F±,
some properties of
PROPOSITION 11.7. associated with (i)
Let
F±
F± w i l l
be shown.
be the generalized Fourier transforms
M .
Then
(11.25)
F±E(B;M)
= X
~T
f±
(11.26)
F±E((O,~);M) = F±
(11.27)
E(B;M)F*
,
= F*;< vZB- ,
(11.28) where
E((O,~,);M)F# = F#, B and
×v"B-
are as in Theorem 11.6.
(ii)
~L2(I,X)
are closed l i n e a r subspace in
(iii)
The f o l l o w i n g three c o n d i t i o n
L2((O,~),X,dk)
(a), (b), (c)
about
.
F+ ( F )
are e q u i v a l e n t : (a)
F+(F_)
(b)
F+F+*= l (F F = l ) , where
(c)
The null space of
PROOF.
maps E((O,,xO;M)L2(I,X)
F+ ( F )
( i ) Let us show (11.25).
l
onto
L2((O,'~),X,dN)
means the i d e n t i t y
consists only of Let
f ~ L2(I,X) 2
K = IIF±E(B;M)f__ - Z F+fll (B- -
0
operator.
0 . .
.
Then
123 2 (11.29) = II F_+E(B;M)fll
2 ÷ II , F÷fll - 2Re (F+E(B;M)f, , _ ×vZT 0 ×¢"~ -- 0
F±f)
= Jl + J2 " 2ReJ3 " From (11.21) and the r e l a t i o n s E(B;M) 2 = E((O,~);M)E(B;M) = E(B;M)
(11.30)
we can easily see that the proof of (11.25). (ii)
(11.26)-(11.28)
Assume that the sequence
sequence in
L2((O,~),X,dk)
assumed to belong to sequence, since F+F+f n
in
F÷
{F+fn} ' fnC
.
Then
is a bounded operator.
has the l i m i t
L2((O,=~),X,dk) . (iii)
L2(I,X ) , be a Cauchy
By taking account of (11.26)
E((O,~);M)L2(I,X)
with the l i m i t
in the same way. (a),
.
K = 0 , which completes
immediately follows from (11.25).
= E((O,~),,4) "~ fn = fn , the sequence
L2(I,X)
{F+f n} in
Jl = J2 = J3 ' and hence
f c L2(I,X ) .
{F+F+f n}
may be
is a Cauchy
Thus, by noting that {fn } i t s e l f
is a Cauchy sequence
Therefore the sequence
F+f , which means the closedness of The closedness of
fn
F_L2(I,X)
By the use of ( i ) and ( i i )
F+L2(I,X)
can be proved quite the equivalence of
(b), (c) can be shown in an elementary and usual way, and hence the
proof w i l l
be l e f t
to the readers. Q.E.D.
When the generalized Fourier transform of the conditions
F+ or
F_ s a t i s f i e s
(a), (b), (c) in Proposition 11.7, i t
one
is called
or~hogor~. The notion of the orthogonality is important in scattering theory. THEOREM 11.8 (orthogonality and (5.5) be s a t i s f i e d . orthogonal.
of I:± ).
Let Assumptions 5.1, (5.4),
Then the generalized Fourier transforms are
124
PROOF.
We s h a l l
show t h a t
F+
Suppose t h a t
F+F = 0
w i t h some
to show t h a t
F = 0 .
Let
using ( I I . 2 7 ) ,
(c) in P r o p o s i t i o n
F • L2((O,,~),X,dk)
B = (a2,b 2)
we o b t a i n from
(ll.31)
satisfy
with
.
11.7.
We have o n l y
0 < a < b < ~
Then,
E(B;M)F+F = 0
F ,,
F = 0 ,
which, together with
11.17),
(11.32)
Ib.q + ( r , k ) F ( k ) d k
gives = 0 .
a
Since
a
exists
a null
(II.33)
and
b
can be taken a r b i t r a r i l y ,
set
e
n+(r,k)F(k)
in = 0
(0,=,) or
noting that
Let
x e D
F(k)f 0 = x
follows
that there
such t h a t
q (r,k)F(k)
and take
( r C T , ke e) ,
= 0
where we have made use o f the c o n t i n u i t y (Theorem 1 0 . 8 ) .
it
of
ri+(r,k)F(k)
fo(r)
by P r o p o s i t i o n
as in ( 8 . 3 0 ) .
8.5 and using ( i i )
in
r c I Then,
o f Theorem
10.6, we have
(F(k),x) X = (F(k),
F(k)fo) x
(II.34) = (rl ( . , k ) F ( k ) , whence f o l l o w s in
that
L2((O,~),X,dk )
F(k) = 0
fo)o = 0
f o r almost a l l
The o r t h o g o n a l i t y
of
(k ~ e, x e D)
k c (0,~,) ~
i.e.,
F = 0
can be proved q u i t e in
the same way. Q.E.D. Finally
we s h a l l
f o r the o p e r a t o r
translate
the expansion theorem, Theorem 11.6
M i n t o the case o f the S c h r ~ d i n g e r o p e r a t o r
us d e f i n e the g e n e r a l i z e d F o u r i e r t r a n s f o r m s
~4
H .
associated with
Let H
by
125
-l
(II .35)
r+= u k
where
Uk = k
N-l 2
x
F+U ,
is a u n i t a r y operator from
Lm((O,°~),X,dk
.
