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 moments. 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]

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͒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

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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 . 2.5 ෈ 1 nϫn ϩ In addition, if Q L2(R;C ), then m˙ l(k,x) and m˙ r(k,x) exist, are analytic in C , continuous in Cϩ, and satisfy the estimates

ʈ ͒ʈр ϩ 2͒ ʈ ͒ʈр ϩ 2͒ ෈ ϩ m˙ l͑k,x C͑1 x , m˙ r͑k,x C͑1 x , k C .

In the following an asterisk will be used to denote complex conjugation. From ͑2.1͒ and Theorem 2.1 we get the following. ෈ 1 nϫn ෈ Corollary 2.2: Assume Q L1(R;C ). Then, for each fixed x R, the four matrix functions Ϫ † Ϫ † Ј Ϫ † Ј Ϫ † ෈ ϩ f l( k*,x) , f r( k*,x) , f 1( k*,x) , and f r ( k*,x) are analytic in k C and continuous in ϩ ෈ 1 nϫn C . Moreover, if Q L2(R;C ), then these functions are differentiable with respect to k ෈Cϩ. The scattering coefficients will be defined in terms of certain Wronskians involving the Jost solutions. We first state a standard result about such Wronskians, which is a consequence of the selfadjointness of Q. Let ͓F;G͔ϭFGЈϪFЈG denote the Wronskian of two square matrix functions F(x) and G(x). Proposition 2.3: For k෈C, let ␾(k,x) be any nϫp solution and ␺(k,x) any nϫq solution of (1.1). Then the pϫq Wronskian matrix ͓␾(Ϯk*,x)†;␺(k,x)͔ is independent of x. ͑ ͒ As a result of Proposition 2.3 the matrices al(k), bl(k), ar(k), and br(k) appearing in 1.4 and ͑1.5͒ can be expressed in terms of certain Wronskians of the Jost solutions as follows:

1 2.6͒͑ ,0͖͕گa ͑k͒ϭ ͓ f ͑Ϫk*,x͒†; f ͑k,x͔͒, k෈Cϩ l 2ik r l

1 2.7͒͑ ,0͖͕گa ͑k͒ϭϪ ͓ f ͑Ϫk*,x͒†; f ͑k,x͔͒, k෈Cϩ r 2ik 1 r

1 2.8͒͑ ,0͖͕گb ͑k͒ϭϪ ͓ f ͑k,x͒†; f ͑k,x͔͒, k෈R l 2ik r l

1 2.9͒͑ .0͖͕گb ͑k͒ϭ ͓ f ͑k,x͒†; f ͑k,x͔͒, k෈R r 2ik 1 r

Alternatively, it is sometimes convenient to use the integral representations

1 ϱ ͑ ͒ϭ Ϫ ͵ ͑ ͒ ͑ ͒ ͑ ͒ al k In dx Q x ml k,x , 2.10 2ik Ϫϱ

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1 ϱ ͑ ͒ϭ Ϫ ͵ ͑ ͒ ͑ ͒ ͑ ͒ ar k In dx Q x mr k,x , 2.11 2ik Ϫϱ

1 ϱ ͑ ͒ϭ ͵ 2ikx ͑ ͒ ͑ ͒ ͑ ͒ bl k dx e Q x m1 k,x , 2.12 2ik Ϫϱ

1 ϱ ͑ ͒ϭ ͵ Ϫ2ikx ͑ ͒ ͑ ͒ ͑ ͒ br k dx e Q x mr k,x , 2.13 2ik Ϫϱ

which follow from ͑1.4͒, ͑1.5͒, and ͑2.1͒–͑2.5͒. Also, with the help of ͑1.2͒–͑1.5͒ and ͑2.6͒–͑2.9͒, we obtain

͒ ͑ گϪ ͒†ϭ ͒ ෈ ϩ ar͑ k* al͑k , k C ͕0͖, 2.14

͒ ͑ گ ϭϪ ͒† ෈͒ br͑k bl͑k , k R ͕0͖, 2.15

͒† ͒ϭ ͒† ͒ϩ ෈ \ ͑ ͒ al͑k al͑k bl͑k bl͑k In , k R ͕0͖, 2.16

͒ ͑ گ ϭ ͒† ͒ϩ ෈͒ †͒ ar͑k ar͑k br͑k br͑k In , k R ͕0͖, 2.17

͒ ͑ گ Ϫ ͒† ͒ϭ Ϫ ͒† ͒ ෈ al͑ k bl͑k bl͑ k al͑k , k R ͕0͖, 2.18

͒ ͑ گ Ϫ ͒† ͒ϭ Ϫ ͒† ͒ ෈ ar͑ k br͑k br͑ k ar͑k , k R ͕0͖. 2.19

We define the transmission coefficient from the left, Tl(k), and the transmission coefficient from the right, Tr(k), by ͒ϭ ͒Ϫ1 ͒ϭ ͒Ϫ1 ͑ ͒ Tl͑k al͑k , Tr͑k ar͑k , 2.20

provided the inverses on the right-hand sides exist, and we define the reflection coefficient from the left, L(k), and the reflection coefficient from the right, R(k), by

͒ϭ ͒ ͒Ϫ1 ͒ϭ ͒ ͒Ϫ1 ͑ ͒ L͑k bl͑k al͑k , R͑k br͑k ar͑k . 2.21 ϩ گ ෈ ͒ ͑ ͒ ͑ From 2.16 and 2.17 we see that al(k) and ar(k) are nonsingular for k R ͕0͖.InC , al(k) and ar(k) are nonsingular except possibly at a finite number of points on the positive imaginary 4 ϭ ϭ axis where both det al(k) 0 and det ar(k) 0; at these points, Tl(k) and Tr(k) have simple 8 ͑ ͒ ෈ 1 nϫn poles corresponding to the bound states of 1.1 . For Q L2(R;C ) the finiteness of the number of bound states has already been established in Refs. 4 and 13. We note that even if ¹ 1 nϫn ෈ 1 nϫn ͑ ͒ Q L2(R;C ) but Q L1(R;C ), the finiteness follows from the operator inequality уϪʈ ʈ 1 Q(x) Q(x) In and the fact that in one dimension a scalar potential in L1(R) can support at most a finite number of bound states. Alternatively, the finiteness of the number of bound states will follow from the results of this paper ͑cf. Theorems 3.1 and 4.6͒, which show that kϭ0 cannot be an accumulation point for poles of either Tl(k)orTr(k). Because of this latter property we will study the asymptotic behavior of the transmission coefficients as k→0 through values in Cϩ. The reflection coefficients, on the other hand, in general do not have analytic extensions off the real ϫ axis, so their asymptotics will be studied for real k only. The n n matrix functions Tl(k), Tr(k), R(k), and L(k) are referred to as scattering coefficients, and the 2nϫ2n matrix

T ͑k͒ R͑k͒ ͑ ͒ϭͫ l ͬ ͑ ͒ S k ͒ ͒ , 2.22 L͑k Tr͑k

is called the scattering matrix. From ͑2.15͒–͑2.17͒,weget

گ ϩ ͒† ͒ϭ ෈͒ †͒ Tl͑k R͑k L͑k Tr͑k 0, k R ͕0͖,

͒ ͑ گ ϩ ͒† ͒ϭ ෈͒ †͒ Tl͑k Tl͑k L͑k L͑k In , k R ͕0͖, 2.23

͒ ͑ گ ϩ ͒† ͒ϭ ෈͒ †͒ Tr͑k Tr͑k R͑k R͑k In , k R ͕0͖, 2.24

S(k) is unitary. Using ͑2.14͒ we obtain ,0͖͕گand hence, for k෈R

͒ ͑ گϭ Ϫ ͒† ෈ ϩ͒ Tr͑k Tl͑ k* , k C ͕0͖, 2.25 ͑ ͒ ͑ ͒ whenever ar(k) is nonsingular, and from 2.18 and 2.19 we get

.0͖͕گL͑Ϫk͒†ϭL͑k͒, R͑Ϫk͒†ϭR͑k͒, k෈R

In order to study S(k) in the small-energy limit, we need to make an important case distinction which involves the solutions to ͑1.1͒ with kϭ0, i.e., the solutions to

␾Љ͑x͒ϭQ͑x͒␾͑x͒, x෈R. ͑2.26͒ ͑ ͒ ͑ ͒ We already know from 2.1 and Theorem 2.1 that f l(0,x) is a solution of 2.26 satisfying ϭ ϩ Ј ϭ →ϩϱ f l(0,x) In o(1) and f l (0,x) o(1/x)asx . According to basic asymptotic results for systems of linear differential equations ͑Theorem 1.5.1 of Ref. 3͒, ͑2.26͒ also has an nϫn matrix ␾ solution, l(x), satisfying ␾ ͑ ͒ϭ ϩ ͑ ͒ ␾Ј͑ ͒ϭ ϩ ͑ ͒ →ϩϱ l x xIn o x , l x In o 1 , x . ␾ Thus the columns of f l(0,x) together with the columns of l(x) form a fundamental set of 2n vector solutions for ͑2.26͒. Any vector solution ␾(x)of͑2.26͒ can be written as

