scattering results for measure potentials with unbounded support

Mathematical Methods in the Applied Sciences, Vol. 17, 1213-1230 (1994) MOS subject classification: 35 P 25; secondary 35 P 10

Scattering Results for Measure Potentials with Unbounded Support

Richard Ford

California Slate University. Chico. US. A.

Communicated by L. E. Payne

The eigenfunction expansion theorem and its application to the scattering operator and the scattering matrix is extended to Schrodinger operators with measure potentials with unbounded support on R" that are known to generate wave operators that are strongly complete. Analyticity conditions of the eigenfunctions and the scattering matrix are presented.

1. Introduction and results

The powerful abstract techniques for studying wave operators developed by Kuroda [12], Schechter [14] and others have been successfully applied (see [6]) to singular potentials realized as measures, V(dx), on R"(n 2 3) satisfying the following criteria:

and

These abstract methods differ significantly from the more direct approaches developed by Ikebe [9], Agmon [l], Alsholm and Schmidt [2], Thoe [16] and others who construct the generalized eigenfunctions and,use them to study the properties of the wave operators through the associated generalized Fourier transform. While the abstract methods proved very useful in determining the existence and completeness of the wave operators, general results have not been obtained regarding the associated scattering operator and scattering matrix. In [7] and [8] the generalized eigenfunctions were constructed for measure potentials with compact support satisfying (1.1) and (1.2). A significant feature of such measure potentials is that the associated

CCC 01 70-42 14/94/15 121 3- 18 0 1994 by B. G. Teubner Stuttgart-John Wiley & Sons Ltd.

Received 20 December 1993

1214 R. Ford

Hamiltonian must be constructed with form techniques and the resulting operator has a domain not contained in that of the self-adjoint extension of the Laplacian. The associated difficulty in constructing the generalized eigenfunctions was overcome by integrating methods used by Davies [5] and Alsholm and Schmidt [2]. Utilizing the properties of the wave operators the generalized Fourier transform was developed and used to study the scattering matrix. The purpose of this paper is to extend these results to measure potentials with unbounded support satisfying (1.1)-( 1.3). Under these conditions it is known that there is a self-adjoint operator, H I , satisfying

( H ~ u , U ) = (u, H u ) + u(x)v(x)V(dx), UED(HI) , U E D ( H ) , (1.4) s where H is the standard self-adjoint realization of - A in R". It is shown in [6] that the associated wave operators

exist and are strongly complete.

tions that will be used in this paper. For the reader's convenience, we summarize below the notation and some defini-

s n - I is the unit sphere in R" and 9' = L 2 ( S " - '). R(z) = (z - H ) - ' and R , ( z ) = (z - H 1 ) - ' denote the resolvents of H and H1. G(x; K) where K' = z denotes the Green's function for R(z) . E ( i ) and El(,?) denote the spectral resolutions of H and If1. E(B) and E l @ ) denote the spectral projections for the Bore1 set B. Ha, = { ~ E L ' ( R " ) such that ( E , ( I ) f , f ) is absolutely continuous}. We denote I c A when I is bounded and ic A. For a set I c R we set I" = {<ER": 1 < 1 2 ~ I } . ( . , . ) , a denotes the inner product in L'(1"). K = L2(R"; I Vl(dx)) and A is the inclusion map from L2(R") to K. L" = L"(R"; I Vl(dx) + dx) where dx is Lebesgue measure. Q(x) = A [ A R ( t ) ] * : K H K, Q * ( r ) = lim,loQ(r f ai). &(a, c) = { a + b i ( im a < b < c, a e l } . f(5) = f ( r o ) = f (r, o) where r = 151 and o = 5/r are polar co-ordinates. We now state our primary results.

Theorem 1.1. Let V satisfy (1.1)-(1.3) and let H and H I be as above with n 2 3. Then there exists a set A c R whose complement is a closed set of measure 0, and there exist two families of bounded functions c$* (x, <) defined on (R" x A") such that

(1) Each c$? satisfies the Lippmann-Schwinger equation:

d , ( X 9 r ) = e - i X ' 5 + 5 G ( X - y ; ~ l < 1 ) Q i ( Y , 5 ) V ( d Y ) . (1.6)

( 2 ) Each c$+(x, 5 ) is a solution of

( Ic;12Q? 9 ) = (-A49 9 ) + Q(x)rnV(dx) , WECom(~"), (1.7) s where - A is the distributional Laplacian.

Scattering Results for Measure Potentials with Unbounded Support 12 15

(3) (4)

For each I c A, 4+ (x , <) is bounded and continuous on Iw" x I" . Define for r = rw in polar co-ordinates

Thenforfixed w ~ S " - l , e irw.x4(x, r, w) is a C(Rn)n L"(Iw")-oaluedjiunction of r that admits a meromorphic extension into the upper half-plane with poles at most in X = { K E C + ( K ~ E R - A } .