L2(~RN,d~)
I f the bounded operators
,q*(r,k),
.
~+
L2(~RN , d~)
are bounded l i n e a r operators
r c- I, k > 0 , on
T~(r,k)
from
onto
L2(IRN,dy)
into
and
X are defined by
~±(r,k)
= r-(N-l)/2k-(N-l)/2~+(r,k)
~*(r,k)
= r -(N-l)12k-(N-l)/2q:(r,k)
(11.36) ,
then we have IR (F+~.)(I~o) = l . i . m . (~+(r,k)@(r-))(w')dr (11.37)R÷~ 0 ~* F R ~, (l:+~;)(y) = l . i . m . . (n+(r,k)f(k.))(s~)dk -
where
F+
R÷<~
JR-l
are the a d j o i n t
be the spectral
in
L2(mN,dE.)
in
L2(]RN,dy)
-
of
F+
measure associated
and
y = rm, ~ = kw' .
with H .
Let
E(';H)
Then we have from Theorems
11.6 and 11.8. THEOREM 11.9. B
be an
Let Assumption 5.1,
arbitrary
Borel set in (O,,x,) .
transforms
T+, defined as above, map
L2(~N,d~)
and
(11.38) where
(5.4) and (5.5) be s a t i s f i e d . Then the generalized
E((O,~);H)L2(~N,dy)
Fourier
onto
E(B;H) = F+X,/_I~_F+ , X
~T
is the c h a r a c t e r i s t i c
{~: ~- LRNI l~ol 2 E B} .
function
of the set
Let
{~1 ~ RN/
126 §12.
The General Short-Range Case
In t h i s section we shall potential ~l(y)
Q1
is a general short-range p o t e n t i a l ,
= O(lyl -l-c)
.
(QO)
i.e.,
Throughout t h i s section we shall assume
ASSUMPTION 12.1. (Q) , (QI)
consider the case that the short-range
The p o t e n t i a l
in Assumption 2.1.
Q(y) = Qo(y) + QI(y)
Further, Qo(y)
satisfies
is assumed to s a t i s f y
in Theorem 5.1. Then a l l
the r e s u l t s of Chapter I and §9 of Chapter I I I
in the present case.
Therefore the l i m i t i n g
good and the Green kernel
G(r,s,k)
are v a l i d
absorption p r i n c i p l e
is w e l l - d e f i n e d .
In order to make
use o f the r e s u l t s of Chapter I I and §I0 - §II of Chapter I I I , approximate
Q1 (y)
by a sequence
IO.in(Y)I = O ( I y i - ~ l )
(n=l,2 . . . . )
To t h i s end l e t us define (12.1) where
Qln(Y)
Qln(Y) = ~ ( [ y [ ~(t)
I
we shall
which s a t i s f i e s
with a constant
c I > max (2 - c, 3) .
by
- n)Ql(y)
(n=l,2 . . . . ) ,
is a r e a l - v a l u e d continuous f u n c t i o n on
~ ( t ) = 1 ( t < O) , = O ( t > I)
(12.2)
{Qln(Y)}
holds
]R
such t h a t
, and l e t us set
Tn = -A + Qo(y ) + 9-I (y) ' d2
LLn = - dr 2 Since the support of
+ B(r) + Co(r) + Cln(r) Qln(Y)
sections can be applied to
is compact, all Ln .
(Cln(r)
= Qln(rW)x)
the r e s u l t in the preceding
According to D e f i n i t i o n
8.3,
i27 Fn(k) , R • ~
- {0} , is w e l l - d e f i n e d as a bounded l i n e a r operator
from
into
F6(I,X)
X .
is also w e l l - d e f i n e d . ~(y,k)
and i t s
The eigenoperator
Z ( y , k ) are common f o r a l l
because the long-range p o t e n t i a l
Q0(y)
w e l l - d e f i n e d as the strong l i m i t a l l the p r o p e r t i e s obtained in
Fn(k)(k • A - { 0 } , F6(I,X) (i)
into
of
Ln, n=l,2 . . . . .
is independent of
show t h a t an operator
PROPOSITION 12.2.
F(k)
Fn(k)
and t h a t
Let Assumption 12.1 be s a t i s f i e d .
n=l,2 . . . . )
satisfies
Let
X defined as above.
Then the operator norm
into
k
llFn(k)ll
is u n i f o r m l y bounded when
, there e x i s t s a bounded l i n e a r operator
moves in a compact set in
A-{0}
.
For each
F(k)
from
X such t h a t
(12.3)
(ii)
is
be the bounded l i n e a r operator from
k • A-{0}
X .
F(k)
L
§8.
and
in
n .
associated with
n=l,2 . . . .
