Chapter VI Inverse Scattering Problem for Time-Dependent Potentials

225

CHAPTER VI INVERS E SCATTER1N G PROBLEM FOR TI M E- DEP EN DENT POTE NTlALS

This chapter is devoted t o the inverse scattering problem for time-dependent potentials q ( t , x) which are not periodic in time. Moreover, q ( t , z) might increase polynomially as t -+ 00. In this situation the local energy can increase too, and, in general, the scattering operator S does not exist. For this reason we introduce t h e generalized scattering kernel K # in Section 6.1. If S exists, then K # coincides with the kernel o f the operator S - Id in t h e translation representation o f Uo(t). We obtain a representation of

I<# involving a solution w ( t , x ; s , w ) of for large negative time equal t o

(0

+ q)w

= 0 with initial d a t a

~5(”-~)/’(t + s- < x , w >).

In Section 6.2

w modulo smooth terms. The uniqueness of t h e inverse scattering problem is established in Section 6.3. Here we expose we construct a parametrix for

a procedure for recovering a stationary potential from the scattering data. Finally, we discuss a link between the inverse scattering problem and t h e asymptotic o f the scattering amplitude .(A,

0 , w ) as

1x1

+ m.

6.1. Generalized scattering kernel Throughout this section we assume n

2 3, n odd,

and we use the notati-

ons o f Chapter V. Let q ( t , x) be a potential satisfying the conditions (a) and

(b) of assumption (H,) in Section 5.1. For such potentials the scattering operator S does noet exist in general, and we cannot define the scattering kernel.

Nevertheless, we introduce a natural generalization o f this kernel

called generalized scattering kernel. For this purpose we first obtain a link between t h e asymptotic wave profiles of the outgoing solutions o f (5.1.1) and those o f the solutions of t h e Cauchy problem for 0 . Introduce the function

Inverse scat t ering problem for time-dependent pot entials

226

and for s 6 R,w

E S"-* consider t h e solution P ( t ,x ; s , w ) of t h e problem

+ q)r+= o ,

{ r+

(6.1.l)

(0

= hl(t

+

3-

< x , w >) .

For the initial data in (6.1.1) we find

+

(hl,Y)(t

5-

< x , w >) E C ( R t x R,x

s : l ;

HAW),

Y ( t ) being the Heaviside function. As we have mentioned in Section 5.1, this guarantees t h e existence o f a solution I?+ with

Define

u + ( t , x ; s,w) = r + ( t , x ; s , w ) - hl(t

+ s-

us+c(t,z;s , w ) = dp+l)/*u+(t,x;s , w )

< x , w >) ,

.

Below we show that utc is closely related t o the kernel of the scattering operator S.

E Ho have translation representation ( R , q ) ( s , w ) E C,"(B x Sn-')with k = 0 for Is1 > a . Then Uo(-t)cp E 0: for t > a + p. We fix t o > a p and set g = U ( 0 , -to)Uo(-to)'p. Then we find Let

'p

+

lim IIU(-t,O)g - Uo(-t)'pII =

t-w

t , it easily follows t h a t g = W-'p. In the general case, we preserve this notation and set g = W-cp. Clearly, W-'p does not depend on t h e choice of to. Introduce In t h e case that q ( t , x ) is periodic with respect t o

(vo(t,x), atvo(t,4) = Uo(t)'p , (v+(t,21, &v+(t,4) = U ( t ,o w - ' p

-

Generalized scattering kernel

227

With the notation vo#(s,w) for the asymptotic wave profile of v o ( t , z ) , we have t h e following. Lemma 6.1.1.

Let cp E (6.1.3)

Ho be such that R,cp E Cr(R x

9-l).

Then

v + ( t , z ) = vo(t,z) - ( 2 ~ ) - ( " - ' ) / ~ .

. J J u+(t,2 ; s, w)a;"+')/2v,#(s, w)dsdw , where the integral must be interpreted in the sense of distributions.

