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Small-Energy Asymptotics of the Scattering Matrix for the Matrix

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Small-Energy Asymptotics of the Scattering Matrix for the Matrix JOURNAL OF MATHEMATICAL PHYSICS VOLUME 42, NUMBER 10 OCTOBER 2001 Small-energy asymptotics of the scattering matrix for the matrix Schro¨ dinger equation on the line Tuncay Aktosuna) Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762 Martin Klausb) Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061 Cornelis van der Meec) Department of Mathematics, University of Cagliari, Cagliari, Italy ͑Received 1 June 1999; accepted for publication 10 July 2001͒ The one-dimensional matrix Schro¨dinger equation is considered when the matrix potential is self-adjoint with entries that are integrable and have finite first mo- ments. The small-energy asymptotics of the scattering coefficients are derived, and the continuity of the scattering coefficients at zero energy is established. When the entries of the potential have also finite second moments, some more detailed asymptotic expansions are presented. © 2001 American Institute of Physics. ͓DOI: 10.1063/1.1398059͔ I. INTRODUCTION Consider the matrix Schro¨dinger equation ␺Љ͑k,x͒ϩk2␺͑k,x͒ϭQ͑x͒␺͑k,x͒, x෈R, ͑1.1͒ where x෈R is the spatial coordinate, the prime denotes the derivative with respect to x, k2 is the energy, Q(x)isannϫn self-adjoint matrix potential, i.e., Q(x)†ϭQ(x) with the dagger standing for the matrix conjugate transpose, and ␺(k,x) is either an nϫ1orannϫn matrix function. We ʈ ʈ ͑ ͒ use • to denote the Euclidean norm of a vector or the operator norm of a matrix. Let 1 nϫn у ϫ Lm(R;C ) with m 0 denote the Banach space of all measurable n n matrix functions f for ϩ͉ ͉ mʈ ʈ ϭ 1 which (1 x ) f (x) is integrable on R.Ifn 1, we denote this space by Lm(R). In this paper 1 nϫn we always assume that Q is self-adjoint and belongs to L1(R;C ). Certain results will be ෈ 1 nϫn obtained under the assumption that Q L2(R;C ), but we will clearly indicate when this stronger assumption is needed. We use Cϩ to denote the upper-half complex plane and write Cϩ for CϩഫR. ϫ ͑ ͒ Among the n n solutions of 1.1 are the so-called Jost solution from the left, f 1(k,x), and the Jost solution from the right, f r(k,x), satisfying the asymptotic boundary conditions Ϫikx ͑ ͒ϭ ϩ ͑ ͒ Ϫikx Ј͑ ͒ϭ ϩ ͑ ͒ !ϩϱ ͑ ͒ e f 1 k,x In o 1 and e f l k,x ikIn o 1 , x , 1.2 ikx ͑ ͒ϭ ϩ ͑ ͒ ikx Ј͑ ͒ϭϪ ϩ ͑ ͒ !Ϫϱ ͑ ͒ e f r k,x In o 1 and e f r k,x ikIn o 1 , x , 1.3 where In denotes the identity matrix of order n. The existence of the Jost solutions can be established as in the scalar (nϭ1) case1,2 by using the appropriate integral equations3,4 ͓cf. ͑2.2͒, ͑2.3͒, and Theorem 2.1 in our paper͔. we have 0͖͕گFor each k෈R ͒ϭ ͒ ikxϩ ͒ Ϫikxϩ ͒ !Ϫϱ ͑ ͒ f l͑k,x a1͑k e bl͑k e o͑1 , x , 1.4 a͒Electronic mail: [email protected] b͒Electronic mail: [email protected] c͒Electronic mail: [email protected] 0022-2488/2001/42(10)/4627/26/$18.004627 © 2001 American Institute of Physics Downloaded 07 Oct 2001 to 130.18.24.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp 4628 J. Math. Phys., Vol. 42, No. 10, October 2001 Aktosun, Klaus, and van der Mee ͒ϭ ͒ Ϫikxϩ ͒ ikxϩ ͒ !ϩϱ ͑ ͒ f r͑k,x ar͑k e br͑k e o͑1 , x , 1.5 ϫ where al(k), bl(k), ar(k), and br(k) are some n n matrix functions of k. These matrix functions enter the scattering matrix S(k) defined in ͑2.22͒, and our primary aim is the analysis of the small-k behavior of S(k). The motivation for this paper comes from our interest in the inverse scattering problem for ͑1.1͒, namely the recovery of Q from an appropriate set of data involving the scattering matrix. As is known from the scalar case, it is important to have detailed information about the behavior of S(k) for small k. For example,1,2 this information is used to characterize the scattering data, so as to ensure that the potential Q constructed from the data at hand belongs to a certain class of 1 1 ͑ ͒ Ͼ functions such as L1(R)orL2(R). The inverse scattering problem for 1.1 when n 1 has been considered by several authors,4–10 but we are not aware of any in-depth study of the small-k behavior of S(k). Not even the continuity of the scattering matrix at kϭ0 seems to have been ෈ 1 nϫn ͑ ͒ ϭ established when Q L1(R;C ); for example, in Ref. 6 p. 294 , the continuity at k 0 of the transmission coefficients is assumed. In the scalar case it is well known1,2,11,12 that the continuity ϭ ෈ 1 ෈ 1 of S(k)atk 0 is easy to establish if Q L2(R), but not if only Q L1(R). In the matrix case, the situation is somewhat different. The decay of Q(x)asx!