Theorem 1.2. Let the conditions of Theorem (1.1) hold and let #+ and A be as in that theorem. Then the following hold:

(1) The wave operators, W , ( H I , H ) = W , exist and are strongly complete. (2) The maps, F , given by

extend to isometries from Ha, to L 2 and furthermore, for all f E L2

CF, w, f I ( < ) = CFf I ( < ) , (1.10)

where

is the unitary Fourier transform. (3) F f is an isometryfrom L2(Iw") onto Ha, and for any fEC?(R"),

C F r f l ( x ) = (24-"'2 J m f ( r ) d < .

(4) For any I c A,

( 1 . 1 1 )

(1.12)

( E l ( l ) f , g ) = ( F , f , F * g ) f ~ , J9EHac. (1.13)

( 5 ) For all fc D ( H , ) n Ha,

F * H l f ( O = 1512F*f(5). (1.14)

Theorem 1.3. Let V satisfy (1.1)-(1.3) and set r = I<[, w = t / r . Let A c R be as in Theorem 1.1. For any r 2 0 such that r 2 E A there exists an operator, S(r), known as the scattering matrix on Y = L 2 ( S n - l ) and the following holds:

( 1 )

(Sf 1 0 . 7 0) = LWf (r, * 11 (0) (1.15)

for all f ( r o ) = f (r, W)E L2(Rn) and where s = FSF* and S = W : W- is the scattering operator, which exists and is unitary by Theorem 1.2.

(2) S ( r ) is unitary and (9- ' f ) ( r , o) = [S(r)* f (r, .)](a). (3) Set

y ( r ) f (o) = - i r u J ' x f ( ~ ) V(dx) , r > 0 (1.16)

1216 R. Ford

and

(1.17)

Then y(r) and y+(r ) are bounded operatorsfrom K = L2(R": I Vl(dx)) to Y = LZ(Sn- l ) . Define S - ( r ) =-I - S(r). Then S - ( r ) is a compact operator admitting the following representation:

(1.18)

(4) I f V satisfies for some rx > 0

~ e u i x l V(dx) < co (1.19)

then S( r ) admits a meromorphic continuation into C,' = { r + is 10 < s < ci, r E R } with poles at most in x n c,' The results above have been proved for ordinary potentials under various conditions that give rise to Hamiltonians with domains contained in that of If. See [1] and the references therein. The above results have also been obtained in [7 ] and [ 8 ] for potentials with bounded support satisfying (1.1) and (1.2). This paper extends the class of Schrodinger operators that give rise to generalized eigenfunctions and scattering matrices with integral representations to highly singular cases with unbounded support for which the existence and completeness of the wave operators (and therefore the unitarity of the scattering operator) is already established.

We will proceed as follows: First we will prove a general theorem regarding analytic Banach space operators that will be useful later. Next we will construct the generalized eigenfunctions using the Banach space theorem and an extended analytic Fred- holm analysis. In section 3 we provide the Fourier results by constructing the kernel of the perturbed Green's function for distant K and applying uniqueness of analytic continuation. To verify the spectral properties of the perturbed Fourier transform we provide a general theorem utilizing the method developed in [8 ] that can be applied when the existence and completeness of the wave operators is known. Finally, in section 4 we demonstrate the existence and form of the scattering matrix and develop its properties using the perturbed transform and modified 'trace' operators defined by (1.16) and (1.17).

2. The generalized eigenfunctions

We will now prove Theorem 1.1 using the results of [7 ] together with several lemmas which we will provide below. We first define the basic Lippmann-Schwinger operator on L":

T K f ( x ) = JG(x - y ; ic)f(y) V(dy), L" = L"(R": I Vl(dx) + dx). (2.1)

The general idea is to truncate the support of our potential so that the results of [7 ] can be applied. By providing a basic theorem regarding operators in Banach space we

Scattering Results for Measure Potentials with Unbounded Support 12 17

will show that the norm convergence of the truncated Lippmann-Schwinger operators yield our key requirements in constructing the eigenfunctions. The same Banach methods will be effective in extending the analyticity results of [a]. We now provide our analytic Banach operator result, which is a modest extension of similar results found in Kato [lo].

Theorem 2.1. Let D be an open domain in C and let { L;};= and { L, } be collections of compact operators on a Banach space, X , analytic for K E D such that ( 1 - L:)- ' exist as bounded operators on X . Suppose that limN-. ID It L; - L, ( 1 = 0 uniformly for K E D. Exactly one of the following must hold:

( I ) Either (1 - L,)-' exists for no K E D or (2) For ALL K E D, 1 - L, has a bounded inverse.

To prove this theorem we will use the following lemma.

Lemma 2.2. Let S be a compact subset of C and let {L:}:= and {L,} he collections of bounded operators on a Banach space, X , continuous for K E S . Suppose that limN+ID I1 L: - L, 1 1 = 0 uniformly for K E S . Suppose further that (1 - L:)-' and (1 - L,)-' exist as bounded operators on X . Then

lim II(1 - - ( 1 - L , ) - ' I I ~ ( ~ ) = O uniformly for K E S . (2.2) N - I )

This lemma is found (essentially) in Kato [lo] and follows easily from the following standard estimate:

Proof of Theorem 2.1. By the analytic Fredholm theorem 1 - L, has a bounded inverse in D except at isolated poles. Thus, if we let r be an arbitrary element of D we can construct an open neighbourhood around ct upon which (1 - L,)- ' exists, except possibly at u. We let S be any closed contour around u and contained in the open neighbourhood. Note that all of the conditions of lemma (2.2) apply to S. Since ( 1 - L:)-' is analytic in D all of the Laurent coefficients must vanish and so for each n:

(2.4)

Since ( 1 - L,"')- ' converges uniformly on S , all of the above contour integrals and hence the Laurent coefficients must also vanish for (1 - L J '. Thus there can be no

0 poles in D and so (1 - L,)-' exists throughout D.