Proof. Denote by V+(t,z)the right-hand side of (6.1.3).

Since u+ = 0 for

< -t - p and v,# = 0 for Is( > a and some a > 0, we obtain V+ = vo for t < -a - p. By Proposition 2.3.2 we have s

/

vo(t,z) = -(27r)-("-')/2

dp-3)/2"0#(< 2 , w

> -t,w)dw

sn-1

Then

V+(t,z) = vo(t, z) + (2r)-("-')/2

-

Consequently, V+(t,z)is a solution t < -a - p. On the other hand,

to > a

+ p.

Thus

s-

< 2 , w >) -

r+(t,z ; s , w ) ] ap+1)/2v,#(s,w)dsdw =

= -(2n)-("-')/2

for

// [ h l ( t +

JJ

r+(t, ; 9, w)ajn+1)/2v,#(s, w)dsdw.

of (5.1.1) with initial data

V+ = vo for

V+ and v+ are solutions of (5.1.1) with equal initial

data, hence V+= v+. The proof is complete. 0

The function u+ is a solution of the equation

Inverse scattering problem for time-dependent potentials

228

~ ~ + (s ,tw ,) ~ = -; q ( t , q + ( t , x ; s , w ) E

E Since u+ = 0 for

t <

c(w,x s;-1

; L;o,(& ; L y R : ) )

.

-s - p,

u+ is obviously outgoing. So by Theorem 2.6.6, we conclude that ut has an asymptotic wave profile u f ( s ’ , 0 ; s , w ) E C(R,x Sz-’; L,”,,(Rnz,, x Sg-l)). Taking the asymptotic wave profiles in (6.1.3), we deduce (6.1.4)

w+#(s‘, 0 ) = v,f(s’, 0 ) -

- (2

JJ u f ( s ’ , 0 ; s,w)d~+’)/’v,#(s,w)dsdw .

-(n-1)/2

Now assume t h a t t h e scattering operator S exists. As in Section 4.1 set

s = R, SR,’

.

Then S is a bounded operator from

L2(Rx Sn-l) into L 2 ( Rx Sn-l).The

relation

shows t h a t t h e asymptotic wave profile v+#(s‘,0) coincides with t h e asymptotic wave profile o f presentation

R,Sq

(Uo(t)Sp)l. The latter differs from the translation re-

only by the factor -2-’/2(-i)(n-’)/2.

So we can rewrite

(6.1.4) in t h e form (6.1.5)

(.?1vo#)(s’, 0 ) = vo#(s’, 0) t

where

and the integral in (6.1.5) is taken in t h e sense of distributions. This observation shows that the Schwartz kernel of t h e operator

3 - I d coincides with

the distribution K # ( s ‘ , O ; s , w ) . On the other hand, we may introduce I<# even if S does not exist. Moreover, we have the equality

Generalized scattering kernel

01

o+#(s/,

= w,j+(s1,0>

229

+ JJ

o ; s,w)vf(s,w)dsdw.

~#(s',

For this reason we introduce t h e following.

Deflnltlon. Assume q ( t , x ) satisfies the conditions (a) and (b) of Section 5.1. Then t h e distribution

K # , given by (6.1.6), is called the generalized scattering kernel

of problem (5.1.1).

Roughly speaking, I@ is t h e asymptotic wave profile of u : = d!ntl)/z~t.

But we cannot directly apply Theorem 2.6.6 to u;,. Next we obtain a representation for I<#. Given cp(s',s) E

C,"(R2),we

have (6.1.7)

// K # ( s ' , O ;

s , w ) ( P ( s ' , s ) ~ s '= ~s

1)/2

.

s , w ) . All integrals are taken in the sense of distributions. In the second equality of (6.1.7) we have used Proposition 2.6.8 and the fact t h a t Out = -qI'+. Thus we have proved the following. Theorem 6.1.5

The generalized scattering kernel

I<#

K#(s',0 ; s , w ) = - d i

JJ

(6.1.8)

admits the representation

d p - 3 ) / 2 q ( tx)w(t, , x ; s,w) .