F6(I,X)
e T , k • A - {0})
Here we should note t h a t the s t a t i o n a r y m o d i f i e r
kernel
We s h a l l f i r s t
q(r,k)(r
F ( k ) l = s - l i m Fn(k)~ n* llF(k)[l F(k)
is bounded when satisfies
k
(1 • F 6 ( I , X ) ) moves in a compact set
(8.12) f o r any
in
A-{0}
l l , 1 2 • F6(I,X ) , i . e . , we
have
(12.4) vj(j
= 1,2)
1 ( F ( k ) l l , F ( k ) 1 2 ) x = 2 - ~ { < / 2 , V l > -
being the r a d i a t i v e function f o r
are also s a t i s f i e d by (iii)
Let
{L,k,vj}
.
(8.13)-(8.15)
F(k) .
x • D and
f o ( r ) = (L - k 2 ) ( ~ e i ~ ( r " k ) x ) defined by (5.8) .
> } ,
Then
let
fo(r) with
be as in
~(r)
(8.30),
i.e.,
defined by (5.36) and
u(y,k)
.
128
(12.5)
F(k)f 0 = x ,
where we s e t
F ( k ) L l f O] = F ( k ) f 0 .
( i v ) As an o p e r a t o r from ]B(L2,,~(l,X),X)-valued each
k c JR- {0} (v)
Let
(12.6) where
L2,,~(I,X)
into
conLinuous f u n c t i o n on
F(k)
f m L2,.(I,X):;
is the r a d i a t i v e
sequence such t h a t
is a
]2- {0} .
Further,
L 2 , .o, ( I , X )
is a compact o p e r a t o r from and l e t
k c JR- {0}
F(k)f = s - linl c-ilJ(rn''k)v(rn n ÷ ,*> v
X , F(k)
r n ....
function and
for
)
for
into
X
Then
in
X ,
{L,k,2[f]}
and
v ' ( r n) - i k v ( r n) • 0
{ r n}
in
X
as
is a n • "~
PROOF. Proceeding as in the proof of Theorem 8.4 we o b t a i n
(~ n(k)4', Fp(k)Z) x
(12.7) - -!-]2ik {
~' L- F~(I,X),,~ and
function
for
n,p=l,2 .....
{Ln,k,%}
o f Theorem 4.5.
({Lp,k,~'}
+
where .
Set
r
.
( t C l n - Clp}Vn,Vp)o} Vn(V p)
p = n
is the r a d i a t i v e
in (12.7) and make use
Then we have
IFn(k)i'l < ~ -
(]2.8)
> -
IIIzlh¢~l Vnll
< c21ilzill26
x
with R-
C = C(k) {0} , where
hand we o b t a i n
which is bounded when II lIB,_6
k
is the norm o f
moves in a compact set H~IB6(I,X)
.
in
On the o t h e r
from (12.7)
2 2 2 }Tn(k)g - / p ( k ) L } x = i F n ( k ) £ } x + }Fp(k)~},x
2Re(Fn(k)g'Fp(k)g)
(]2.9) -
as
]k Im({Cln
- C]p]Vn'Vp)O
n,p + ,~ , because by Theorem 4.5 both
vn
and
÷ 0
Vp
converge to the
129 radiative function
v
for
{L,k,£}
in
L2,_~(I,X)
and
Cln(r)
- Clp(r )
satisfies llCln(r) (12.g)
- Cmp(r)il = I ~ ( r - n )
IICln(r) - C l p ( r ) I i - ~ 0
follows.
F(k)L = s - l i m Fn(k)
llF(k)ll < C , where
(12.12)
Thus
(ii)
-<
~I'
Vn2 > } '
{Ln,k,~[ j }
has been proved.
f o n ( r ) =(L n - k 2 ) ( ~ e i ~ ( r ' ' k ) x )
Then i t
X and we
n -~ ~,, in the r e l a t i o n
Vnl >
being the r a d i a t i v e f u n c t i o n f o r
being made use of.
T)
C is as in (12.8), whence ( i )
1 ( F n ( k ) Z i , F n ( k ) # 2 ) x = 2ik {<~2'
Vnj,j=l,2,
c0(1 + r) -E~I ( r ~
e x i s t s in
(12.4) immediately follows by l e t t i n g
(12.11)
<
(n,p . . . . ) .
Therefore the strong l i m i t have the estimate
,]cl(r) II
- ~(r-p)i
and Theorem 4.5
Let us show ( i i i ) .
Set
(n=l,2 . . . . ) .
follows from Proposition 8.5 that
(12.13)
Fn(k)fon(r)
Further, (12.14)
it
(n=l,2 . . . . ) .
is easy to see from (5.20) in Lemma 5.5 t h a t f O n + fo
(n -~ ~)
(12.5) is obtained by l e t t i n g (12.14)
= x
n -~ ~
in
L2,5(I,X)
.
in (12.13) and using ( i ) and
Re-examining the proof of Theorems 8.6 and 8.7, we can see that
( i v ) and (v) are obtained by proceeding as in the proof of Theorems 8.6 and 8.7, r e s p e c t i v e l y .
This completes the proof.
q.E.D. Let us now define the eigenoperator (12.15)
q(r,k)x = F(k)2[r,x]
(r e I,
q(r,k)
associated with
k ~ ]R - {0} , x'.~ X) ,
L
by
130
where
L'Lr,x]
is as in ( 9 . 2 ) ,
HI~B(I,X)
.
proposition, operator
i.e.,
At the same t i m e ,
n(r,k)
qn(r,k)
as w i l l
associated with
Ln
be the e i g e n o p e r a t o r a s s o c i a t e d w i t h
with
(r,k)
C = C(k)
- {0~
.