We will also use the following theorem.

Theorem 2.3. Let I c R be an open set and suppose L, is a collection of compact operators on X analytic in the set Cp = { a + bi I aE R, b > B } . Suppose ( 1 - L,)-' exists for each K E Cp. Then there is an open set, A E [w whose complement has Lebesgue measure zero and (1 - L,)- ' exists for each a + pi such that u E A.

The above theorem is well known in the case f i = 0 (see Kuroda [ l l , p. 1043). For arbitrary ,4 we simply set T, = L,+P and apply the f l = 0 case to T,.

1218 R. Ford

Returning to the construction of the generalized eigenfunctions, we have the following lemma.

Lemma 2.4. Let T, be as above (2.1) and define T ( p , o, K ) for K and p E C and w E S"- by

T ( p , w , ~ ) f(x) = e'P"'"[T,e-'P".(.)f](x) (2.5) Let VN(dx) be the truncated version of the measure, V(dx), i.e. the support of VN is restricted to the sphere of radius N in R". We let T," and T N ( p , o, K ) be dejined as in (2.1) and (2.5) with V replaced by VN. Then the following hold:

(1) T(p , w, K ) and T N ( p , w, K ) are bounded operators on L" =

L"(R"; IVl(dx) + dx) for all KEC', PEF such that 0 < imp < imK (2) Let 1 g R\{O), and let c >O. Then for each jixed p e p , we have

I I T N ( p , o , ~ ) - T ( p , o , ~ ) I I ~ O as N- too uniformlyfor ~ ~ C , ( p , c ) = { a + bi laEI; imp < b < c}. In addition, 11 TN(x, o, K ) - T ( K , w, K ) I I -, 0 as N -t 00 uniformly for K E w o , c)

(3) For each jixed p E@+, T ( p , o, K ) is uniformly continuous for K E CI(p, c) and T ( K , o, K ) is uniformly continuous for KEE~(O, c).

ProoJ It is shown in [8] that (1) and (3) hold for T N ( ~ , w, K ) and T N ( p , o, K ) , and so (2) will imply (3). To prove (2) and the remainder of (1) we consider

IcT(P,w, 4- TN(PdW)l f (x ) l

N = 1; + I 2 .

Using standard estimates on G(x; K ) and condition (1.1) and (1.3) we will show that 1: and 1," are finite and vanish appropriately. Considering 1; we have by Thoe [16],

Ix - Y12-"I VI(dy) = Ilf II&i(N), (2.7) s I : < const. ~ l f l l ~ e ~ ~ p IYI > N ; IX - YI < 1

where

We also have for 1," that

Scattering Results for Measure Potentials with Unbounded Support 1219

where

( X - I VI(dy). (2.10) I c 2 ( N ) = const. lyl > N, Ix - yl > I

By our conditions (1.1) and (1.3) and the proof of Lemma 2 of Alsholm and Schmidt [2] we have that E~ (N) and g 2 ( N ) are finite and vanish uniformly in x as N + 00. Note that estimates (2.7) and (2.9) also hold for p = K . This gives (2) and the balance of (1)

Remark 2.5. Taking p = 0 in Lemma 2.4 shows that the conclusions all hold for T,.

Lemma 2.6. For each fixed p E @+, T ( p, o, K ) is analytic for im K > imp. I n addition, T (K , o, K ) is analytic for K E C +. The proof of the above follows from Lemma 2.4 and the analyticity of the associated kernels. See [7] Lemma 2.4 for details.

completing the proof. 0

Lemma 2.7. For each KEC' and p ~ @ + such that 0 < im p < im K, T ( p , o, K ) is a compact operator on L"(R"; I Vl(dx)). In particular, T ( K , o, K ) is compact for all K E @ + .

Proof Using Lemma 2.4 it is enough to show that T N ( p , o , K) is compact. By condition (1.2) and standard estimates on G(x; K ) (e.g. see [16]) we first choose a1 ~(0, 1) small enough so that

(2.1 1)

and so

leiw'(x-y)G(x - y ; K)I I VNI(d<) < E. (2.12)

We then set e l p "G(x; K )

e'P"'"G(6,/4; K ) otherwise. if 1x1 > 6,/4, (2.13)

Then we claim that G6,(x) is a continuous function decreasing to 0 uniformly on R" and hence it is uniformly continuous. The continuity is obvious and to verify the uniform decay we note that from Thoe [16] again there exists a constant, C, such that for 1x1 > 1,

leipw.xG(x; K ) ~ ~ ~ i m p l x l 1 ~ 1 ( n - 3 ) / 2 ~ - i m ~ l x l ( l - n ) / Z 1x1 \ < C I K I ( ~ - ~ ) / ~ I X J ( ~ - ~ ) / ~ for imK 2 imp. (2.14)