. d::-')/%(t

+

SI-

< x,0 > ) d x d t ,

Inverse scattering problem for time-dependent potentials

230

where the integral is interpreted as in (6.1.7).

Moreover,

K#

i s a C"

function of 0, w with values in the space of distributions in s', s. Notice that for

1x1 5

p t h e function

has compact support with respect t o t ,

d$'-1)/2d~"t')/2~(< z,0 > - t , s ) 2

and s. So integration with re-

t , z and s in (6.1.7) can be taken in the ordinary sense since q(t, z)r+(t2 , ; s , w ) E C ( R t x Esx SZ-l; L?oc(IR;)).On the other hand, if q ( t , z) E C"(R"+*), a similar argument implies that K# is a C" function

spect t o

of s , 0 , w with values in the space of distributions in s'. a

To study the support of I{#, we need some preparation. First we obtain representation for U ( t , s ) when t and s are bounded.

Lemma 6.1.4.

t

For fixed

E

IR, s E IR and f

.Hpthe series

E

+C 00

(6.1.9) U ( t , s ) = Uo(t - S)

K(t,s)

k=l

with

1 1 t

Vk(t,S)

=

a1

do1

8

a

d

~

...2

8

7'

dUk

8

Uo(t - a i ) & ( ~ i ) ~ o( 00 2i ) ... U O ( 0 k - 1 - 0 k ) Q ( n k ) U o ( 0 k

k 2 1, is convergent

in

- s)f

Hp. Moreover, the convergence is uniform when t

and s run over bounded intervals.

where t

R ~ ( t , s ) f= (-l)N

J

dal

I

Let

CA

,

> 0 and 7 > 0 be constants

...

7-l

daNUo(t

8

for which

- 0 1 ) Q ( o 1 ) ...

Generalized scattering kernel

231

Then

IIRN(t,s)ll 5 CA It - s l N y N / N !5 CA(2Ay)N/1 and the convergence of (6.1.9) follows immediately. 0

Lemma 6.1.5.

We have

Proof. Clearly, the initial data f of I?+

a t the time tl

in the half space l-I's,w = { ( t , x ) : < x , w

< -s - p are supported

> 5 t + s } . We

claim that the

same is true for

W ~ ( t , t ~= ) fUo(t - .~)Q(.~)...&(.N)~O(.N

- t1)f

Indeed, assume ( t o , x O )E s u p p U o ( a ~- tl)f and consider the backward cone

c, = { ( t , Z )

:

Ix - 501 5 t o - t , t < t o }

.

By a finite propagation speed argument the cone Co must intersect the space

l-I's,w. Thus there exists

(i,i) E nS,,n Co which implies

< z0,u > 5 < i , w > Repeating this argument for

+ li - 201 5 c + s + t o - t^ = t o + s .

Q(ak)Uo(ak

- Ok-l)g, k = 2, ..., N

claim is proved. Application of (6.1.9) finishes the proof. 0

Obviously, Lemma 6.1.5 yields

- 1, the

Inverse scattering problem for time-dependent potentials

232

(6.1.11)

c ( ( t , z ) : < z,w > 5 t + s} .

supp w t,x

After this preparation we turn t o the analysis of s u p p K # . Theorem

6.1.6.

For fixed

0,w

we have

Proof. Assume (d,0 ; s , w )

E suppK#.

In view of (6.1.8) there exists

(i,2 )

such that

lil I p ,

(i,2 ) E supp w(t, z ; 5,w ) , i + sf - < i,0 > = 0 .

By (6.1.11) we find

and the proof is complete. 0

In the next section we shall construct a parametrix for possible t o examine t h e behaviour o f

K # as 0

-t

w which makes it

w.