T x (m-{o})
is an X - v a l u e d ,
q(r,k)
is a bounded
( r e-T) k
moves in a COnlpact set in
continuous function
on
We have
(12.17)
2ik(q(s,k)X,'l(r,k)x')
f o r any
x,x'
(iii) (12.18)
3R - { 0 }
rl(r,k)
x x .
(ii)
with
e T x (~ -{0})
which is bounded when
q(r,k)x
Let
L .
l l q ( c , k ) l l < C(I + r ) ~
(12.16)
o f the e i g e n -
Let Assumption 12.1 be s a t i s f i e d .
Then f o r each
for
be shown in the f o l l o w i n g
can be d e f i n e d as the s t r o n g l i m i t
PROPOSITION 12.3.
(i)
= (x,~(r)) x
X = ({G(r,s,k)
c X, k -e ~ - { 0 } , The o p e r a t o r norm
and
llqn(r,k)ll
l l r l n ( r , k ) l l <__"C(l + r) ~
C = C(k) .
f o r any t r i p l e (iv)
Let
T . is estimated as
R moves in a compact set i n
we have
q(r,k)x (r,k,x)
=
s
-
tim n n ( r , k ) x
in
X
e T x (~R - [ 0 } ) x X .
f c L2,6(I,X)
X
( r (< I - , n = l , 2 . . . . )
which is bounded when
Further,
(12.19)
r,s,c
- G(r,s,-k)}x,x')
and l e t
k c ~ - {0} .
Then
;31
(12.20)
F ( k ) f = s - lira R ......
PROOF.
(i)
and ( i i )
I
R Ti(r,k)f(r)dr
the uniform boundedness of to
n(r,k)x
I t is s u f f i c i e n t
L
n
llrln(r,k)H
,
and the strong convergence of
f o l l o w s from P r o p o s i t i o n
12.2.
Let us show ( i v ) .
to show
(12.22) for
Since we have
rin(r,k)x = Fn(k)gls,kJ
nn(r,k)x
X .
can be e a s i l y obtained by proceeding as in the
proof of Theorems I 0 . 4 and I 0 . 8 . (12.21)
in
0
F(k)f = I 'I
f c L2(I,X )
.~l(r,k)f(r)dr
w i t h compact support in
Applying Theorem I 0 . 5 to
I .
, we o b t a i n F f-n(k) = i l rln(r, k) f ( r ) d r
(12.23)
whence (12.22) f o l l o w s by l e t t i n g
,
n ~ =: Q.E.D.
Let
r1*(r,k)
w i t h respect to
be the a d j o i n t o f
:1*(r,k)
Tl(r,k)
.
Almost a l l
the r e s u l t s
obtained in §lO are also v a l i d f o r our
rl*(r,k) PROPOSITION 12.4. n*(r,k)
Let Assumption 12.1 be s a t i s f i e d .
be as above and l e t
operator (i)
rln(r,k) let
r~n(r,k)
associated w i t h
R e_ IR-{O} .
Then
be the a d j o i n t o f the eigen-
Ln T l * ( ' , k ) x ~= L 2 , _ d ( I , X )
w i t h the e s t i m a t e (12.24)
Let
l l n * ( ' , k ) x l l _ 6 <_- C'x' 'X
(x ~ X) ,
f o r any
xcX
132
where
is bounded when
C = C(k)
k
moves in a compact set in
A-{O
We have (q*(.,k)x,f)o
= (x,F(k)f)x
(12.25) (k (- R - { O ! , Let
(ii)
where
x c D . Then
q*(r,k)x
(12.26)
function
for
1
~(r)eip(r"k)x
{L,-k,('[fo]} fo(r)
(12.27)
a compact set (12.28)
C = C(k) ,sucn
IIq~(.,k)xil.
{q~(',k)x}
I 0 . 7 and
h(.,k,x)
= ~-~-~ l (L - k 2 ) ( ' , e l"" I r' " k ) x )
IR-!O}
in
- h(r,k,x)
converges to
is tile r a d i a t i v e
with
There e x i s t s
(iii)
each
=
is as in P r o p o s i t i o n
~(r)
x (- X, f r - L 2 , , i ( I , X )
-C
.
, which is bounded when
k
moves in
tnat
< C~x I Z~
'
r/*(',k)x
( x m X)
.
X
in
L 2 , _ 5 ( I , X ) :-" H l ~ B ( I , X ) l o c
for
x c X .
(iv
q*(.,k)x
Hl~B(I,X)loc
(L - k 2 ) v ( r )
(1Z.29) rl*(r,k)x
is an x - v a l u e d ,
PROOF.
(i)
and ( i i )
= 0
from the r e l a t i o n
and s a t i s f i e s
( r cI [ , k (- m - { 0 } , x
continuous function
on
the e q u a t i o n
c X) .
T x (~--{0})
x X .
can be o b t a i n e d by proceeding as in the p r o o f
o f Theorem I 0 . 6 and P r o p o s i t i o n the p r o o f o f ( i i i ) .
r~ D(I
I0.7,
respectively.