␾ ͒ϭ ͒␩ ϩ␾ ͒␩ ͑ ͒ ͑x f l͑0,x 1 l͑x 2 , 2.27 ␩ ␩ ෈ n ␾ ͑ ͒ where 1 , 2 C are uniquely determined by (x). It follows from 2.27 that a vector solution ͑ ͒ →ϩϱ ␩ ϭ ␾ ϭ ␩ of 2.26 is bounded as x if and only if 2 0, i.e., if and only if (x) f l(0,x) 1 for some ␩ ෈ n ␾ ϭ␩ 1 C . Moreover, in this case limx→ϩϱ (x) 1 exists. This means that if a solution is bounded at ϩϱ, then it also has a limit as x→ϩϱ. Also, ͑2.27͒ implies that any solution of ͑2.26͒ that is o(x)asx→ϩϱ is necessarily bounded at xϭϩϱ and any solution that is o(1) as x→ ϩϱ must be the zero solution. Similar results hold at xϭϪϱ; in particular, any solution of ͑2.26͒ that is o(x)asx→Ϫϱ is necessarily bounded at xϭϪϱ and has a limit as x→Ϫϱ, and any solution that is o(1) as x→Ϫϱ must be the zero solution. ͑ ͒ ͑ ͒ From 2.1 – 2.3 we see that f l(0,x) and f r(0,x) obey the integral equations

ϱ ͑ ͒ϭ ϩ ͵ ͑ Ϫ ͒ ͑ ͒ ͑ ͒ ͑ ͒ f l 0,x In dy y x Q y f l 0,y , 2.28 x

x ͑ ͒ϭ Ϫ ͵ ͑ Ϫ ͒ ͑ ͒ ͑ ͒ ͑ ͒ f r 0,x In dy y x Q y f r 0,y . 2.29 Ϫϱ

In the subsequent analysis the two Wronskian matrices

⌬ ϭ ͒† ͒ ⌬ ϭϪ ͒† ͒ ͑ ͒ l ͓ f r͑0,x ; f l͑0,x ͔, r ͓ f l͑0,x ; f r͑0,x ͔, 2.30 ⌬ ⌬ ͑ ͒ will play a key role. By Proposition 2.3, l and r are independent of x, and from 2.30 it follows that

⌬ ϭ⌬† ͑ ͒ l r . 2.31

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The importance of these Wronskians lies in the fact that they are related to the transmission coefﬁcients via ͑2.20͒ and

⌬ ϭ ͒ ⌬ ϭ ͒ ͑ ͒ l lim 2ikal͑k , r lim 2ikar͑k , 2.32 k→0 k→0

where the limits are taken from within Cϩ; ͑2.32͒ follows from ͑2.6͒, ͑2.7͒, and Corollary 2.2. Evaluating the ﬁrst Wronskian in ͑2.30͒ as x→Ϫϱ and using ͑2.28͒ we obtain

ϱ ⌬ ϭ Ј͑ ͒ϭϪ͵ ͑ ͒ ͑ ͒ ͑ ͒ l lim f l 0,x dy Q y f l 0,y . 2.33 x→Ϫϱ Ϫϱ

Similarly, from ͑2.29͒ and ͑2.30͒, letting x→ϩϱ, we get

ϱ ⌬ ϭϪ Ј͑ ͒ϭϪ͵ ͑ ͒ ͑ ͒ ͑ ͒ r lim f r 0,x dy Q y f r 0,y . 2.34 x→ϩϱ Ϫϱ

From ͑2.28͒ and ͑2.29͒ we also infer that

͒ϭ ⌬ ϩ ͒ →Ϫϱ f l͑0,x x l o͑x , x ,

͒ϭϪ ⌬ ϩ ͒ →ϩϱ ͑ ͒ f r͑0,x x r o͑x , x . 2.35

Now we are ready to introduce the distinction between the exceptional case and the generic case. Let

Nϭ ␰෈ n ͒␰ ͑ ͒ ͕ C : f l͑0,x is bounded on R͖. 2.36

Then we say that the generic case occurs if Nϭ͕0͖ and we say that the exceptional case occurs if NÞ͕0͖. These two cases can be characterized in other ways. We choose the above deﬁnition as our starting point and will arrive at some other characterizations as we go along. We observe that the generic case occurs if and only if ͑2.26͒ has no bounded nontrivial solution. The exceptional case occurs if and only if there exists at least one nontrivial bounded solution. As the next theorem shows, we can alternatively characterize the two cases by means of the subspace

Mϭ ␹෈ n ͒␹ ͑ ͒ ͕ C : f r͑0,x is bounded on R͖. 2.37

Then the generic ͑exceptional͒ case occurs if and only if Mϭ͕0͖ (MÞ͕0͖). ϭ We mention that when n 1 the exceptional case occurs if and only if f l(0,x) and f r(0,x) are linearly dependent, i.e., the Wronskian ͓ f r(0,x);f l(0,x)͔ is zero. In our paper we generalize this characterization to the matrix case. In the scalar case it is also known that the generic ͑exceptional͒ ϭ Þ case occurs if Tl(0) 0(Tl(0) 0). This will also turn out be true in the matrix case, but we do not use this property as our primary characterization because it is implicitly based on the assump- ϭ tion that Tl(k) is continuous at k 0, something we ﬁrst need to prove. The next theorem further clariﬁes the relations among the two cases, the Wronskians in ͑2.31͒, and the subspaces N and M. ෈ 1 nϫn Theorem 2.4: Assume Q L1(R;C ). Then we have ͑ ͒ ⌬ ⌬ i The generic case occurs if and only if l , or equivalently r , is nonsingular. ͑ ͒ Nϭ ⌬ Mϭ ⌬ ii Ker l and Ker r . ͑iii͒ dim Nϭdim M. ⌬ ͑ ͒ ͑ ͒ Proof: If l is nonsingular, then 2.35 implies that every solution of 2.26 of the form ␩ ␩෈ n →Ϫϱ ⌬ f l(0,x) , with some nonzero vector C , becomes unbounded as x . Hence, if l is nonsingular, then ͑2.26͒ has no bounded nontrivial solutions; so the generic case occurs. Con-

͑ Nϭ ͒ ⌬ ͑ ͒ versely, suppose the generic case i.e., ͕0͖ occurs and l is singular. Then, by 2.35 , for any ␰෈ ⌬ ␰ϭ →Ϫϱ ͑ ͒ nonzero Ker l , we have f l(0,x) o(x)asx . Hence, by the remarks following 2.26 ͑ ͒ ␰ ␰෈N NÞ and 2.27 , f l(0,x) is bounded, i.e., and thus ͕0͖. This is a contradiction. Therefore, in ⌬ ͑ ͒ ⌬ ͑ ͒ the generic case, l cannot be singular. This proves i for l . In view of 2.31 , the assertion also ⌬ ⌬ ͑ ͒ ␰෈ ⌬ ͑ ͒ ␰ holds if l is replaced by r . To prove ii , suppose that Ker l . Then, by 2.35 , f l(0,x) ϭ →Ϫϱ ␰ ␰෈N ␰෈N ␰ o(x)asx . Hence f l(0,x) is bounded and so . Conversely, if , then f l(0,x) ͑ ͒ ⌬ ␰ϭ ͑ ͒ is bounded and, therefore, again by 2.35 , l 0. This proves the ﬁrst equality in ii . The second equality is proved similarly. Finally, ͑iii͒ follows immediately from ͑2.31͒. ᭿

III. SMALL-k BEHAVIOR IN THE GENERIC CASE In this section we analyze the behavior of the scattering coefﬁcients near kϭ0 in the generic case. In order to state the next theorem, which is the main result of this section, we introduce the matrices

ϱ ϱ ϭ ͵ ͑ ͒ ͑ ͒ ϭ ͵ ͑ ͒ ͑ ͒ ͑ ͒ El dx xQ x ml 0,x , Er dx xQ x mr 0,x , 3.1 Ϫϱ Ϫϱ