6.2. Construction of a parametrix for w ( t , z ; s , w ) Throughout this section we use freely t h e notations of the previous section and we assume that q ( t , z )

E Cm(Rnt') is a potential satisfying t h e

conditions (a) and (b) of Section 5.1. Introduce the functions

Notice t h a t hk(E) E

Hkc(R).

We shall construct a parametrix u, for w of the form

Construction of a parametrix for w ( t , x ; s , w )

(6.2.1) u , ( t , x ; s , w )

Applying ( 0

233

)+

=S(t+s-

+ q ) to urn we find j=O m

+2 C j=O

where

((at

+

< w,

02

>)Aj)hj-l

7

h-l = 6. Determining A j inductively from the equations

we find

-W

/

0

(6.2.2)

1 Aj+l(t,x,w) = - 2 --oo j=o,l,

Clearly,

((0+ q ) A j ) ( t + n , x

+ a w , w ) d ~,

...,m - 1 ,

A j = 0 for

< x,w > <

urn= 6 ( t + s -

- p , j = 0,1, ..., rn,hence

< x,w >) f o r t < - s - p

Thus we get

It is straightforward t o see that

hj(t

+s -

< x,w >) E C ( R t x ZR, x s y ; H:,,(a:)) ,

j

21

Inverse scattering problem for time-dependent potentials

234

Moreover, the support of ((0 {(t,X)

Since 0

: -p

+ q)A,)h,

is contained in the set

5 < z,w > 5 t +s} .

+ q is a strictly hyperbolic operator,

t h e energy estimates for t h e

solutions of the Cauchy problem (6.2.3) yield the following. ProDosition 6.2.1 We have

w ( ~ , xS;, W )

(6.2.6)

m

=

C

- 6(t + s - < X , W

>) =

A j ( t , ~ , w ) h j (+tS - c X , W >)

j=O

where

R,

E

+ R,(~,x;

S,W)

,

C ( R t x R,x SG-’; Hmt’(Rz)).

In the same way we can show that

R,

has N derivatives with respect t o

t , E , s, w , provided rn is sufficiently large. 6.3. Uniqueness of t h e inverse scattering problem for time- dependent potentials Throughout this section we assume n = 3 and consider potentials q(t, x) satisfying

(i) (ii)

4(t,Z) E

C Y R 4 )9

there exists p > 0 such that q ( t , X) = 0

for 1x1 2 p, t E

(Q)

(iii) there exist C

R,

> 0, N > 0 such that

Iq(t,z)l 5 C ( l +

for all ( t , x ) E R4.

Our aim in this section is t o prove that if the generalized scattering ker-

(Q) coincide, then the potentials coincide, too. W e start with a useful formula for I<#(s’,O; s , w ) . nels of two potentials satisfying

Proposition 6.3.1.

For 0 # w we have

Uniqueness of the inverse scattering problem

K#(s', 0 ; s , w ) =

(6.3.1)

+J

2,O >

q(<

- s',X)u;(<

Proof. Let cp(s',s)

=

+

+

=

I2

Clearly, t h e integrals

R3 as

s' - s.

yields

K#(s', 0 ; S,W)cp(S',S)dS'ds

I1

< 2, 0 - w > =

E C,"(IR2). Since w ( t , z ; s , w ) = usfc(t,x; s , w ) + S ( t

< x , w >), application o f (6.1.8)

JJ

1,

2,0 > - s f , 2 ; s,w)da:

where dS, denotes the measure in the hyperplane

s-

235

Il

and

I2 are taken over

compact sets. In

I* we

regard

a union of planes

IJ

{z : < x , O - w > =

s'-s}

S'ER

and thus write

Il

in the form

On the other hand, according t o Proposition 6.2.1, for fixed w E S2 we have

L ~ o , ( I R t x I R , x I R ~Changingthevariable< ). z,O > 12,we obtain (6.3.1). The proof is complete.

uL(t,z;s,w) E = s' in 0

-

t

=

Inverse scattering problem for time-dependent potentials

236

Now we pass t o the main result o f this chapter. Theorem 6.3.2.