The u n i f o r m boundedness o f
Let us t u r n i n t o
li~n(',k)xll_r ~
follows
133
(12,30)
(qn(.,k)x,f)o
= (X,Fn(k)f)x (f c L2,~(I,X),
and the uniform boundedness of
llFn(k)[l
.
Now t h a t (I0.28)
established,
in order to show the convergence of
~*(.,k)x
L2,_s~(I,X)
in
(12.31) in
is s u f f i c i e n t
L2,_6(I,X)
.
Applying Proposition
~*',in~r,k)x : ~ I
function {f0,n }
for
{Ln,-k,Zffn]}
10.7 to
, where
and
{rln(-,k)x}
to show f o r
~(r)eiU(r"k)x
x e D , (r,k) c T x ~-{0}
has been to
x E D
(k E ~ - { 0 } )
riiT(.,k)x ÷ q*(.,k)x
(12.32) for
it
n = 1,2 . . . . )
L n , we have
- hn(r,k,x) hn(.,k,x)
is the r a d i a t i v e
f0,n = (Ln - k2) ( ~~ e i U ( r ' ' k ) x )
can be e a s i l y seen to converge to
Since
fo = (L - k 2 ) ( c e i U ( r " k ) x )
L 2 , 6 ( I , X ) , Theorem 4.5 is applied to show t h a t {h ( ' , k , x ) }
ir
converges
n
to
h(',k,x)
qn(r,k)x interior
in
satisfies
L2,_6(I,X)
, which implies
the equation
the proof of ( i i i ) .
Noting t h a t
(L n - k2)v = 0 , we make use of the
estimate and the convergence, of
show the convergence of
(10.31).
{q~(.,k)x}
in
{Ti*(.n , k ) x } Hl~B(l,X)loc
in
L2,_6(I,X)
to
, which completes
We have (q*(',k)x,
(L - k2)~) 0 = 0
(12.33) (k e m - { O } ,
by l e t t i n g
n -~ ~
x c X, <~ e C'~(Z,X))
in the r e l a t i o n (qn(',k)x,
(L n
k2)¢.)0 = 0
(12.34) (n=l,2 . . . . . k e ] R - { O } , x c X,~b e CO(I,X))
134
The f i r s t
h a l f of (iv) follows from (10.31) and Proposition 1.3.
c o n t i n u i t y of Tl*(r,k)x
The
can be shown in quite the same way as in the
proof of Theorem 10.8, which completes the proof. Q.E.D. Now the Propositions 12.2, 12.3, 12.4 have been established, we can show the expansion theorem (Theorem l l . 6 ) of the generalized Fourier transforms
F±
same way as in the proof of Theorems l l . 6
with the orthogonality
(Theorem l l . 8 ) and l l . 8 .
in quite the
Therefore we shall
omit the proof of the following THEOREM 12.5.
Let Assumption 12.1 be s a t i s f i e d .
alized Fourier transforms and the a d j o i n t tively.
F~
F±
F±
Then the gener-
is w e l l - d e f i n e d by D e f i n i t i o n l l . 4 .
can be represented as ( l l . 1 6 )
are orthogonal and ( l l . 2 1 )
holds good.
and ( l l . 1 7 ) ,
F±
respec-
Concluding Remarks 1°
For the proof of the unique continuation theorem (Proposition
1.4) we r e f e r r e d to the unique continuation theorem f o r p a r t i a l differential
equation.
On the other hand J~ger 131 gives the unique
continuation theorem f o r an a b s t r a c t o r d i n a r y d i f [ e r e n t i a l
operator
with operator-valued c o e f f i c i e n t s which can be r e f e r r e d to in our case. 2°
Our proof of the l i m i t i n g absorption p r i n c i p l e was along
the l i n e of Saito [3J.
In Ikebe-Saito l l J
SchrSdinger operator was d i r e c t l y
and Lavine I I I
the
treated and the l i m i t i n g absorption
p r i n c i p l e f o r the Schr~dinger operator with a long-range p o t e n t i a l was proved. 3° modifier
In 15 we introduced the kernel ~(y,k)
(13.1)
(13.2)
4°
- Z(y) 2 - g ( y ) }
Z(y) 2 + ~(y) = (grad ~)2 ,
:
O(Jy! -j-F)
(j=O,l,
?>l)
we can r e w r i t e (13.1) as
~ , ..~ Dj {2k ~--F.,yT-qo(y)- (grad X) 2} = O(;yJ -J-~-)
(j=O,l,
(;>I)
Our proof of Theorems 5.3 and 5.4 is a u n i f i c a t i o n of the ones
in Saito 13I and 161. J~ger [3]
of the s t a t i o n a r y
as a s o l u t i o n of the equation
Dj {2kZ(y) - ~ ( y )
Noting t h a t
Z(y,k)
The method of the proof has i t s o r i g i n in
in which he treated
an d i f f e r e n t i a l
valued, short-range c o e f f i c i e n t s .
operator with operator-
136
5°
In ~5 we defined the s t a t i o n a r y m o d i f i e r by a s o r t of
successive approximation method.