ϱ ϱ ϭ ͵ ͑ ͒ ͑ ͒ ϭ ͵ ͑ ͒ ͑ ͒ Gl dx Q x m˙ l 0,x , Gr dx Q x m˙ r 0,x . Ϫϱ Ϫϱ

The quantities El and Er will also play a role in Sec. IV. 1 nϫn ϭ Theorem 3.1: Assume Q is a generic potential in Lm(R;C ) for m 1 or 2. Then the scattering coefﬁcients satisfy the following: ͑i͒ If mϭ1, then ͑ ͒ϭ ⌬Ϫ1ϩ ͑ ͒ ͑ ͒ϭ ⌬Ϫ1ϩ ͑ ͒ → ϩ Tl k 2ik l o k , Tr k 2ik r o k , k 0inC , ͒ϭϪ ϩ ͒ ͒ϭϪ ϩ ͒ → R͑k In o͑1 , L͑k In o͑1 , k 0inR. ͑ii͒ If mϭ2, then ͑ ͒ϭ ⌬Ϫ1ϩ 2⌬Ϫ1͓ ϩ ͔⌬Ϫ1ϩ ͑ 2͒ → ϩ Tl k 2ik l k l 4In 2iGl l o k , k 0inC , ͑ ͒ϭ ⌬Ϫ1ϩ 2⌬Ϫ1͓ ϩ ͔⌬Ϫ1ϩ ͑ 2͒ → ϩ Tr k 2ik r k r 4In 2iGr r o k , k 0inC , ͑ ͒ϭϪ ϩ ͓ ϩ ͔⌬Ϫ1ϩ ͑ ͒ → L k In 2ik In El l o k , k 0inR, ͑ ͒ϭϪ ϩ ͓ Ϫ ͔⌬Ϫ1ϩ ͑ ͒ → R k In 2ik In Er r o k , k 0inR. ⌬ ⌬ ͑ ͒ Proof: Using the fact that in the generic case l and r are invertible, i is a consequence of ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ෈ 1 nϫn 2.6 – 2.9 , 2.20 , 2.21 , 2.32 , and Corollary 2.2. When Q L2(R;C ), expanding the inte- grals in ͑2.10͒–͑2.13͒ as

1 i a ͑k͒ϭ ⌬ ϩI ϩ G ϩo͑1͒, k→0inCϩ, ͑3.2͒ l 2ik l n 2 l

1 i b ͑k͒ϭϪ ⌬ ϩE Ϫ G ϩo͑1͒, k→0inR, ͑3.3͒ l 2ik l l 2 l

1 i a ͑k͒ϭ ⌬ ϩI ϩ G ϩo͑1͒, k→0inCϩ, ͑3.4͒ r 2ik r n 2 r

1 i b ͑k͒ϭϪ ⌬ ϪE Ϫ G ϩo͑1͒, k→0inR, ͑3.5͒ r 2ik r r 2 r

and using ͑2.20͒ and ͑2.21͒ we obtain ͑ii͒. ᭿

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෈ 1 nϫn For later use we remark that when Q L2(R;C ), El and Er can be expressed in terms of certain Wronskians, namely

ϩ ϭ ˙ ͒† ͒ Ϫ ϭϪ ˙ ͒† ͒ ͑ ͒ In El i͓ f r͑0,x ; f l͑0,x ͔, In Er i͓ f l͑0,x ; f r͑0,x ͔. 3.6 ͑ ͒ ˙ ˙ Note that the Wronskians in 3.6 are independent of x because f l(0,x) and f r(0,x) are also solutions of ͑2.26͒. The expressions in ͑3.6͒ follow easily from ͑2.28͒, ͑2.29͒, and the correspond- ˙ ˙ ͓ ͑ ͔͒ ϭϪ † ing integral equations for f l(0,x) and f r(0,x) cf. A.20 . Moreover, we have Gr Gl and Er ϭ †ϩ † ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ El iGl , as can be seen by using 2.14 , 2.15 , and 3.2 – 3.5 . ϭ Theorem 3.1 shows that if the generic case occurs, then Tl(0) 0. In the next section we will see that the converse is also true.

IV. SMALL-k BEHAVIOR IN THE EXCEPTIONAL CASE Recall that in the exceptional case ͑2.26͒ has at least one bounded nontrivial solution. In this section we analyze how this affects the small-k properties of S(k), and we prove in the exceptional ϭ ෈ 1 nϫn case the continuity of S(k)atk 0 when Q L1(R;C ) and its differentiability when Q ෈ 1 nϫn ෈ 1 nϫn L2(R;C ). It turns out that when Q L1(R;C ) the exceptional case gives rise to certain technical complications that necessitate a careful study of certain asymptotic expansions. Since the proof of one result, namely Proposition 4.2, is especially long, that proof is given in the Appendix. Recall the deﬁnitions of the subspaces N and M given in ͑2.36͒ and ͑2.37͒, respectively. There is a natural mapping from N to M, which we denote by ⌫, deﬁned as follows. For every ␰෈N, let ␹ϭ ͒␰ ͑ ͒ lim f l͑0,x , 4.1 x→Ϫϱ

and put

␹ϭ⌫␰. ͑4.2͒ ͑ ͒ ␰ ͑ ͒ Note that, by 2.36 , f l(0,x) is bounded and hence, by the discussion below 2.27 , the limit in ͑4.1͒ exists. To see that ⌫ maps N into M, we note that ͑4.1͒ implies ͒␰Ϫ ͒␹ ϭ lim ͓ f l͑0,x f r͑0,x ͔ 0. x→Ϫϱ ␰Ϫ ␹ ͑ ͒ →Ϫϱ Hence f l(0,x) f r(0,x) is a solution of 2.26 which approaches zero as x ; therefore, it must be identically zero and we have ͒␰ϭ ͒␹ ෈ ͑ ͒ f l͑0,x f r͑0,x , x R. 4.3 ␹ ␹෈M Hence f r(0,x) is bounded, which implies . ෈ 1 nϫn ⌫ N M Proposition 4.1: Assume Q L1(R;C ). Then is a bijection between and . Proof: We have already seen that ⌫ maps N into M. The map ⌫ is injective, for if ⌫␰ϭ0, ͑ ͒ ͑ ͒ ␰ϭ ෈ ␰ϭ then, by 4.2 and 4.3 , f l(0,x) 0 for all x R and hence 0. It is also onto, because for every ␹෈M ␹ϭ␰ ͑ ͒ ␹ϭ⌫␰ ᭿ , lim f r(0,x) exists, and hence 4.3 holds; thus . x→ϩϱ The mapping ⌫ will make its appearance as a restriction to N of certain linear transformations n ͑ ͒ deﬁned on all of C . One such representation immediately follows from 4.3 . We can pick any x0 for which f r(0,x0) is invertible and write ⌫ϭ ͒Ϫ1 ͒ ͉ ͑ ͒ ͓ f r͑0,x0 f l͑0,x0 ͔ N , 4.4

where the symbol ͉N denotes the restriction to the subspace N. Recall that when nϭ1, ⌫ becomes a constant, so that ͑4.4͒ expresses the fact that, in the exceptional case, the two Jost solutions at

ϭ ͑ ͒ ෈ 1 nϫn kk 0 are linearly dependent. Clearly, 4.4 is valid whenever Q L1(R;C ). Another repre- ⌫ ෈ 1 nϫn sentation of that will play a role in this section is only valid when Q L2(R;C ). It follows from ͑2.28͒ which, for any ␰෈N, implies

ϱ ␹ϭ ͑ ͒␰ϭ␰ϩ ͵ ͑ ͓͒ ͑ ͒␰͔ ͑ ͒ lim f l 0,x dy yQ y f l 0,y , 4.5 x→Ϫϱ Ϫϱ

͑ ͒ ⌬ ␰ϭ where we have also used 2.33 and the fact that l 0. Note that the integral on the right-hand ͑ ͒ ෈ 1 nϫn ␰ side of 4.5 exists when Q L1(R;C ) because f l(0,y) is bounded. However, without the vector ␰ in the integrand, the integral in general does not exist as a matrix-valued integral, because →Ϫϱ some column vectors of the matrix f l(0,y) may grow linearly as y . In fact, according to ͑ ͒ ⌬ ϭ ෈ 1 nϫn 2.35 , this is always the case unless l 0. On the other hand, if Q L2(R;C ), then the integral in ͑4.5͒ without the vector ␰ in it exists as a matrix-valued integral and, in view of ͑3.1͒, ␹ϭ ϩ ␰ we can write (In El) . In other words, we have

⌫ϭ͑ ϩ ͉͒ ෈ 1͑ nϫn͒ ͑ ͒ In El N provided Q L2 R;C . 4.6

We will also need representations for ⌫Ϫ1. To this end we assume, without loss of generality, that f l(0,0) is invertible. If not, we can perform a shift of the origin and use the fact that f l(0,x) is invertible for x sufﬁciently large. We deﬁne

Rϭ ͒Ϫ1 ͒ ͑ ͒ f l͑0,0 f r͑0,0 , 4.7

and note that, by ͑4.3͒,

Ϫ1 R͉Mϭ⌫ . ͑4.8͒

⌫Ϫ1 Another representation for is obtained by using the integral relation for f r(0,x) given in ͑ ͒ ෈ 1 nϫn ␹෈M ͑ ͒ ͑ ͒ ⌬ ␹ 2.29 .IfQ L2(R;C ), then, for any , by using 2.29 , 2.34 , and the fact that r ϭ0, we obtain

ϱ ␰ϭ lim f ͑0,x͒␹ϭ␹Ϫͫ ͵ dy yQ͑y͒f ͑0,y͒ͬ␹, r r x→ϩϱ Ϫϱ

͑ ͒ ␰ϭ Ϫ ␹ and thus, by 3.1 , (In Er) . Therefore,

⌫Ϫ1ϭ͑ Ϫ ͉͒ ෈ 1͑ nϫn͒ In Er M provided Q L2 R;C .