By the uniform continuity of Gal and the fact that the VN(dx)-volume of R" is finite we can choose a d2 so small that

I X - yl < 6 2 IG~,(x - <) - G ~ , ( Y - 01 I VNl(dS) < C. (2.15) 5 We now set B , = {t: Ix - <I < 6,/4} so that

61 I x - yl < - * B x u B , c (5: I x - ( 1 < 61AIy- 51 < 61) 2

1220 R. Ford

< 5 E I I f 1100. (2.16)

0

Lemma 2.8. For all I C E @ + and p E C f such that 0 < imp < im IC, 1 - T ( p , o, IC) and 1 - T ( K , o, K ) have bounded inverses. In addition, there exists an open set, A E R whose complement has Lebesgue measure 0 and 1 - T, has a bounded inverse for each r such that r 2 E A

Proof: First we will show that the result holds for 1 - T(p, w, IC) when imp < imx and K is chosen sufficiently far away from the real and imaginary axis. We then will apply Theorem 2.1 to extend all K such that imp < im IC and im ic2 # 0. Finally, we will apply the Fredholm alternative and Theorem 2.3 to complete the lemma.

Now suppose that p is fixed. Then there exists a constant, M such that if im IC > M and Iretc21 > M then 11 T(p , w, K ) I I < 1. To see this we use the following estimates found in [6]: For any E > 0 there are constants, a1 and a2 such that

We conclude via Ascoli's theorem that T ( p , w, IC) is compact.

sup J IG(x - y; I C ) I I Vl(dx) d E + allrelc21-1/2 (2.17) y x - y l G 1

and

IG(x -- y; I C ) ~ I Vl(dx) < C I ~ I I C I ( " - ~ ) / ~ ~ - ' ~ ~ . (2.18) sup y J; X - Y l > l

Applying these at the appropriate moment below we have

< {eimp(E + allreK21-1/2) + a2~IC~("-3)/2eim~-im~ 1 Ilf /la,.

(2.19)

From this estimate we can see that for E chosen small enough so that eimpe < 1/2, we can then find M so that if im K > M and Ire IC' I > M then 1 1 T ( p , o, K) 1 1 < 1. This assures the existence of [l - T( p, o, IC)] - for such distant K . Note that this argument only applies when imp < im K . It is shown in [8] that 1 - T N ( p , a, IC) has a bounded inverse for all p, IC such that 0 < im p 6 im IC provided im IC' # 0. By Lemmas 2.4,2.6


and 2.7 we can apply Theorem 2.1 with D = @@\{K: imK2 = 0} where fi = imp to conclude that 1 - T(p , w, K) has a bounded inverse for all K such that im K > imp and imK2 # 0. By Theorem 2.3 we can further conclude that our inverse exists for all K = a + pi such that a avoids only a closed set of measure 0. We therefore choose a K such that im K = fl and (1 - T(p , w, K ) ) - exists. We claim that the inverse must also exist when p is replaced by any p' such that imp' = 6 and in particular when p' = K. This will show that (1 - T ( K , o, K ) ) - ' exists for some K E D = @ + \ { K :

i m ~ ' = 0). Since Lemmas 2.4, 2.6 and 2.7 apply to T ( K , o, K) we can again apply Theorem 2.1 to conclude that (1 - T ( K , o, K ) ) - ' exists for all K E D . To verify our claim we let p = a l + pi. Now let p' = z2 + pi and observe that

~ ( p ' , 0, K ) = eiP'o.xTKe -ip'w.x

- ei(az - a l ) w . x ipo.xT - ipw.xe- i (z , - 2 , ) w . x e Ke - = e i ( ~ z - * , ) W . X T ( p , o, K)e - i @ 2 - Z * ) Q . X (2.20)

Suppose that forfrz L" that T(p' , w, K) f = f i If we set

f (4 E L", g(x) = -h - * , )w .x

then (2.20) implies T ( p , o, K)g = g. We conclude that g = 0 hence j-= 0. By the Fredholm alternative 1 - T(p' , o, K) has a bounded inverse and the claim is verified. This completes the first statement in the lemma. Finally, by taking p = 0, Theorem 2.3 assures the existence of an open set, A, such that 1 - T, has a bounded inverse for all r such that r 2 E A. U

Proof of Theorem 1.1. We set d?(x, 5 ) = (1 - T*lcl).-' eic'(')(x) assuring that part ( I ) of the theorem holds. Part (2) follows from (1) and a straightforward calculation that is provided in [7]. Part (3), the verification of the continuity of ++(x, <) in R" x I " for I cc A, is standard and follows [7, p. 6013. To address part (4) we now set Y(x, p, w ) = [(l - T ( p , o, p) ) - T (p , o, p) 1( .)](x), where 1 (.) is the unit constant function OI! R". Note that

Y(X, p, 0) = T ( p , 0, p ) ( ' y ( . , p, 0) + l ( . ) ) (x)

@(x, p, o) = e-iP'o.XIY(x, p , a,) + 13.