Let qi(t, z), i = 1,2, be two potentials satisfying t h e assumption (Q) and let

.I 0 so t h a t

(6.3.2) Kf(s‘, O ; S , W ) = Kf(s‘, 0 ; S, w )

( w - w o ( < 6 , Is’-s1<6.

for 1 0 - w 0 ( < 6 ,

Then q l ( t , z )= q 2 ( t , 2 ) for all

Proof.

For fixed

( t , z ) E R4.

O # w the distribution

is a smooth function of s with values in t h e space of distributions in s‘. Since u:, = 0 for < z,w > > t s, it is straightforward t o check t h a t

+

M(s’,O; s , w ) = O

for s ’ - s

>plO-w(.

Mi(s’,O; s , w ) , i = 1,2, the functions related t o q ; ( t , x ) , i = 1,2. Since I
d (6.3.3) - ( M i - Mz) = 0 dS’

# w , (0- wo( < 6,(w - u g ( < 6,1s’ - s( < 6. Fix 0 # w so that 10 - woI < E , I w - wo( < E , where 0 < E 5 rnin(:,

for 0 Then

MI = M2 = 0 for

s’

> s + 5/2.

Combining this with

6).

(6.3.3),we arrive

at

10 - w o J < E , Ju-wol < 6 , 1s’ - sI < E . Now fix E > 0. We want t o examine the behaviour of 10, - wI h.I, as 0, -t w , s h -+ s for m -t 00. The functions Mi are not conitnuous in

for

Uniqueness of the inverse scattering problem s’,

0 , s, w. So limm+m 10,

{Om}, {&I.

Take a E S2, < a , w

--wI

237

Mi depends on the choice of the sequences

> = 0 and set

O ( p ) = w cos p

+ a sin p ,

s’(p) = s

+ a! sin p .

Obviously,

-s‘(p)+s

P = [

2

-cr]sinp

and hence,

(6.3.5)

lim lO(p) - wI M,(s’(p),O ( p ) ; s , w ) = )r-0 P#O

=a

Indeed, taking into account representation (6.2.6) for u:~, we conclude t h a t

ut remains bounded for fixed w E S2 as t , z and s run over a compact set. This observation enables us t o deduce the boundedness of the integrals

- s ‘ ( p ) , p ; s,w)dz as p

00.

---f

U L ,denotes ~ the solution related t o q,, i = 1,2. Conse-

Here

quently, from (6.3.4) and (6.3.5) we derive

1

(6.3.6)

(41 - q2)(< z , w

> - s,z)dSx = 0

=Cl

for all s E

R,

(Y

E

R

and all w , a E S2 satisfying

IW-WIJI<&,

By q we denote q1 q ( t , z ) = 0 for

- q2.

=

0.

Our next purpose is t o show that (6.3.6) implies

( t , z ) E R4. Fix s and w so that I w - woI < E . Given a E S2, < a , w > = 0, choose b E S2 so that the triple (w, a , a) forms an orthonormal basis in E3.Setting z = (YU + Pb + yw,from (6.3.6) we find the equality

23 8

Inverse scattering problem for time-dependent potentials

11

(6.3.7)

q(y - s , a u

+ p b + yw)dpdy = 0

for all a E R

.

Define

Qdz) =

J

q(r - 8

+ p b E {x

,

+ 7w)dy , ~

< z , w > = 0). Choosing arbitrary orthogonal unit vectors a , b in t h e plane < x , w > = 0, we conclude that t h e integral of Q s , w ( z )over each straight line in t h e hyperplane < x , w > = 0 vanishes.

where z = a a

:

Now using t h e inverse Radon transform, it is easy t o see that Q S , J z ) = 0 for all z , i.e.

1

(6.3.8)

q(y - s , a a

R,CY

for all s E

E

+ p b + yw)dy = 0 IR, Iw - woI < E and basis in < x , w > = 0.