After these preparations we are ready to begin the analysis of the small-k asymptotics of S(k) in the exceptional case. We ﬁrst consider the Wronskian

͒ϭ Ϫ ͒† ͒ ෈ ϩ W͑k ͓ f r͑ k*,x ; f l͑k,x ͔, k C , ͑ ͒ ͑ ͒ which appears in 2.6 and, as seen from 2.20 , is related to the transmission coefﬁcient Tl(k)by

͒ϭ ͒Ϫ1 ͑ ͒ Tl͑k 2ikW͑k . 4.9

The method employed here to study W(k) is patterned after that used in Ref. 12 in the scalar case. Unless otherwise stated, we will assume that k is real. This suffices for all the auxiliary results leading up to our main result given in Theorem 4.6. There we will extend the asymptotics from the real axis to Cϩ with the help of a Phragmeń–Lindelo¨f theorem. † ϭ Using ͓ f l(0,x) ; f l(0,x)͔ 0 we first write W(k) in the form

͒ϭ Ϫ ͒† ͒† Ϫ1⍀ ϩ⍀ ͒Ϫ1 ͒ W͑k f r͑ k,0 ͓ f l͑0,0 ͔ 1 2 f l͑0,0 f l͑k,0 ,

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where we have deﬁned ⍀ ϭ ͑ ͒† Ј͑ ͒Ϫ Ј͑ ͒† ͑ ͒ 1 f l 0,0 f l k,0 f l 0,0 f l k,0 , ⍀ ϭ ͑Ϫ ͒† Ј͑ ͒Ϫ Ј͑Ϫ ͒† ͑ ͒ 2 f r k,0 f l 0,0 f r k,0 f 1 0,0 . ⍀ ⍀ ␸ The quantities 1 and 2 can be written as Wronskians by means of a new solution, (k,x), of ͑1.1͒, which is deﬁned by the initial conditions ␸͑ ͒ϭ ͑ ͒ ␸Ј͑ ͒ϭ Ј͑ ͒ ͑ ͒ k,0 f 1 0,0 , k,0 f l 0,0 , 4.10 so that ␸ ͒ϭ ͒ ͑ ͒ ͑0,x f 1͑0,x . 4.11 Then we have ͒ϭ Ϫ ͒† ͒† Ϫ1 ␸ ͒† ͒ ϩ Ϫ ͒† ␸ ͒ ͒Ϫ1 ͒ W͑k f r͑ k,0 ͓ f l͑0,0 ͔ ͓ ͑k,x ; f l͑k,x ͔ ͓ f r͑ k,x ; ͑k,x ͔ f l͑0,0 f l͑k,0 . ͑4.12͒ We mention that the particular choice of the solution ␸(k,x) is motivated by the fact that there is a crucial estimate, namely ͑A8͒ of the Appendix, for the difference ͓␸(k,x)Ϫ␸(0,x)͔␰ with ␰ ෈N, which plays a key role in the proof of the next proposition. Since the proof of this proposition is lengthy, it is given in the Appendix. ෈ 1 nϫn ϭ → Proposition 4.2: Assume Q Lm(R;C ) for m 1 or 2. Then, as k 0 in R we have m ␸ ͒† ͒ ϭ j ϩ m͒ ͑ ͒ ͓ ͑k,x ; f l͑k,x ͔ ͚ k Y j o͑k , 4.13 jϭ1

where

ϱ ϭ ϭ ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔ Y 1 iIn , Y 2 dz f l 0,z f l 0,z In , 0

and

mϪ1 Ϫ ͒† ␸ ͒ ϭ j ϩ mϪ1͒ ͑ ͒ ͓ f r͑ k,x ; ͑k,x ͔ ͚ k X j o͑k , 4.14 jϭ0

with ϭ⌬ ϭ ϩ X0 l , Xl i͓In El͔. For ␰෈N we have

m Ϫ ͒† ␸ ͒ ␰ϭ j ˇ ␰ϩ m͒ ͑ ͒ ͓ f r͑ k,x ; ͑k,x ͔ ͚ k X j o͑k , 4.15 jϭ1

0 ˇ ϭ ⌫ ˇ ϭ ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔⌫ X1 i , X2 dz f r 0,z f r 0,z In . Ϫϱ

The notational differences between ͑4.14͒ and ͑4.15͒ are justified by the fact that in ͑4.15͒ the ˇ ϭ ͑ ͒ coefficient X1 is used when m 1, while in 4.14 the corresponding coefficient X1 is used only ϭ ϭ ˇ ϭ ͉ ͑ ͒ when m 2. Of course, if m 2, then X1 X1 N ,by 4.6 . Our first goal is to find the leading terms in the asymptotics of W(k)Ϫ1 as k→0. For this purpose it is convenient to temporarily replace the factors multiplying the Wronskians in ͑4.12͒ by their limits as k→0. That is, we consider the simpler expression

͒ϭR† ␸ ͒† ͒ ϩ Ϫ ͒† ␸ ͒ ͑ ͒ Z͑k ͓ ͑k,x ; f l͑k,x ͔ ͓ f r͑ k,x ; ͑k,x ͔, 4.16

where we have used ͑4.7͒ via its adjoint. In order to further motivate the use of Z(k), we note that on account of ͑4.12͒ and ͑4.16͒ we can write

͒Ϫ1ϭ ͒Ϫ1 ͒ ͒ϩ⌰ ͒ϩ⌰ ͒ Ϫ1 ͑ ͒ W͑k f l͑k,0 f l͑0,0 ͓Z͑k 1͑k 2͑k ͔ , 4.17

where

⌰ ͒ϭR† ␸ ͒† ͒ ͒Ϫ1 ͒Ϫ ͑ ͒ 1͑k ͓ ͑k,x ; f l͑k,x ͔͓f l͑k,0 f l͑0,0 In͔, 4.18

⌰ ͒ϭ Ϫ ͒† ͒† Ϫ1ϪR† ␸ ͒† ͒ ͒Ϫ1 ͒ ͑ ͒ 2͑k ͕f r͑ k,0 ͓ f l͑0,0 ͔ ͖͓ ͑k,x ; f l͑k,x ͔ f l͑k,0 f l͑0,0 , 4.19

provided the second inverse on the right-hand side of ͑4.17͒ exists. The existence of this inverse will be established below, where we show that Z(k)Ϫ1 exists for sufﬁciently small k and satisﬁes Z(k)Ϫ1ϭO(1/k)ask→0. This, together with the fact that, in view of ͑4.13͒ and Corollary 2.2, ⌰ ⌰ → 1(k) and 2(k) are both o(k)ask 0, implies ͒Ϫ1ϭ ͒Ϫ1 ͒ ͒Ϫ1 ϩ ⌰ ͒ϩ⌰ ͒ ͒Ϫ1 Ϫ1 ͑ ͒ W͑k f l͑k,0 f l͑0,0 Z͑k ͕In ͓ 1͑k 2͑k ͔Z͑k ͖ , 4.20

where the inverse of the matrix inside the braces exists provided k is sufﬁciently small. This explains why we focus on Z(k) in the next result, which is an immediate consequence of ͑4.16͒ and Proposition 4.2. ෈ 1 nϫn ϭ → Corollary 4.3: Suppose that Q Lm(R;C ) for m 1 or 2. Then, as k 0 in R

mϪ1 ͒ϭ j ϩ mϪ1͒ ͑ ͒ Z͑k ͚ k V j o͑k , 4.21 jϭ0

ϭ⌬ ϭ ϩ ϩR† V0 l , V1 i͓In El ͔.