(2.21)

(2.22)

For p z E A, p > 0 we have by (2.5) and (2.21) that Y(x, p, o) = eipW.' T,[@(. , p, o)](x). Thus by (2.22) again we have eipw'x@(x, p o ) - 1 = eiP" T, [@( . , p, a) ] (x) and, therefore, Q, satisfies the Lippmann-Schwinger equation:

and so Y is continuous in x by (2.16). We next set

~ , ( x , p , o ) = e-iPw.x + ~ , [ ~ , ( - , p , o ) ] ( x ) . (2.23)

By the invertibility of (1 - T,) for p2 E A, Q, must agree with 4 + (x, p, w ) when p > 0. Similarly, when p < 0, Q, must agree with &(x, - p , -o). By the analyticity of T( p, w, p), Y (x, p, w ) is also analytic except when p2 E R - A. The exceptional points of [ l - T ( p , (0, p)]- T ( p , o, p ) correspond to those of (1 - T ( p , w, p ) ) - ' . By Lem- mas 2.6 and 2.7 we can apply the analytic Fredholm theorem to conclude that the exceptional points of (1 - T ( p , o, p ) ) - are isolated poles. Since our inverse only fails for p2 real these poles must lie along the positive imaginary axis. Thus by (2.22), eipo'XQ,(x, p, w ) is a La-valued function of p, meromorphic for p in d=+ with poles at most at the isolated exceptional points of(1 - T(p , o, p ) ) - ' along the imaginary axis. This completes our proof. U

1222 R. Ford

3. The generalized transform

We now begin our study leading to the proof of Theorem 1.2. We first provide the following lemma.

Lemma 3.1. Suppose that H and H I are selfadjoint operators satisfying (1.4) and assume that the conclusions of Lemmas 2.6 and 2.8 hold. In addition assume that there is an open neighbourhood, D c C+ and a constant aE(0, 1) such that for all K E D ,

sup IG(x - y ; K ) I I Vl(dx) d a. (3.1) Y 5

Then for each ~ E R " and all K such that i m ~ 2 0, K'EC' u A there exists a unique bounded function, b(x, t; K ) satisfving the following:

For im K' > 0 we have

[ F R , ( K ' ) f I ( t ) = ( K 2 - lr12)-1"~)-"'2 j b ( % t; K)f(X)dX. (3.3)

Pro05 Lemma 2.8 allows us to define 4 by

+(x, 5, K ) = [ ( l - T K ) - . l e - ' ( ' ) ' t ] ( x ) .

R ~ ( z ) = R(z) + [AR(F)]*[ l - Q(z)] . 'AR(z) ,

(3.4)

(3.5)

Clearly with this definition (3.2) holds. From (1.4) (see [6]) we have that

where, for z = K', im K > 0, AR(z) and Q(z) are bounded operators given by

(3.7)

By (3.1) it is easy to show that II Q(K') )I < 1 when K E D and so (3.5) can be written as

(3.8)

From this it is shown in [7] that for such K , R , ( z ) has an integral kernel, Gl(x, y; K )

that is in L 1 ( R n ) in either variable, and that

R l ( z ) = R(z) + [ A R ( i ) ] * ( n = O f Q(z).)AR(z), z = K'.

IIGI(.,Y;K)IIL~ d(1 - a) 'IIG(.;K)IILL.

In addition, we have the resolvent equation [14]:

R l ( z ) = R(z) + [ A R , ( t ) ] * A R ( z ) .

Therefore, for K E D, Gl(x, y; K ) satisfies

Gf(x, Y ; K ) = G(x - Y ; K ) + G(x - I ; K ) G I ( L Y; K ) V(dI). s (3.9)

(3.10)


The integrability of G and GI allow us to apply the Fourier transform to obtain

[FG,(x,(*); K ) ] ( l ) = (27t)-"'2(K2 - 1 t 1 2 ) - 1 C - i x ( ' ) c

+ [G(x - ~ ; K ) [ F G I ( L ( . ) ; ~ ) I ( t ) V ( d l ) . J

By uniqueness and (3.2) we see that

(2n)""(K2 - IcIZ)[FGi(x, (-1; K ) l ( t ) = $(X, 5; K ) , VKED. (3.1 1)

This equation shows that (3.3) holds for I C E D . By Lemma 2.6 both sides of (3.3) are analytic for K~ E Q=+. Our lemma follows by uniqueness of analytic continuation. 0

Remark 3.2. We remark that by the results of Section 2 the conditions of Lemma 3.1 hold under our assumptions on V.

Proof of Theorem 1.2. Part (1) of Theorem 1.2 is proved in [6] . Now, since W,, are strongly complete, W, : L 2 ( R n ) I+ Ha, are isometries. It is shown in [6] that there is a set, A' c R whose compliment is a closed set of measure 0 and for each I g A' we have E , ( I ) W,f= W,E(l)ffor allfEL2(R"). We set A" = A n A' where A is from Theorem 1.1 . Then all of the results of Theorem 1.1 and [6] apply using A". We now rename this set as A. We define the generalized eigenfunctions by 4, (x, 5 ) = [ (I - T,,,,)-'e-'(').C](x) where I{J'EA. We form thegeneralized transforms, F,,

by the rule:

F,f(t) = (2n)-"'2 jO*(X, t)f(x)dx, fGC,".