R,p

forming an orthonormal

E

for all unit vectors a , 2,

The condition (iii) guarantees t h a t q ( t , z ) E J’(R4). Hence t h e Fourier transform

@(T,<)E

Given cp(r,<)E

Cr(R4),we

have

< i(T,E),cp(?-,E)> =

(6.3.9)

= where

J‘(R4)exists.

J q ( t , x) [J e-i*T-i(p(T, < ) d ~ d < dtdx ] ,

< , > denotes t h e action

of

G(T,<)

butions. L e t a , b , w form an orthonormal variables

we obtain

v(T,<)in t h e sense of distribasis in R3.Introducing the new on

Uniqueness of the inverse scattering problem

239

then by (6.3.8) we can conclude that (6.3.10) < ~ ( T , ( ) , V ( T , ( )> = 0

.

To make this argument precise, we must show that the set c3,,,s has a non-empty interior. Consider an orthonormal basis a , b,wo in R3. Introduce a ( p ) = a cos p - w,-, sin p

,

w ( p ) = wo cos p

+ a sin p .

Clearly, for Ipl < ~ / we 2 have Iw(p) - woI < E . The vectors a ( p ) , b, w ( p ) form an orthonormal basis in R3 and by (6.3.8) we find (6.3.11)

J

Q(Y - % 4 P )

+ Pb + V J ( P ) ) d Y= 0

R,/3 E R and Ipl < ~ / 2 . Given a fixed T~ E R,choose toso that

for all s E

R,cr

E

Consider the equation (6.3.12)

= aa(p)

+pb - .~(p)

A simple calculation yields

By the implicit function theorem we determine the functions p ( ~t ,) as solution of (6.3.12) for

0 < E~ being sufficiently small. Clearly, we can choose

€0

t ) ,P(T, t),

CY(T,

so that

Now choose V ( T ,<) E C,"(R4)with suppcp c Co. Consider the integral with respect to (T,<)in (6.3.9) and pass to the new variables

+,

(7,

0, P ( T , E l , P ( 7 , t ) ),

Then in view of (6.3.13) we find

(TrO

E

co

'

Inverse scattering problem for time-dependent potentials

240

with ( ( ~ , a , , b , p= ) aa(p)

+ ,bb - ~

( p ) The . integration in

T,

a,p, p is

over some small neighbourhood of (TO,ao,@O,O).

We wish t o change the order of integration in (6.3.14). For this purpose consider

Next for fixed p , Ipl 2,

< ~ / 2 ,in the

above integral with respect t o

t and

we apply the change of variables

tIf(t,s,p)l = y - 3 ,5 CN(1 z = a'a(p) + Itl’)-’+ p'b* + y f d ( p ). This justifies thefunction change of the order o f integration: Then t h e phase

tr

+

< x , ( ( ~ , a , / ? , p ) >= a ' a + P ' P - s r

does not depend on y. Thus we are in position t o apply (6.3.11) which yields

Uniqueness of the inverse scattering problem

$(T)

=0

for

241

IT - rol > E O ,

(J e-""vt>@(t)d<) dtdx

=

J

=

J ~ " - + c (J q ( t , x M ( t ) d t ) @(<)dt.

q(t,x)&t)

=

$ ( t ) is the Fourier transform of $ with respect t o T and 3"-+( denotes the Fourier transform with respect t o x . Since Q ( < ) is an arbitrary test function with support in {[ : I< - to[< E ~ } we , derive Here

(6.3.15) FX+t

(/ q ( t, x) &t) d t) = 0

On t h e other hand, according t o (ii),

for

I< - (01 < EO .

Inverse scattering problem for time-dependent potentials

242

F(O = ~ L - E

(J q ( t ,z ~ t ) d t )

is an analytic function of

J

I . By (6.3.15)we deduce

q(t,z)q(t)dt = o

for 1x15 p

.

