Spectral Representations for Schrdinger Operators with Long-Range Potentials by Yoshimi Saito

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




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 + 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 )

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 -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 )¢) = 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~,

=

(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

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 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~,

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. __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 (~ + . 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-~,)(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~
'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 xdrO 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~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 (½O 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 <x~

and the

functional

Therefore

,

[

As we have seen above,

function

and we have



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

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 { }

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_ '

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 = - . " '

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)

÷

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 = - ,

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)

(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-> 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 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


',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 (~: -~-+ ~,

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 .


~ + # , 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÷ 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 > -

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)

-


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 {