Moreover, for ␰෈N, we have

m ͒␰ϭ j ˇ ␰ϩ m͒ ͑ ͒ Z͑k ͚ k V j o͑k , 4.22 jϭ1

ˇ ϭ ⌫ϩR† V1 i͓ ͔,

ϱ 0 ˇ ϭR† ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔ϩ ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔⌫ V2 dz f l 0,z f l 0,z In dz f r 0,z f r 0,z In . 0 Ϫϱ

Now our task is to identify those matrix elements of Z(k)Ϫ1 that dominate as k→0. To do this ⌬ ␬ we choose a Jordan basis for l as follows. We assume that there are Jordan chains indexed by ␣ ␣ϭ ␬ ϭ for 1,..., , each consisting of n␣ vectors u␣ j , with j 1,...,n␣ , satisfying the relations

͑⌬ Ϫ␭␣͒u␣ ϭ0, ͭ l 1 ͑ ͒ ⌬ Ϫ␭ ͒ ϭ ϭ 4.23 ͑ l ␣ u␣ j u␣͑ jϪ1͒ , j 2,...,n␣ . ␭ ⌬ ␣ Here ␣ is an eigenvalue of l ,u␣1 is the corresponding eigenvector belonging to the th chain, Þ and the vectors u␣ j with j 1 are the generalized eigenvectors. We assume that the eigenvalue 0 ⌬ ␮ ␯ ⌺␮ ϭ␯ ␮ of l has geometric multiplicity and algebraic multiplicity ; thus ␣ϭ1n␣ and ϭdim Nу1. We arrange the vectors of the Jordan basis in a list which is ordered according to the ␣Ͻ␤ ␣ϭ␤ Ͻ rule that u␣ j comes before u␤s if and only if or and j s. In other words, this is the ‘‘dictionary order’’ of the two two-letter words ␣ j and ␤s. It is necessary to specify an order on the Jordan basis because later we will have to perform certain permutations on these basis vectors.

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␮ ⌬ We further assume that the ﬁrst Jordan chains belong to the eigenvalue 0 of l so that ⌬ ͕u11 ,u21 ,...,u␮1͖ forms a basis for the kernel of l . We will also need the adjoint Jordan basis ͕ ͖ ␣ϭ ␬ † ϭ␦ ␦ ␦ w␣ j whose vectors, for 1,..., , satisfy w␣ ju␳t ␣␳ jt , where ␣␤ denotes the Kronecker delta, and

͑⌬ Ϫ␭*͒w␣ ϭ0, ͭ r ␣ n␣ ⌬ Ϫ␭ ͒ ϭ ϭ Ϫ ͑ r ␣* w␣ j w␣͑ jϩ1͒, j 1,...,n␣ 1.

Thus the set ͕w ,...,w␮ ͖ forms a basis for the kernel of ⌬ . The transition matrix from the 1n1 n␮ r standard basis to the Jordan basis will be denoted by S. Given any nϫn matrix M in the standard basis, we use M˜ , where M˜ ϭSϪ1MS, to denote the matrix representation of M in the Jordan basis ͑ ͒ ⌬˜ ͕u␣ j͖. Then from 4.23 it follows that l has the appearance

␬ ⌬˜ ϭ ͑␭ ͒ ͑ ͒ l Jn␣ ␣ , 4.24 ␣ϭ1

␭ ␭ where Jn␣( ␣) is the Jordan block with ␣ appearing on the diagonal and 1 on the ﬁrst superdi- agonal. In the notation introduced above we can view the pair ␣ j as a ‘‘block index’’ in the sense that ␣ ͑ ͒ indicates the Jordan block resp. the Jordan chain to which the vector u␣ j belongs, and j indicates the position within that block. Generalizing this notation, we will sometimes use block indices to designate the matrix elements of matrices represented in the Jordan basis ͕u␣ j͖. Then the matrix elements of M˜ ϭSϪ1MS in block index notation are given by

˜ ϭ † ͑ ͒ M ␤s;␣ j w␤sMu␣ j . 4.25

An important observation about ˜Z(k) is that it has ␮ columns, namely those with ‘‘addresses’’ ␣1 for ␣ϭ1,...,␮ which are O(k), and these are the only columns with this property. Any other column contains at least one element that tends to a nonzero limit as k→0. Now, as we shall see below, the entries of ˜Z(k) which determine the leading asymptotic behavior of ˜Z(k)Ϫ1 as k→0 form a submatrix of ˜Z(k) consisting of columns ␣1 and rows ␤n␤ , where ␣ and ␤ both belong to ͕1,...,␮͖. It is, therefore, convenient to perform suitable permutations of the columns and rows of ˜Z(k) in order to collect these particular matrix elements in a ␮ϫ␮ diagonal block of a new matrix, called Z(k). The formal deﬁnition of these permutations and their implementation are as ␲ follows. Let 1 be the permutation ͒ ۋ␲ ␯͒ 1 : ͑1,..., ͑q1 ,...,q␯ ,

where

n ϩ ϩn␶Ϫ ϩ1, ␶ϭ1,...,␮, ϭͭ 1 ¯ 1 ͑ ͒ q␶ 4.26 ␶Ϫ␮ϩ␣, ␶ϭ␮ϩ1,...,␯,

and ␣෈͕1,...,␮͖ is the unique integer such that, for given ␶ and ␮,

ϩ ϩ ϩ Ϫ␣ϩ ϭ␶Ϫ␮ n1 n2 ¯ n␣Ϫ1 j , ෈͕ ͖ у ϩ ϩ ϩ Ϫ␣ for some j 2,...,n␣ . Note that, since n␣ 1, the quantity n1 n2 ¯ n␣Ϫ1 is a nonde- ␣ ␲ creasing function of . Similarly, let 2 be the permutation

͒ ␴ ␴ ۋ␲ ␯͒ 2 : ͑1,..., ͑ 1 ,..., ␯ ,

where

n ϩ ϩn␣ , ␣ϭ1,...,␮, ␴ ϭͭ 1 ¯ ͑ ͒ ␣ 4.27 ␣Ϫ␮ϩ␳Ϫ1, ␣ϭ␮ϩ1,...,␯,

and ␳෈͕1,...,␮͖ is the unique integer such that, for given ␣ and ␮

ϩ ϩ ϩ Ϫ␳ϩ ϭ␣Ϫ␮ n1 n2 ¯ n␳Ϫ1 s , ෈ ϭ ␯ for some s ͕2,...,n␳͖. To implement these permutations we let eˆ j for j 1,..., denote the column ␯ ⌸ ␯ϫ␯ vectors of the standard basis in C and let 1 be the permutation matrix whose jth column † vector is eˆ , and let ⌸ be the ␯ϫ␯ permutation matrix whose kth row vector is eˆ ␴ . Now q j 2 k ␯ϫ␯ ⌸ ⌸ observe that, if M is any matrix, then the matrix 2M 1 can be thought of as being ␲ obtained from M by a permutation of the columns according to 1 and a permutation of the rows ␲ ˜ according to 2 . In order to apply these operations to Z(k) we deﬁne ϭ ⌸ ϭ ⌸ P1 diag͕ 1 ,InϪ␯͖, P2 diag͕ 2 ,InϪ␯͖,

Z ͒ϭ ˜ ͒ ϭ SϪ1 ͒S ͑ ͒ ͑k P2Z͑k P1 P2 Z͑k P1 , 4.28

and we partition Z(k)as

A͑k͒ B͑k͒ Z͑k͒ϭͫ ͬ, ͑4.29͒ C͑k͒ D͑k͒

where A(k) has size ␮ϫ␮ and, consequently, D(k) has size (nϪ␮)ϫ(nϪ␮). Then A(k) ˜ ␣ coincides with the submatrix of Z(k) consisting of the elements in columns 1 and rows sns , where 1р␣р␮ and 1рsр␮. As we have already indicated above, the matrix A(k) determines the leading asymptotic behavior of Z(k)Ϫ1 as k→0. The next two propositions provide the necessary information about the behavior of the four matrix blocks in ͑4.29͒. ෈ 1 nϫn ϭ A B Proposition 4.4: Assume Q Lm(R;C ) for m 1 or 2. Then the matrices (k), (k), C(k), and D(k) appearing in (4.29) behave near kϭ0, with k෈R, as

m mϪ1 A ͒ϭ jA ϩ m͒ B ͒ϭ jB ϩ mϪ1͒ ͑ ͒ ͑k ͚ k j o͑k , ͑k ͚ k j o͑k , 4.30 jϭ1 jϭ1

m mϪ1 C ͒ϭ jC ϩ m͒ D ͒ϭ jD ϩ mϪ1͒ ͑ ͒ ͑k ͚ k j o͑k , ͑k ͚ k j o͑k , 4.31 jϭ1 jϭ0

B ϭ A D where in the expansion for (k) the sum is absent when m 1. Moreover, 1 and 0 are invertible. B Proof: We give the proof only for (k); the proofs for the other matrices are similar. Let e j for jϭ1,...,n denote the standard basis vectors in Cn. Let s෈͕1,...,␮͖ and ﬁrst suppose that p ෈͕1,...,␯Ϫ␮͖. Then we have

eˆ B͑ ͒ ϭ †Z͑ ͒ ϭ † ˜ ͑ ͒ ϭ͓ † ͔˜ ͑ ͒ͫ q␮ϩpͬ k sp e k e␮ϩp e P2Z k P1e␮ϩp eˆ ␴ 0 Z k s s s 0