We rewrite this in the form

J Ft,,

q(t, z)d(t)dt =

< 3 t + , q ( t , 5),$(4 > = 0

being the Fourier transform with respect t o

Since 7.0

7

t . This yields

E B is arbitrary, we obtain q ( t , z ) = 0 for all ( t , z ) E R4and t h e

proof is complete. 0

Finally, we discuss briefly the stationary case in which q ( z ) does not depend on

t . First, it is easy t o see t h a t K # ( s ’ , O ; s , w ) depends on - s. In fact, let wO(t, z ; w) be the solution of the problem

the

difference s’

(6.3.16)

(0

+ q(z))wo = 0

WO lt<-,

7

< z,w >)

=qt -

Then we obtain 2U(t,z; s , u ) = w r J ( t + s , z ; w)

and changing the variables in

(6.1.8),we find

‘J ‘J

(6.3.17)K # ( s ’ , O ; s , w ) = 8n2

Thus

q(z)&wO(< z,0

>

+ s - s’,z;

w)dx

K#(s’, 0 ; s , w ) = S#(s‘ - s, 0 , w ) , where

(6.3.18) S#(s,O,w) = 7

q(z)dtwo(< z , 0

is a smooth function of 0 , w E

S2with

8n

-

S,Z;

w)ds

values in t h e space of distributions

6.3.2says that if the scattering to the potentials q ; ( x ) , i = 1,2,satisfy

in s. In this situation Theorem

i = 1,2,related

>

kernels S?,

.

Uniqueness of the inverse scattering problem S,#(S,

243

0 , w ) = S,#(S,0 , w )

for Is1 < 6, 10 - woI

< 6,Iw - uoI < 6,

then q I ( z ) = q2(z).

For stationary potentials q ( z ) there exists a simple procedure for recovering q ( z ) from S#(s,O,w).

Assume that we know the distribution

S # ( s , O , w ) related t o q ( z ) for Is1 < S and all 0 E S2, w E S 2 . Since S#(s,O,w) = d , M ( s , O , u ) , we can determine M ( s , O , w )for Is1 < S

&

from the expression S

< ~ ( s , ~ , w ) , p (>s = )

-ST~

< ~ # ( s , ~ , w-) ,J

p(cr)da

>,

--oo

where 'p E

C,"(lsl < 6). For a E S2, < a,w >= 0 we set

~ ( p=) Q sin p

,

O ( p ) = w cos p

+ a sin p

and define

In physics literature, t h e inverse scattering problem is connected with t h e scattering amplitude. T o discuss this connection, consider t h e case t h a t t h e operator

L = -A

+ q(z), introduced in Section 5.5, has no point spectrum.

Then t h e global energy of the solutions with initial data and for fixed z , w E to

t.

f E Ho

is bounded

S 2 , w o ( t , x ; w ) is a tempered distribution with respect

This implies that for fixed 0 , w E S2, S # ( s , O , w ) is a tempered

distribution with respect t o s. We define the scattering amplitude by

2T .(X,O,w) = - Fs+xS#(s,O,w) ,

zx

.Fs.+x being the Fourier transform with respect t o s. It is easy t o see t h a t

3 , , ~ ( d ~ w ~z( <, >~ - s , z ; w ) ) = -ixewhere v0(X, z ; w ) = Fs',,xwo(s, z ; u)and

iX

Vo(-X,

z ; 0)

,

Inverse scattering problem for time-dependent potentials

244 (A

+ X2 - q(z))vo(X,

Thus

1 (6.3.19) a(X, 0 , w ) = - -

4n

2

1

;W ) = 0

.

e-ix<"o>q(z)vo(-X,

z ; w )dz

.