Ϯϱ plays an important role, but there are further complications due to the particular structure of the solution space of ͑1.1͒ at k ϭ0. From the scalar case it is known1,2,11 that the behavior of the solutions of ͑1.1͒ at kϭ0 makes it necessary to distinguish between two cases, the generic case and the exceptional case, and that the small-k behavior of S(k) is different in each case. If nϾ1, the situation is more complicated because the exceptional case gives rise to a variety of possibilities depending on the Jordan structure of a certain matrix associated with the solution space of ͑1.1͒ at kϭ0. In this paper we clarify the connection between the solutions of ͑1.1͒ at kϭ0 and the behavior of S(k) near k ϭ0. As a result, we are able to prove the continuity of the scattering matrix at kϭ0 when Q ෈ 1 nϫn ෈ 1 nϫn L1(R;C ) and to obtain more detailed asymptotic expansions when Q L2(R;C ). The inverse problem is not considered here; we may report on it elsewhere. This paper is organized as follows. In Sec. II we establish our notations and review some basic known results on the solutions of ͑1.1͒. Since this material is standard, we refer the reader to the literature for proofs and more details. In Sec. II we also give various characterizations of the generic and exceptional cases. In Sec. III we prove the continuity of the scattering matrix at k ϭ0 in the generic case, and we obtain some more detailed asymptotic results when Q ෈ 1 nϫn L2(R;C ). The exceptional case is treated in Sec. IV; the main results are contained in ෈ 1 nϫn ෈ 1 nϫn Theorem 4.6 when Q L1(R;C ) and in Theorem 4.7 when Q L2(R;C ), where we prove the continuity and differentiability of S(k)atkϭ0, respectively. In Sec. V we discuss some special cases that illustrate the results of Sec. IV. Finally, the Appendix contains the proof of Proposition 4.2, which is a key result needed to establish Theorems 4.6 and 4.7. II. SCATTERING COEFFICIENTS AND A CASE DISTINCTION In this section we review some basic results about those solutions of ͑1.1͒ that are relevant to scattering theory, and we define the scattering coefficients and some related quantities. We also elaborate on the distinction between the generic case and the exceptional case which will play an important role in the subsequent sections. We define the Faddeev functions ml(k,x) and mr(k,x)by ͒ϭ Ϫikx ͒ ͒ϭ ikx ͒ ͑ ͒ ml͑k,x e f l͑k,x , mr͑k,x e f r͑k,x . 2.1 From ͑1.2͒, ͑1.3͒, and ͑2.1͒ it follows that 1 ϱ ͑ ͒ϭ ϩ ͵ ͓ 2ik͑yϪx͒Ϫ ͔ ͑ ͒ ͑ ͒ ͑ ͒ ml k,x In dy e 1 Q y ml k,y , 2.2 2ik x Downloaded 07 Oct 2001 to 130.18.24.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp J. Math. Phys., Vol. 42, No. 10, October 2001 Small-energy asymptotics 4629 1 x ͑ ͒ϭ ϩ ͵ ͓ 2ik͑xϪy͒Ϫ ͔ ͑ ͒ ͑ ͒ ͑ ͒ mr k,x In dy e 1 Q y mr k,y . 2.3 2ik Ϫϱ Some properties of the matrix functions ml(k,x) and mr(k,x) are summarized in the next theorem and its corollary. The proofs of these results can be obtained as in the scalar case and we refer the reader to the literature;2–4,11 in particular, see Theorem 1.4.1 in Ref. 3 and Theorem 1 in Ref. 4. We denote differentiation with respect to k by an overdot and use C for suitable constants that do not depend on x or k. ෈ 1 nϫn ෈ Theorem 2.1: If Q L1(R;C ), then, for each x R, the functions ml(k,x), mr(k,x), Ј Ј ෈ ϩ ෈ ϩ ml (k,x), and mr (k,x) are analytic in k C and continuous in k C ; moreover ͑ ͒ϭ ϩ ͑ ͒ Ј͑ ͒ϭ ͑ ͒ !ϩϱ ͑ ͒ ml k,x In o 1 , m1 k,x o 1/x , x , 2.4 ͑ ͒ϭ ϩ ͑ ͒ Ј͑ ͒ϭ ͑ ͒ !Ϫϱ mr k,x In o 1 , mr k,x o 1/x , x , ʈ ͒ʈр ϩ Ϫ ʈ ͒ʈр ϩ ෈ ϩ ͑ ͒ ml͑k,x C͓1 max͕0, x͖͔, mr͑k,x C͓1 max͕0,x͖͔, k C .
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