† ϭe␴ ˜Z͑k͒e ϭ˜Z͑k͒␴ ϭ˜Z͑k͒ ␣ , s q␮ϩp sq␮ϩp sns ; j

where ␣ and j are determined by ͑4.26͒ with ␶ϭ␮ϩpр␯; hence 2р jрn␣ and 1р␣р␮. Thus ͑ ͒ B ϭ ϭ B ϭ B ϩ it follows from 4.25 and Corollary 4.3 that (k)sp o(1) if m 1 and (k)sp k 1,sp o(k)if

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† mϭ2. Speciﬁcally, we have B ϭw V u␣ , where V is given in ͑4.21͒. It remains to consider 1,sp sns 1 j 1 ෈ Ϫ␯ϩ ␯Ϫ␮ ϭ the matrix elements with p ͕n 1,..., ͖. Since P1e␮ϩp e␮ϩp , we obtain

† B͑k͒ ϭ˜Z͑k͒␴ ͑␮ϩ ͒ϭ˜Z͑k͒ ␣ ϭw Z͑k͒u␣ , sp s p sns ; j sns j ␣ ϩ ϩ ϩ ϭ␮ϩ ␮ϩ Ͼ␯ where and j are determined by the equation n1 ¯ n␣Ϫ1 j p; note that p and ␣у␮ϩ р␮ B ϭ ϭ thus 1. Since s , by using Corollary 4.3, we conclude that (k)sp o(1) if m 1, and † B(k) ϭkB ϩo(k) with B ϭw V u␣ if mϭ2. sp 1,sp 1,sp sns 1 j A ෈ ␮ To prove that 1 is invertible we ﬁrst note that for s and j ͕1,..., ͖, we have

† A͑k͒ ϭ˜Z͑k͒␴ ϭ˜Z͑k͒ ϭw Z͑k͒u , sj sq j sns ; j1 sns j1

and thus, by ͑4.21͒

A ϭw† V u ϭiw† ͓⌫ϩR†͔u . ͑4.32͒ 1,sj sns 1 j1 sns j1 ␮ ۋA ␮ We show that the kernel of the transformation 1 :C C is trivial. Suppose there is a vector ⌺␮ A ϭ ϭ ␮ ␰ϭ⌺␮ ␹ϭ⌫␰ ͓ ͑ ͔͒ (c1 ,...,c␮) such that jϭ1 1,sjc j 0 for s 1,..., . Let jϭ1c ju j1 and cf. 4.2 . † Since ␹෈M, it is a linear combination of the vectors w ,...,w␮ and hence ␹ V ␰ϭ0. On the 1n1 n␮ 1 other hand, by using ͑4.7͒, we obtain

␹† ␰ϭ ␹† ⌫ϩR† ␰ϭ ʈ␹ʈ2ϩʈ␰ʈ2͒ V1 i ͓ ͔ i͑ , ϭ ϭ ϭ A ͑ ͒ ͑ ͒ which is nonzero unless c1 ¯ c␮ 0. Hence 1 is invertible. Finally, from 4.28 , 4.29 , and Corollary 4.3, we get

D ϭdiag͕I␯Ϫ␮ ,J ,...,J ͖, ͑4.33͒ 0 n␮ϩ1 n␬ ͑ ͒ D ᭿ where Jn␣ are the matrices appearing in 4.24 . Clearly, 0 is invertible. Next we study the behavior of the inverse of the matrix deﬁned in ͑4.29͒ near kϭ0. ෈ 1 nϫn ϭ → Proposition 4.5: Assume Q Lm(R;C ) for m 1 or 2. Then as k 0 in R we have the following: ͑i͒ If mϭ1, then ͑ ͒AϪ1ϩ ͑ ͒ ͑ ͒ 1/k 1 o 1/k o 1/k Z͑k͒Ϫ1ϭͫ ͬ. ͑4.34͒ ϪDϪ1C AϪ1ϩ ͑ ͒ DϪ1ϩ ͑ ͒ 0 1 1 o 1 0 o 1 ͑ii͒ If mϭ2, then 1 Ϫ1 Z͑k͒ ϭ ZϪ ϩZ ϩo͑1͒, ͑4.35͒ k 1 0

where

Z ϭ ͕AϪ1 ͖ ͑ ͒ Ϫ1 diag 1 ,0 , 4.36

ϪAϪ1A AϪ1ϩAϪ1B DϪ1C AϪ1 ϪAϪ1B DϪ1 1 2 1 1 1 0 1 1 1 1 0 Z ϭͫ ͬ. ͑4.37͒ 0 ϪDϪ1C AϪ1 DϪ1 0 1 1 0

Proof: We exploit the fact that

Ϫ1 I␮ ϪB͑k͒D͑k͒ I␮ 0 ͫ ͬZ͑ ͒ͫ ͬϭ ͕U͑ ͒ D͑ ͖͒ ͑ ͒ k ϪD ͒Ϫ1C ͒ diag k , k , 4.38 0 InϪ␮ ͑k ͑k InϪ␮

where

U͑k͒ϭA͑k͒ϪB͑k͒D͑k͒Ϫ1C͑k͒.

By ͑4.30͒, ͑4.31͒, and Proposition 4.4, we have

B ͒D ͒Ϫ1C ͒ϭ ͒ A ͒ϭ A ϩ ͒ ͑k ͑k ͑k o͑k , ͑k k 1 o͑k , A Þ U with det 1 0, and hence we conclude that, for small enough nonzero k, (k) is invertible and ͑ ͒AϪ1ϩ ͑ ͒ ϭ 1/k 1 o 1/k , m 1, U͑k͒Ϫ1ϭͭ ͑4.39͒ ͑ ͒AϪ1ϪAϪ1A AϪ1ϩAϪ1B DϪ1C AϪ1ϩ ͑ ͒ ϭ 1/k 1 1 2 1 1 1 0 1 1 o 1 , m 2.

As a result, from ͑4.38͒ we obtain

U͑k͒Ϫ1 ϪU͑k͒Ϫ1B͑k͒D͑k͒Ϫ1 Z͑k͒Ϫ1ϭͫ ͬ, ϪD͑k͒Ϫ1C͑k͒U͑k͒Ϫ1 D͑k͒Ϫ1C͑k͒U͑k͒Ϫ1B͑k͒D͑k͒Ϫ1ϩD͑k͒Ϫ1

and hence ͑4.34͒–͑4.37͒ follow by using ͑4.30͒, ͑4.31͒, and ͑4.39͒. ᭿ The primary conclusion of Proposition 4.5 is that Z(k)Ϫ1 hasa1/k-singularity at kϭ0 if dim Nу1. Therefore, ˜Z(k)Ϫ1 and Z(k)Ϫ1 have a similar behavior. Indeed, from ͑4.28͒ and ͑4.35͒ we infer that

mϪ1 ͒Ϫ1ϭ jϪ1 ϩ mϪ2͒ → ͑ ͒ Z͑k ͚ k Z jϪ1 o͑k , k 0inR, 4.40 jϭ0

where

ϭS Z SϪ1 ϭS Z SϪ1 ͑ ͒ ZϪ1 P1 Ϫ1P2 , Z0 P1 0P2 . 4.41

This leads us to the main result of this section. We will lift the restriction that k be real and allow k෈Cϩ in the asymptotics of the transmission coefﬁcients. ෈ 1 nϫn Nу Theorem 4.6: Assume Q L1(R;C ) and dim 1. Then the scattering coefﬁcients are continuous at kϭ0, and we have

͑ ͒ϭ ϩ ͑ ͒ ͑ ͒ϭϪ † ϩ ͑ ͒ → ϩ ͑ ͒ Tl k 2iZϪ1 o 1 , Tr k 2iZϪ1 o 1 , k 0inC , 4.42

͒ϭ ⌬ ͒ϭ ⌬ ͑ ͒ Im Tl͑0 Ker 1 , Ker Tl͑0 Im 1 , 4.43

͒ϭ ⌬ ͒ϭ ⌬ ͑ ͒ Im Tr͑0 Ker r , Ker Tr͑0 Im r , 4.44

͒ϭϪ ϩ⌫ ͒ϩ ͒ ͒ϭϪ ϩ⌫Ϫ1 ͒†ϩ ͒ → ͑ ͒ L͑k In Tl͑0 o͑1 , R͑k In Tl͑0 o͑1 , k 0inR, 4.45

ϩ ͒ ϭ ͒ ͒ϭ ϩ ͒ ͑ ͒ Ker ͕In L͑0 ͖ Ker Tl͑0 , Ker Tr͑0 Ker ͕In R͑0 ͖, 4.46

ϩ ͒ ϭ ͒ ϩ ͒ ϭ ͒ ͑ ͒ Im ͕In L͑0 ͖ Im Tr͑0 ,Im͕In R͑0 ͖ Im Tl͑0 . 4.47

Proof: For k෈R, the continuity of the transmission coefﬁcients and ͑4.42͒ follow immediately from ͑2.25͒, ͑4.9͒, ͑4.20͒, ͑4.40͒, and ͑4.41͒. To extend the asymptotic formulas in ͑4.42͒ from k෈R to k෈Cϩ we ﬁrst note that