The condition
(.0_0) was assumed
so that the successive approximation process may be e f f e c t i v e . As a r e s u l t our long-range p o t e n t i a l the estimates f o r the d e r i v a t i v e s a r a t h e r large number,
.O.o(y)
is assumed to s a t i s f y
DJ,.O.O, j = O , l , 2 . . . . . m and
m is
llere i t should be mentioned t h a t our proof of
Theorems 5.3 and 5.4 is e f f e c t i v e as f a r as we can construct a s t a t i o n a r y m o d i f i e r which s a t i s f i e s rll
(1) of Remark 5.9.
constructed the time-dependent m o d i f i e r
W(y,t)
H~rmander
f o r a type of e l l i p t i c
operators with more general long-range c o e f f i c i e n t s than
ours.
Kitada
r21 showed that a s t a t i o n a r y m o d i f i e r s a t i s f y i n g (1) of Remark 5.g can be constructed by s t a r t i n g with H[irmander's time-dependent m o d i f i e r . I f we use Kitada's s t a t i o n a r y m o d i f i e r we can replace (0.0) .
There e x i s t constants
Qo(y)
is a
C4
Dj
(j=1,2,3)
0 < e. < 1
by
such that
function and
IDJQo(y)I < C o ( l + l y l ) - d ( j ) where
CO and
(Q~)
(y~RN,j=O,I,2,3,4)
denote an a r b i t r a r y d e r i v a t i v e of
j-th
order and
d ( j ) = J+~o
, d(4) > 0 , d(1) + d(4) ". 5 .
But in t h i s l e c t u r e I adopted the p r i m i t i v e method of successive approximation,
because we need f u r t h e r preparations with respect to the
theory of p a r t i a l d i f f e r e n t i a l
equation in order to introduce the more
minute method which s t a r t s with H~rmander I l l . 6°
Theorem 8.7 was f i r s t
stated and proved e x p l i c i t l y
by
Ki tada [?].
7°
Ikebe 141 gave the proof of the orthogonality of the generalized
Fourier transforms by treating the Schrbdinger operator directly and making use of the Lippmann-Schwinger equation.
Our proof of the
137 orthogonality of F±
is d i f f e r e n t
in using Proposition 8.5 instead
of the Lippmann-Schwinger equation. 8°
The modified wave operators.
wave operators potential Matveev
WD, +
f o r the SchrSdinger operator with a long-range
we defined by Alsholm-Kato I I I ,
[I]
Alsholm [ I I
and Buslaev-
as WD ,+ = s-lim eitHe-itHo - i X t ,
(13.3)
-
where
The time-dependent modified
Xt
t+~
is a function of
H0 .
On the other hand from the viewpoint
of the stationary method the s t a t i o n a r y wave operator
WD,~
should
be defined by (13.4) FO,~
WD, t = F
FO, t
,
being the generalized Fourier transforms associated with
H0 .
From the orthogonality of the generalized Fourier transforms the completeness of WD,+ = WD,~ [lJ,
WD,~
follows immediately.
is shown by Kitada [ l ] ,
Recently the r e l a t i o n
[21, [3]
and Ikebe-lsozaki
whence follows the completeness of the time-independent modified
wave operator 9°
WD,+ .
In §12 we treated the case that
range p o t e n t i a l . where
Qln(Y)
Then we approximated has compact support in
Ql(y ) is a general short~(y)
RN .
by a sequence ~o I (y)} But there is another
method which s t a r t s with the relations (13.5) where
(L-k2) -l =(Ll-k2) -l Ll = -
d2 dr 2
+ B(r) + Co(r )
1
{I-C l(L-k2) - I } is the i d e n t i t y operator and
138
v = (L-k2)-If f(k)
means the r a d i a t i v e
function for
{L,k,~[fl}
.
Then
can be defined by
(13.6)
F(k) : F l ( k ) { l - C l ( L - k 2 ) - l }
This method was adopted in Ikebe 131. I0 ° transforms
In Theorem 11.9 we introduced the generalized Fourier F~ associated with
H.
Let
Fourier transforms associated with of -A .
FO, +
H0 ,
In t h i s case the Green kernel
be the generalized
the s e l f - a d j o i n t Go(r,s,k)
realizatior
can be represented
by the use of the Hankel f u n c t i o n and the exact form of
FO, +
is
known as N
(13.7)
in
(Fo,+~)(~) = CI(N)(2~) 2 R-~l'i'm" j I
L2(RN,d~)
where -'I
(13.8)
CI(N) = -e+¼(N-l)i
For proof see Saito [2I, ~7.
This means that
the usual Fourier transforms and the
f:+
"F are essentially 0,_+ are the generalization of
the usual Fourier transforms in t h i s sense. II °
As was stated in the I n t r o d u c t i o n , Q~(y)
any assumption.
As f o r the SchrSdinger operator with an o s c i l l a t i n g
12 °
Finally
%(y)
= ~-F sinlyJ
long-
range p o t e n t i a l
long-range p o t e n t i a l
such as
an o s c i l l a t i n g
does not s a t i s f y
we can r e f e r to Mochizuki-Uchiyama I I ] ,
121.
l e t us qive two remarks on the condition
(5.2)
in Assumption 5.1 on a long-range p o t e n t i a l "
~ -
In order to give 0
"
139 a u n i f i e d treatment f o r in the case of
0 < e < 1 we assumed the c o n d i t i o n
½ < e C 1 we can adopt a weaker c o n d i t i o n
(5.2).