We first will show that for any I e A and f, g E C,"(R") that

(3.12)

(J%(I)W*f,g) = (CFfI(09 CF,d(t))P. (3.13)

To see this we follow the method in [7] and set

ij(t; z) = (2n)-"" $(x, 5 ; K)g(x)dx, K' = z, im K > 0. (3.14)

Then by Remark 3.2 equation (3.3) holds. This together with results from Chapter 8 of Schechter [13], Fubini's theorem, Parceval's identity, and the intertwining relations we provide the following calculation:

5

rb

(3.15)

1224 R. Ford

It is easy to see that by the uniform continuity of (1 - T'J' for t c 2 ~ Z I ( O , c) that J ( 5 ; z) is continuous and uniformly bounded in I" x &(O, c) (see [7]). Thus we can apply the Dominated Convergence theorem to obtain

_ _ _ _

n n h

where

(3.16)

(3.17)

If we note that XI(<2)F = FE(I ) , where Xs is the characteristic function of S c R, then we have for any f and g in C?(R"),

(3.18)

On the other hand, by using the intertwining relations and Parseval's identity, we have,

(3.19)

Since the set of all u e L 2 ( R " ) such that u = FE(I)fwithfE C,"(R") is dense in L2(R") we see that F W f g = F + g for all gEC?(R"). By the completeness of the wave operators and the-unitarity of the Fourier transform we see that F , extend to all of Ha, verifying part (2) of Theorem 1.2. Part (3) of the theorem follows easily from part (2) and the completeness of the wave operators. To see part (4), assume that f; g E Ha, and let {fj} and { g j } be sequences in C;(R") converging in Lz(R" ) to f and g , respectively. Next set uj = W f f j so that by (3.18), we have

(El (1) w* f ; 9) = (FA Fi Slrn = (%(4 z ) F f ; Fi 9) = (FElOf, F , 9).

(El (I )W*f , 9) = ( E m f , w: 9) = ( F E ( I ) f ; FWf 9).

( E ~ ( z ) f ; 9) = lim (El(I)fjy g j ) j

= lim (E l (1) W, uj, g j ) i

= lim (Fuj, F , g j ) p i

= lim (FWrf j , F , g j ) r n j

= ( F , f , F , d r n . (3.20)

Finally, part ( 5 ) follows easily from the intertwining relations:

F , H 1 = FWfH1 == FHW: = 14;I2FWf = 14;lZF+. (3.21)

This completes the proof of Theorem 1.2.

4. The scattering matrix

To conduct our study of the scattering matrix we will make use of modified trace operators similar to those -used by Kuroda [12]. We recall some definitions from the


The necessary tools to prove Theorem 1.3 are provided by the following lemmas and their corollaries.

Lemma 4.1. LRt I C R+ and set K,(x) = G(x; - r ) - G(x; r). Then K,(x) is bounded, unifbrmly continuous for (r, x) E ( I x W), V-integrable, and in K for each r E R \ { 0). In addition, we have the following:

K,(x) = K,(-X) = K-,(x), (4.3)

sup IIKr(x - ( ' ) ) l~L1(Rn:~P '~ (dx) ) < cl(I)* VrEI , (4.4)

SUP l l ~ r ( x - ( . ) ) l l ~ G Cz(I), (4.5)

X

X

where C , ( I ) and C,(I) are constants depending only on I E R\{O} and the dimension, n.

Proot The fact that K,(x) is bounded follows from the following formula given in Alsholm and Schmidt [2]:

(4.6)

Also in [2] we have

(G(x; +r)l < Clr l (n-3)12 1x1 ( 1 - n ) / 2 1x1 2 1, C = const. (4.7)

Thus we can conclude that for r E 1 as above,

IKr(x)l G C(I)(1 + I x ~ ) ( ~ - ~ ) " , (4.8)

where C(1) is a constant depending on I. This together with (1.3) gives (4.4) and (4.5). The uniform continuity follows from (4.6) and (4.8) while the identities (4.3) are given

We now let A to be the inclusion operator, (Af)(x) = j ( x ) mapping from L 2 ( R n ) to

in [2]. 0

K, and set Q(z) = A[AR(Y) ]* . Thus we have that

[Q(z)f l (x) = I G ( x - Y ; K ) ~ ( Y ) V ( ~ Y ) , K' = z. (4.9)

It is shown in [6] that Q(z) is a compact analytic operator, continuous in p\{O} and that [I - Q(z)]-' exists and is uniformly continuous in C,(O, c) for I C A. (Note by intersection we can take the set A from [6] to be the same as that in Lemma 2.8.) We recall standard Green's function estimates [2]:

(4.10)

where C is a constant depending only on n. When fEC,"(Rn) and r > 0 we have by

1226 R. Ford

( 1 . 1 ) and (1.3) that IG(x - y; f r ) f ( y ) l is V-intcgrable and with (4.10) we can apply Lebesgue dominated convergence:

Qi ( r 2 ) f (x) = lim G(x - y: K ) f ( y ) V(dy), K' = r 2 f is & 10 I

(4.1 1 )

and hence the boundary values of Q(z ) also have integral representations. The importance of Q(z ) is seen in the following lemma.