͒ϭ ͒ ϩ ͒ ϭ Z ͒ ϩ ͒ ϭ ␮ ϩ ͒ → det W͑k ͓det Z͑k ͔͓1 o͑1 ͔ ͓det ͑k ͔͓1 o͑1 ͔ C0k ͓1 o͑1 ͔, k 0inR, ϭ Ϫ ␯Ϫ␮ A D Þ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ where C0 ( 1) (det 1)(det 0) 0. This follows from 4.20 , 4.28 , 4.30 , 4.36 , 4.38 , ϭ Ϫ ␯Ϫ␮ Ϫ␮ → Proposition 4.4, and the fact that (det P1)(det P2) ( 1) . It follows that k det W(k) C0 as

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k→0 along the real axis. Since det W(k) extends as an analytic function to Cϩ, there is a constant C such that ͉kϪ␮ det W(k)͉рC͉k͉Ϫ␮ for k near 0 in Cϩ. Appealing to some theorems of Phragme´n– ͑ ͒ Ϫ␮ → → Lindelo¨f e.g., Theorems 1.4.1–1.4.4 in Ref. 14 we conclude that k det W(k) C0 as k 0in ϩ ϩ C . Thus there is a set ⌺⑀ϭ͕k෈C :0Ͻ͉k͉Ͻ⑀͖, with ⑀ sufﬁciently small, on which ͉det W(k)͉ у ͉ ͉␮ C1 k for some constant C1 . Recalling the cofactor representation of the inverse of a matrix we conclude that

ʈ ͒Ϫ1ʈр ͉ ͉Ϫ␮ ෈⌺ W͑k C2 k , k ⑀ , → → for some constant C2 . Since Tl(k) Tl(0) as k 0 along the real axis, we can apply a Ϫ1 ͑ ͒ → → Phragme´n–Lindelo¨f theorem to 2ikW(k) and conclude that, by 4.9 , Tl(k) Tl(0) as k 0in Cϩ. This, together with ͑2.25͒, completes the proof of ͑4.42͒. To prove ͑4.43͒ we note that ͑4.24͒ and ͑4.26͒ imply Ker ⌬˜ ϭSpan͕e ,...,e ͖. Thus, in l q1 q␮ Z ͑ ͒ view of the form of Ϫ1 given in 4.36 , we have

u ͕ Z ͖ϭ ͭͫ ͬ ෈ ␮ͮ ϭ ͕ ͖ϭ ⌬˜ Im P Ϫ P P : u C P Span e ,...,e␮ Ker . 1 1 2 1 0 1 1 l

⌬ ϭS⌬˜ SϪ1 ͑ ͒ ͑ ͒ ͑ ͒ Since l l , the ﬁrst equality in 4.43 follows from 4.41 and 4.42 . To prove the second equality we note that

⌬˜ ϭ ¹ ␴ ␴ Im l Span͕ek : k ͕ 1 ,..., ␮͖͖,

which follows from ͑4.24͒ and ͑4.25͒. Therefore,

0 ͕Z ͖ϭͭ ෈ n ϭͫ ͬ ෈ nϪ␮ͮ Ker Ϫ P w C : P w , v C 1 2 2 v

ϭ͕ ෈ n † ϭ ϭ ␮͖ w C : ek P2w 0, k 1,...,

n † ϭ͕w෈C : e␴ wϭ0, kϭ1,...,␮͖ k

ϭ ⌬˜ Im l . ϭ ⌬ ͑ ͒ This implies Ker Tl(0) Im l and thus the second equality in 4.43 is proved. The equalities in ͑4.44͒ follow from ͑4.43͒ by taking adjoints and using the fact that (Ker M)ЌϭIm M† for any nϫn matrix M. To prove the remaining assertions we use

͒ ͑ گ ϭ Ϫ ͒ϩ ͒ ͒ ෈͒ ͒ f l͑k,x Tl͑k f r͑ k,x f r͑k,x L͑k , k R ͕0͖, 4.48

͒ ͒ϭ Ϫ ͒ϩ ͒ ͒ ෈ \ ͑ ͒ f r͑k,x Tr͑k f l͑ k,x f l͑k,x R͑k , k R ͕0͖, 4.49 ͑ ͒ ͑ ͒ ͑ ͒ which can be derived with the help of 1.4 and 1.5 . From 4.48 and the continuity of Tl(k)it immediately follows that L(k) is continuous at kϭ0 and we have

͒ ϩ ͒ ϭ ͒ ͒ ͑ ͒ f r͑0,x ͓In L͑0 ͔ f l͑0,x Tl͑0 . 4.50 ͑ ͒ Ϫ1 Now choose x such that f r(0,x) is invertible and multiply 4.50 from the left by f r(0,x) . Owing ͑ ͒ ͑ ͒ Ϫ1 ⌫ to 4.4 and the ﬁrst equation in 4.43 , we can replace f r(0,x) f l(0,x)by . Hence the ﬁrst relation in ͑4.45͒ follows. Similarly, the second relation in ͑4.45͒ is obtained from ͑4.49͒. The two equalities in ͑4.46͒ are immediate consequences of ͑4.45͒. Finally, ͑4.47͒ follows from ͑4.43͒, ͑4.45͒, and Proposition 4.1. ᭿

From ͑4.43͒, ͑4.44͒, Proposition 2.4͑i͒, and Theorem 3.1 we infer that the exceptional case Þ ͑ ͒ ͑ ͒ occurs if and only if Tl(0) 0. Moreover, 4.43 , 4.46 , and Theorem 3.1 show that L(0) and Ϫ ⌬ Þ ͑ ͒ ͑ ͒ R(0) each have eigenvalue 1 if and only if l 0. In view of 2.23 and 2.24 we also have ʈ ʈϭʈ ʈϭ ⌬ Þ ⌬ ϭ L(0) R(0) 1 if and only if l 0. The case l 0 can be called the purely exceptional case because then we have NϭMϭCn. This case is further analyzed in Example 5.4 of the next section. ෈ 1 nϫn Nу Theorem 4.7: Assume Q L2(R;C ) and dim 1. Then the scattering coefﬁcients are differentiable at kϭ0 and

͒ϭ ͒ϩ ˙ ͒ϩ ͒ → ϩ ͑ ͒ Tl͑k Tl͑0 kTl͑0 o͑k , k 0inC , 4.51

with

˙ ͑ ͒ϭ ͓ Ϫ Ϫ1͑ ͒˙ ͑ ͒ ϩ ϩ ͔ ͑ ͒ Tl 0 2i Z0 f l 0,0 f l 0,0 ZϪ1 iH1 iH2 , 4.52

where ZϪ1 and Z0 are given in (4.41) and

ϭ R† Ϫ1͑ ͒˙ ͑ ͒ ϭ ˙ ͑ ͒†͓ ͑ ͒†͔Ϫ1 H1 ZϪ1 f l 0,0 f l 0,0 ZϪ1 , H2 ZϪ1 f r 0,0 f l 0,0 ZϪ1 .

Moreover,

͒ϭ ͒†Ϫ ˙ ͒†ϩ ͒ → ϩ Tr͑k Tl͑0 k*Tl͑0 o͑k , k 0inC ,

͒ϭϪ ϩ ϩ ͒ ͒ϩ ˙ ͒ϩ ͒ → L͑k In ͑In El Tl͑0 kL͑0 o͑k , k 0inR,

͒ϭϪ ϩ Ϫ ͒ ͒ϩ ˙ ͒ϩ ͒ → R͑k In ͑In Er Tr͑0 kR͑0 o͑k , k 0inR,

where El and Er are as in (3.1) and

˙ ͒ϭ ϩ ˙ ͒ϩ ˙ ͒† ˙ ͒ ͒ L͑0 ͓In El͔Tl͑0 i͓ f r͑0,x ; f l͑0,x ͔Tl͑0 ,

˙ ͒ϭ Ϫ ˙ ͒Ϫ ˙ ͒† ˙ ͒ ͒ R͑0 ͓In Er͔Tr͑0 i͓ f l͑0,x ; f r͑0,x ͔Tr͑0 .