Setting

U ~ c ( t , z ; w ) = w o ( t , z ; w ) - S (t ),

we obtain

(6.3.20) a(X,O,w) = -

-

!- eixX<"9W-o' q(x)dx 4x 4n

J

e-iX

-

q ( x ) v s c ( - X , z ; w)dz

'

Consider the function u + ( t , z ; w ) = rt(t,x ; 0 , w ) - hl(t - < x , w >) with and hl as in Section 6.1. Omitting the dependence of rt on s and w , we

conclude that u+ is an outgoing solution of the equation

nu+ = -q(z)r+(t,a:j . Conseq uent I y, t

u+ = -

J

( ~ o (t

(0, q ( x ) r + ( T ,x)))

d~ =

-M M

= -

J

( U 0 ( 4 ( 0 ,n(.)r+(t

-

.,4)), do

*

0

Choose p ( X ) E J ( R )and denote by v+(X,x; w ) the Fourier transform of u + ( t , z ; w ) with respect to t . Then (6.3.21)

< v+(X,x; w),p(X) > = < u + ( t , x ;w ) , J e-ixtcp(X)dX > = M o o

=-

J J -03

0

(uo(0)(0, q ( x ) r + ( t- 0,~ 1 ) ) ~.

245

Uniqueness of the inverse scattering problem When z runs over a compact set in

R3,t h e integration with

respect t o c in

(6.3.21) is over a bounded interval. For such z we can change the order of integration and get

v,,(--X,x; w ) =

w) =

-X%+(-X,X;

03

=

J

( ~ o ( o )(0, q(x)eixovo(o, ; w ) ) ) d c

.

0

Repeating t h e argument of t h e proof o f Theorem 2.6.9 with p = -iX, we can transform the latter integral and we obtain a representation o f vSc(-X, z ; w ) by the outgoing Green function

rZixgiven

Thus we see that

in (2.6.12).

vsc( -A, x ; w ) is t h e unique (-iX)-outgoing solution of the equation

(A

+ X2 - q ( x ) ) v s c =

Furthermore,

q(X)e

iX

v,, satisfies t h e outgoing Somrnerfeld condition (2.6.17) and

we can find a link between u(X,O,w) and the asymptotic behaviour of

v,,(X,x; w ) as z = r 0 , r = 1x1 --+

00.

In physics literature t h e term

is called Born approximation. Letting

X(O - w )

-+

1x1

-+

00,

10 - wI

-+

0, so t h a t

- p , we can find from the Born approximation the Fourier

transform i ( p ) . On the other hand, it is possible t o show that the second term in the right-hand side of (6.3.20) goes t o 0 as

1x1

respect t o 0 , w E S2. The reader should consult Saito

-+

00

uniformly with

111 for results in this

direction. Thus the Born approximation determines uniquely t h e potential Q(X).

Inverse scattering problem for time-dependent potentials

246 Notes

The idea t o consider a generalized scattering kernel I<# for scattering by moving obstacles is due t o Cooper and Strauss 121, 141. Lemma 6.1.1

[4]. The representation 6.1.3,and Lemma 6.1.4have been obtained by Stefanov [8]. In Sections 6.2 and 6.3 we closely follow the above work of Stefanov, where Theorem 6.3.2 is proved. This theorem generalizes t h e previous results o f Ferreira and Perla Menzala [l] and Stefanov [5], [6],

is an analogue o f a result of Cooper and Strauss

of

I<#, given

in Theorem

where t h e uniqueness of t h e inverse scattering problem for two potentials

qi(t,z), i = 1 , 2 , has been treated under the assumption q l ( t , z )2 q l ( t , z ) . For stationary potential t h e representation

(6.2.6)has

been obtained and

111, Newton [l], Rose, Cheney and De Facio [l],[2]. A procedure for recovering q ( t , x) from t h e solution of t h e Newton-Marchenko

studied by Morawetz

equation for time-dependent potentials has been proposed by Stefanov 171. Thus he generalizes the previous results for stationary potentials.

The inverse scattering problem for stationary potentials has been examined by many authors. We refer t o t h e books o f Marchenko and Chadan and Sabatier

[3],Levitan [l]

111 for the results and history concerning this pro-

blem. The recent results for t h e multidimensional inverse scattering problem are exposed in t h e important work o f Henkin and Novikov

[l].