But
m0 = 2.
In f a c t in t h i s case we have Z(y) = 2-~ sr QO( t ~ ) d t 0
(13.9) and the f i r s t (13.10)
(Y = r ~ ) ,
c o n d i t i o n of (5.41) can be weakened as
l(DJZ)(y)]
~
C(l+lyl) -j-~
(j=0,1,2),
because the c o n d i t i o n
(13.11)
ID3Z(y)I ~ C(l+lYI)-3-E
is used to estimate the term
((Z
Lemma 6.5 only and we can d i r e c t l y use of (13.11) in the case of
+ P)u,Bu) x
in the proof of
estimate t h i s term w i t h o u t making
~1 < c ~ 1
Thus (5.2) can be replaced
by (13.12)
(0 < c <_
and
Next l e t us consider the case the
Qo(y)
i.e.
m0 > ~ -
1
Qo(y) = Q o ( l y l )
is also s p h e r i c a l l y ~(y;~)
,
.
(
< ~ <= I)
is s p h e r i c a l l y
symmetric and the operator
P(y;~) are a l l
identically
Moreover (5.35),
zero.
M ,
.
symmetric,
In t h i s case the s t a t i o n a r y m o d i f i e r
Theorem 5.4 becomes much simpler. Lemma 6.6.
m0 : 2
Z(y)
the functions
Therefore the proof o f
For example, we do not need
by which
Z(y) is defined, takes the
f o l l o w i n g simpler form:
i
(13.13) and we set
Z(y) = Zn0
Zl (Y) = 2 T 1 Qo(y ) , Zn+l(y) = 2~{0.o(y) + (Zn(y)) 2} , where
n O is the l e a s t i n t e g e r such t h a t
IzO
(n O + l ) c > 1 .
Noting t h a t the right-hand side of (13.13) does not
contain any d e r i v a t i v e of
Zn(y) ,
(13.14)
(if
m0 = 2
we can replace (5.2) by
Qo(y) i s s p h e r i c a l l y symmetric).
REFERENCES S. Agmon [I]
Spectral properties of Schr~dinger operators and scattering theory, Ann. Scuola Nor. Sup. Pisa (4) 2 (1975), 151-218.
P.K. Alsholm [I]
Wave operators for long-range scattering, Thesis, UC Berkeley (1972).
P.K. Alsholm and T. Kato [I]
Scattering with long-range potentials, Proc. Symp. Pure Math. Vol X I I I (1973), 393-399.
N. Aronszajn
[l]
A unique continuation theorem for solutions of e l l i p t i c partial d i f f e r e n t i a l equations or inequalities of second order, J. de Math. 36 (1957), 235-247.
E. Buslaev and V. B. Matveev [I]
Wave operators for the Schr~dinger equations, with a slowly decreasing potential, Theoret. and Math. Phys. 2(1970), 266-274.
J.D. Dollard
[l]
Asymptotic convergence and the Coulomb interaction, J. Mathematical Phys. 5 (1964), 729-738.
A. Erd~lyi and others [I]
Higher transcendental function I I , McGraw-Hill, New York, 1953.
L. H~rmander [I]
The existence of wave operators in scattering theory; Z. 149(1976), 69-91.
Math.
T. Ikebe
[I]
Eigenfunction expansions associated with the Schr~dinger operators and t h e i r application to scattering theory, Arch. Rational Mech. Anal. 5(1960), 1-34.
[2]
Spectral representations for the Schr~dinger operators with long-range potentials, J. Functional Analysis 20 (1975), 158-177.
[3]
Spectral representations for the Schr~dinger operators with long-range potentials, I I , Publ. Res. Inst. Math. Sci. Kyoto Univ. I I (1976), 551-558.
142
[4]
Remarks on the orthogonality of eigenfunctions for the Schr~dinger operator in Rn J. Fac. Sci. Univ. Tokyo 17 (1970), 355-361.
T. Ikebe and H. Isozaki ~]
Completeness of modified wave operators for long-range potentials, Preprint.
T. Ikebe and T. Kato [l]
Uniqueness of the s e l f - a d j o i n t extension of singular e l l i p t i c d i f f e r e n t i a l operators, Arch. Ratinal Mech. Anal. 9 (1962), 77-92.
T. Ikebe and Y. Sait~ [I]
Limiting absorption method and absolute continuity for the Schr~dinger operator, J. Math. Kyoto Univ. 7(1972), 513-542.
W. J~ger
[I]
Uber das Dirichletsche Aussenraumproblem fur die Schwingungsgleichung, Math. Z. 95 (1967), 299-323.
[2]
Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Aussengebieten, Math. Z. 102 (1967). 62-88.
[3]
Das asymptotische Verhalten von L~sungen eines typus von D i f f e r e n t i a l gleichungen, Math. Z. 112 (1969), 26-36.