Lemma 4.2. For all r E R+, y(r) is a bounded operator from K to 9, unijormly continuous for r e 1 c R+.

Proof: We have by (4.6), (4.9), (4.1 l), and the definition of K , ( x ) in Lemma 4.1 that for f € C,"(W):

( y ( r ) f ; y ( r ) f ) y =

- = c(r)-

= c(r)- ( [Q - ( r 2 ) - Q - 1 ( r ' ) I f , f )K,

KAY - x ) f ( Y ) f ( 4 V(dJJ) V(dx)

(4.12) J

where c(r) = (i/4n)(r/2n)"-' and where Q i ( r 2 ) = limcio Q(r2 f iE). From this and the fact that Q k ( r 2 ) is bounded for reX\{O} we see that y(r) extends to a bounded operator from K to Y and we have the following estimate:

(4.13)

We complete the lemma by noting that uniform continuity follows from (4.12) and the

Remark 4.3. y(r)* is a bounded integral operator from Y to K , uniformly continuous for r e 1 c R\{O} and enjoying the estimate (4.1 1). Furthermore, y(r)* is given by

(4.14)

Lemma 4.4. Define yN(r ) by (4.1) with V(dx) replaced with VN(dx). Then yN(r ) is bounded and uniformly continuous for r e 1 C R + and we have the following uniform limits in operator norm for r E I c R \ {O};

IIy(r)II d c(r)- '" l lQ-(r2) - Q+(r2)I11/'.

continuity of Q (r'). 0

y ( r ) * f ( x ) = JelroJ "j(o) d o .

lim 11 y(r) - yN(r ) 1 1 = 0. N - a

(4.15)

ProoL The boundedness and continuity are clear from the proof of Lemma 4.2. We verify the norm limit (4.15) by replacing y(r) with [y(r) - yN(r ) ] in (4.12) and apply the result shown in [6] that Q N ( z ) converges in norm to Q(z) uniformly for

We will soon demonstrate that y + (r) enjoy the same properties as y(r). We now define for r such that r z E A;

2 E C,(O, c). 0

7; ( r ) f (x) = 4 2 (x, ro)f (W)do. (4.16) 5---


For K such that the kernel, G,(x, y; K) , of R1(rcZ) exists (see Lemma 3.1) we can define T,(Ic) for f~ L" by Tl(rc)f(x) = Gl(x , y; ~ ) f ( y ) V(dy). Then by (3.10) and the V- integrability of G and G1 we have

(4.17) Tl (JC)f(x) = T K f ( X ) + r, Tl (JC)f(x). From this it is easy to check that

11 - TKI-lf(x) = f ( x ) + J G d x , Y ; K)f(Y)V(dY).

With the same argument we can verify that forfEK,

- Q ( ~ ~ ' ) l - ' f ( x ) = f W + ~ C , ( X , Y ; JC)~(Y)V@Y) .

We can now prove a useful lemma relating y(r)* and y ; (r).

(4.18)

(4.19)

Lemma 4.5. For all r such that r ' E A , y;(r) are bounded operators from Y to K, uniformly continuous for r E I c A and;

~ ' d r ) f ( x ) = C1 - Q + ( r 2 ) 1 - 1 ~ ( r ) * f ( ~ )

(1 - TTr)-' e-i'm'(')(x)f(o)do. = J (4.20)

Proof: We first assume that ~ E Y , and set g(x) = [y(r )* f] (x ) . Next set Ql(z) = A[AR1(5)]* . It follows from (3.5) and the above discussion that Ql(z) is a bounded operator on K. Using (3.9) we can see that 1 + Ql(z) = [l - Q(z)]-' for all Z E @ + u A. We also have by the proof of Lemma 3.1 that for distant K and all geK,

Qi(JC')&) = Gi(x, Y ; K)g(y)V(dy). (4.21) s Thus for such JC and t,b E C,"(R"),

= ( [ 4 ( x , 7-04 - W ( o ) d o , $ . (4.22)

The first and last members of (4.22) are both analytic functions of ICE C+, continuous for all JC' E @ + u A. By analytic continuation we can conclude equality holds for IC' EA. Since II is arbitrary, and $(x, ro ; f r ) f ( o ) d o is well-defined pointwise, it must be in K since it is equivalent (pointwise) to [l - Q* ( r 2 ) ] - ' g EK. Recalling that 4(x, ro ; f r ) = 4+(x, rw) we see from (4.16) that [ y r T ( r ) f ] ( x ) ~ K . Equation (4.20) follows by the continuation of the K-valued arguments in (4.22). Continuity follows

0

Corollary 4.6. For all r E R+ such that r z E A, y* ( r ) are bounded operatorsfrom K to Y, uniformly continuous for r E I C A. Furthermore, we have

~ ; ( r ) = ~ * ( r ) * and ~ + ( r ) = ~ ( r ) ( l - Q+(r'))- ' . (4.23)

)

from Remark 4.3, equation (4.20) and the continuity of Q*(r') (see [6]).