[4]
Ein gew~hnlicher Differentialoperator zweiter Ordnung fur Funktionen mit Werten in einem Hilbertraum, Math. Z. 113 (1970) 68-98.
T. Kato [I]
Perturbation Theory for Linear Operators, Second Edition, Springer, Berlin, 1976.
H. Kitada [I]
On the completeness of modified wave operators, Proc. Japan. Acad. 52 (1976), 409-412.
[2]
Scattering theory for Schr~dinger operators with long-range potentials, I, J. Math. Soc. Japan 29 (1977), 665-691.
[3]
Scattering theory for Schr~dinger operators with long-range potentials, I I , J. Math. Soc. Japan 30 (1978)9 603-632.
143 S. T. Kuroda [1]
Construction of eigenfunction expansions by the perturbation method and i t s application to n-dimensional Schr~dinger operators, MRC Technical Summary Report No. 744 (1967).
R. Lavine
[i]
Absolute c o n t i n u i t y of positive spectrum for Schr~dinger operators with long-range potentials, J. Functional Analysis 12 (1973), 30-54.
K. Mochizuki and J. Uchiyama
[i]
Radiation conditions and spectral theory for 2-body Schr~dinger operators with " o s c i l l a t o r y " long-range potentials, I , the principle of l i m i t i n g absorption, to appear in J. Math. Kyoto Univ.
[2]
Radiation conditions and spectral theory for 2-body Schr~dinger operators with " o s c i l l a t o r y " long-range potentials, I I , spectral representation, to appear in J. Math. Kyoto Univ.
Y. Saito
[l]
The principle of l i m i t i n g absorption for second-order d i f f e r e n t i a equations with operator-valued coefficients, Publ. Res. Inst. Math. Sci. Kyoto Univ. 7 (1971/72), 581-619.
[2]
Spectral and scattering theory for second-order d i f f e r e n t i a l operators with operator-valued c o e f f i c i e n t s , Osaka J. Math. 9(1972), 463-498.
[3]
Spectral theory for second-order d i f f e r e n t i a l operators with long-range operator-valued coefficients, I, Japan J. Math. New Ser. 1 (1975), 311-349.
[4]
Spectral theory for second-order d i f f e r e n t i a l operators with long-range operator-valued c o e f f i c i e n t s , I I , Japan J. Math. New Ser. 1 (1975), 351-382.
[5]
Eigenfunction expansions for d i f f e r e n t i a l operators with operator-valued coefficients and t h e i r applications to the Schr~dinger operators with long-range potentials, I n t e r national Symposium on Mathematical Problems in Theoretical Physics. Proceedings 1975, Lecture Notes in Physics 39(1975), 476-482, Springer.
[6]
On the asymptotic behavior of the solutions of the Schr~dinger equation (-A + Q(y)-k2)V = F, Osaka J. Math. 14(1977), 11-35.
144 ~]
Eigenfunction expansions for the Schr~dinger operators with long-range potentials Q(y) = O(lyl-C), E > O, Osaka J. Math. (1977), 37-53.
D. W. Thoe [I]
Eigenfunction expansions associated with Schrbdinger operators in ~n, n ~ 4, Arch. Rational Mech. Anal. 26(1967), 335-356.
K. Yosida [I]
Functional Analysis, Springer, 1964.
NOTATION
INDEX
A, A N
12
mr)
6
B(Y, z)
12
B(x)
12
bj, bj(e)
56
@+
I0
C(r)
6, 21
Co(r), Cl(r)
21
c(~,
B,
..o
)
12
Co(J, x)
I0
Cac(J, X)
II
D
12 12
D(J)
18
p(w)
12
E(o, M)
117 119
146
f_+
125
F(~:)
87
F, (k)
119
F,(~)
125
F(k, 2)
54
F~(a, X), F(J, X)
ll
O(r, s, k)
101
H
ll6
El, 0, Bs(J, X), HI'B(J, X)
l0
H 1,B(l, X) loc
12
Hj(~~)
12
I,
l0
L
6
L2, s(J, X), L2(J , X)
l0
L2(I, X)Io c
12
147
M
57, 117
m O, m I
53
Mj
56
P(y)
57
T
1
U v(.,
k, ~)
23
WD, +~ WD~ +
137
X
5, I0
Y(y, k)
62
z(y, k)
55, 62
ak(r), ~k(r)
80
~(r, k), ~*(r, k)
106
~e(r, k), ~±(r, k)
ll9
~±(r, k) , ~*(r, k)
125
148
12
~. (y, k)
55, 62
~(y, k)
54
~(y~ ~)
57
SUBJECT INDEX
Eigenoperator
I06
Generalized Fourier transform
4, 119
Green formula
96
Green kernel
lO1
Interior estimate
1.6
Laplace-Beltrami
operator
6, 56
Limiting absorption principle
8, 22
Long-range potential
4
Ortho~onality
123
Polar coordinates
6, 56
Radiative function
22
Regularity theorem
17
Schr~dinger operator
I
Short-range potential
4
Spectral representation theorem
121
Stationary modifier
9, 56
Stationary wave operator
137
Time-dependent modifier
136
Time-dependent modified wave operator
137
Unique continuation theorem
19