1228 R. Ford

In addition the following limit holds in operator norm, ungormly in I C A;

ProoJ: The first part of (4.23) follows from the preceding lemma and equations (4.2) and (4.16). The second part o f (4.23) follows from (4.20) and the fact that Q(z)* = Q(2). The norm limit (4.24) follows from Lemma 4.4, equation (4.20) and the fact shown in [6] that Q(z)" converges uniformly in operator norm to Q(z ) for ZE Z,(O, c) for I c A. 0

We now define z(r) by

z(r) = c(r)y+(r)y(r)*, where c(r) = (4.25)

We have the following lemma.

Lemma 4.7. Let f E C;(R") and set f-(r, o) = [F- f ] (YO). Then

ProoJ

Then by Fubini's theorem, (4.6), and the proof of Lemma 4.5 we have

g(x) = c(r)(271)-"'2

= ~(r)(271)-"/~ JJ [(l - T',)-'e-irW~(~)](y)eirw'xf(y)dody, S"-l

We now set

(4.28)

then by (4.28), Fubini, the V-integrability of K,(x), and the proof of Lemma 4.5 we have


= lim (2n)-"/z J[(I - T - ~ ) - I ( T ! ~ - T ~ N ) # + ( . , rw)l(y)f(y)dy

= (2n)Fn/' [ ( I - T-r)-'(T-r - T, )#+( . , rw) I (y ) f (~ )d~ . (4.29)

N - 0 3

I The last step is justified by Lebesgue dominated convergence and the estimate,

I(1 - T- r ) - (T!r - TrN) 4 t (. 7 rw) (y)f(y) I < S U P II(1 - T-r)-'(T!r- T7)II II#+(.,rrn)IIrnIf(~)Iy (4.30)

which is integrable by Lemma 2.4. This verifies (4.26) and completes the proof of the lemma. 0

Proofof Theorem I .3. To prove part (1) we set S(r) = 1 - T(r). Then by (4.26) we have for all f~ C," that

N

[.r(r)F-f(r;)](w) = (1 - T-,)-'(7'- , - T r ) # + ( . , rw)(x)f(x)dx

(1 - T-r)-l(T-r - T,)(1 - T,)-'e-irW'(')(x)f(x)dx

I = I = j [#- (x , 10) - #+(x, rw)lf(x)dx

= I [ ( 1 - T-r)-l - (1 -- T , ) - ' ] e - i r W ' ( ' ) ( x ) f ( x ) d x

= F- f(rw) - F+ f(rw). (4.3 1)

This shows that T(r)F- extends to all of H,, and that

F+ f(rcl)) = [ ( l - .r(r))F- f(r, .)I (0). (4.32)

This together with the fact that s = F , Ff gives (1.15). The unitarity of S(r) follows by (1.15) and the continuity of S(r) (Remark 4.3 and Corollary 4.6) which allows us to duplicate the proof of Kuroda [12, p. 5.533. The rest of part (2) is shown by making the appropriate changes in the calculation given in (4.31) as applied to T(r)*. To prove part (3) we set S-(r) = T(r) and apply (4.32) which gives the representation, (1.18). Next we let TN(r) = c(r)y;(r)yN(r)*. It is shown in [7] that zN(r) is a compact integral operator. Then by Lemma 4.4 and Corollary 4.6 we have that r N ( r ) converges to T(r) in norm hence T(r) is compact.

To prove part (4) we assume (1.19) holds so we have

4+(x, rw)eiro"xf(w')dw' V(dx)

= IT(r, w, w')f(w')do',

where

.r(r, w, 0') = c(r) 4 (x, rw) eirW'" V(dx), 5 (4.33)

(4.34)

1230 R. Ford

which exists by (1.19). By inserting the continued functions, O(x, p, w ) from Theorem 1.1 in place of b + (x, 10) we claim that (4.33) continues analytically to C: . To see this we have from (2.22) that

< Ic(p)I j l e -Pw" ['Y(x, p, w ) + 13 ePw"xl 1 Vl(dx)

< Ic(p)I jle-J'"J'x['Y(x,p,w) + 1]eP"""lIVl(dx)

< Ic(p)I IIY(x, p, o) + 1 II eimplxll Vl(dx). (4.35) i Thus by (1.19) r ( p , w, 0') is a well-defined kernel for all p such that imp < c1 and such that p is not an exceptional point of (1 - T ( p , w, p ) ) - I . Since this kernel is analytic in p. 1 - r ( p ) defines a meromorphic continuation of S(r) into C,' with poles at most in

0 X. This completes the proof of Theorem 1.3.

Acknowledgements

The author would like fo thank Dr. Martin Schechter for his numerous suggestions in the preparation of this work. This work has been supported by a California State University Research Award.

References

I .

2.

3.

4.

5.

6. 7. 8. 9.

10. 11.

12.

13. 14. 15.

16.

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scattering results for measure potentials with unbounded support

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