Proof: To prove ͑4.51͒ and ͑4.52͒ for k→0inR, we ﬁrst note the expansions

͒Ϫ1 ͒ϭ Ϫ ͒Ϫ1˙ ͒ϩ ͒ f l͑k,0 f l͑0,0 In kfl͑0,0 f l͑0,0 o͑k ,

Ϫ ͒† ͒† Ϫ1ϭR†Ϫ ˙ ͒† ͒† Ϫ1ϩ ͒ f r͑ k,0 ͓ f l͑0,0 ͔ kfr͑0,0 ͓ f l͑0,0 ͔ o͑k ,

⌰ ͒ϭϪ 2R† ͒Ϫ1˙ ͒ϩ 2͒ 1͑k ik f l͑0,0 f l͑0,0 o͑k ,

⌰ ͒ϭϪ 2˙ ͒† ͒† Ϫ1ϩ 2͒ 2͑k ik f r͑0,0 ͓ f l͑0,0 ͔ o͑k ,

which follow from ͑4.18͒, ͑4.19͒, together with ͑4.7͒ and Proposition 4.2. Inserting these expansions in ͑4.20͒ and using ͑4.9͒ we obtain ͑4.51͒ and ͑4.52͒. As with ͑4.42͒ we can use a Phragmeń–Lindelo¨f argument to extend the result to Cϩ. To find the expansions for L(k) and ˙ ͑ ͒ ͑ ͒ R(k) we first note that the existence of Tl(0), together with 4.48 and 4.49 , implies the existence of L˙ (0) and R˙ (0). Differentiating ͑4.48͒ with respect to k and taking the Wronskian with ˙ † f r(0,x) , we obtain

˙ ͒† ͒ ˙ ͒ϭ ˙ ͒† ˙ ͒ ͒ϩ ˙ ͒† ͒ ˙ ͒ ͑ ͒ ͓ f r͑0,x ; f r͑0,x ͔L͑0 ͓ f r͑0,x ; f l͑0,x ͔Tl͑0 ͓ f r͑0,x ; f l͑0,x ͔Tl͑0 , 4.53

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˙ † ˙ ϭ ͑ ͒ ˙ where we have used ͓ f r(0,x) ; f r(0,x)͔ 0. Using the integral relation 2.29 and that for f r(0,x) ͓ ͑ ͔͒ ˙ † ϭϪ ͑ ͒ ͑ ͒ cf. A.20 we obtain ͓ f r(0,x) ; f r(0,x)͔ iIn . Inserting this together with 3.6 in 4.53 and using ͑4.47͒ we get the expansion for L(k). The proof of the expansion for R(k) is similar. ᭿

V. EXAMPLES In this section we consider some special cases that illustrate various details of the analysis in Sec. IV. With the exception of Example 5.4 we only consider Tl(k). ϭ ෈ 1 Example 5.1: Let n 1 with Q L2(R) and assume the exceptional case occurs. Then Z(k) ϭ˜Z(k)ϭZ(k)ϭA(k), and these are all scalar functions. We choose ␰ϭ1ϭw and put ␥ϭ⌫ ϭ ␥ ϭ f l(0,0)/ f r(0,0), where now is a real nonzero constant. Since Tl(k) Tr(k), we denote the ͑ ͒ A ϭ ␥2ϩ ␥ transmission coefﬁcient by T(k). By 4.32 we have 1 i( 1)/ ,

ϱ 0 A ϭ␥Ϫ1 ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔ϩ␥ ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔ 2 dz f l 0,z f l 0,z In dz f r 0,z f r 0,z In , 0 Ϫϱ

so that

2␥ 2ik␥⌶ T͑k͒ϭ ϩ ϩo͑k͒, k→0inCϩ, ͑5.1͒ ␥2ϩ1 ͑␥2ϩ1͒2

where we have deﬁned

ϱ 0 ⌶ϭ␥͓˙ ͑ ͒ ˙ ͑ ͔͒ϩ ͵ ͓ ͑ ͒2Ϫ ͔ϩ␥2 ͵ ͓ ͑ ͒2Ϫ ͔ f r 0,x ; f l 0,x dz f l 0,z 1 dz f r 0,z 1 . 0 Ϫϱ

In deriving ͑5.1͒ we have used the identity

˙f ͑0,x͒ ˙f ͑0,x͒ l ϩ r ϭϪ ͓˙ ͑ ͒ ˙ ͑ ͔͒ ͑ ͒ ͒ ͒ i f r 0,x ; f l 0,x , 5.2 f r͑0,x f l͑0,x ˙ which can be veriﬁed as follows. Since f r(0,x) and f r(0,x) are linearly independent solutions of ͑2.26͒, we can write

˙ ͒ϭ ͒ϩ ˙ ͒ f l͑0,x c1 f r͑0,x c2 f r͑0,x ,

and evaluate c1 and c2 as

1 ϭϪ ͓˙ ͑ ͒ ˙ ͑ ͔͒ ϭϪ c1 i f r 0,0 ; f l 0,0 , c2 ␥ ,

͑ ͒ ͑ ͒ ෈ 1 so that 5.2 follows. It seems that the expansion 5.1 is new under the assumption Q L2(R). ෈ 1 nϫn ⌬˜ Example 5.2: Assume Q L1(R;C ) and suppose that l consists of one single Jordan у ␬ϭ ␮ϭ ϭ ϭ␯ block of size n 2 associated with the eigenvalue 0. Thus 1, 1, and n n1 . ϭ ϭ In this case we can simplify the notation by setting u1 j u j , for j 1,...,n. Then u1 is the ⌬ Nϭ eigenvector for the eigenvalue 0 of l , that is Span͕u1͖. The adjoint basis is ͕w1 ,...,wn͖ and Mϭ ⌫ ⌫ ϭ we have Span͕wn͖. The mapping maps u1 to a multiple of wn , i.e., u1 c3wn for some Þ A ͑ ͒ c3 0. Moreover, (k) is a scalar function and from 4.32 we obtain

i A ϭ ͉ ͉2ʈ ʈ2ϩʈ ʈ2͒ 1 ͑ c3 wn u1 , c3* ͑ ͒ R ϭ ͑ ͒ where we have used 4.7 via wn (1/c3)u1 . The permutation matrices appearing in 4.28 are given by

00...01 10...00 ϭ ϭ P1 In , P2 ͫ ]]]]ͬ . 00...00 00...10

Using ͑4.41͒ and ͑4.42͒ we obtain

00...02/c4 0 0 ... 0 0 ˜ ͑ ͒ϭ Tl 0 ͫ ]]] ]ͬ , 0 0 ... 0 0 0 0 ... 0 0

where

1 ϭ ͉ ͉2ʈ ʈ2ϩʈ ʈ2͒ ˜ ͒ϭSϪ1 ͒S c4 ͑ c3 wn u1 , Tl͑0 Tl͑0 . c3*

⌬˜ ͑ ͒ Example 5.3: This example illustrates the situation where l in 4.24 consists of two Jordan ෈ 1 nϫn ϭ ␮ϭ ϭ ϭ ␯ϭ ␬ϭ ⌬ blocks. We assume Q L1(R;C ) and let n 3, 2, n1 1, n2 2, 3, and 2, so that l has the Jordan form

000 ⌬˜ ϭ l ͫ 001ͬ . 000

N The Jordan basis is ͕u11 ,u21 ,u22͖, where ͕u11 ,u21͖ is a basis for , and the adjoint basis is M ˜ ͕w11 ,w21 ,w22͖, where ͕w11 ,w22͖ is a basis for . In this case the rows of Z(k) need to be ۋ ␲ permuted according to 2 : (1,2,3) (1,3,2), whereas no permutation of the columns is required. Thus we have

100 ϭ ϭ P1 I3 , P2 ͫ 001ͬ . 010

Then A(k)isa2ϫ2 matrix and

w† V u w† V u A ϭͫ 11 1 11 11 1 21ͬ 1 † † , w22V1u11 w22V1u21

͑ ͒ where V1 is given in 4.21 . Hence we obtain

† Ϫ † w22V1u21 0 w11V1u21 1 ˜ ͑ ͒ϭ Ϫ † † T1 0 ͫ w22V1u11 0 w11V1u11 ͬ . A det 1 000

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Example 5.4: This is the purely exceptional case mentioned above Theorem 4.7. We assume ෈ 1 nϫn Ͼ ⌬ ϭ ␮ϭ␯ϭ␬ϭ NϭMϭ n Q L1(R;C ) with n 1. We have l 0, which implies n. Then C , and ͑ ͒ ͑ ͒ Rϭ⌫Ϫ1 thus no restrictions are necessary in 4.4 and 4.8 ; that is, we have . Moreover, P1 ϭ ϭ P2 In . It follows that A ϭ SϪ1 ⌫ϩ ⌫Ϫ1͒† S 1 i ͓ ͑ ͔ , Z ϭAϪ1 and thus, since Ϫ1 1 , we obtain ͒ϭ S SA ͒Ϫ1ϭ ⌫† ⌫⌫†ϩ ͒Ϫ1 Tl͑0 2i ͑ 1 2 ͑ In .

For the reflection coefficients, after some straightforward manipulations, we find

͒ϭ ⌫⌫†Ϫ ͒ ⌫⌫†ϩ ͒Ϫ1 ͒ϭ Ϫ⌫†⌫͒ ⌫†⌫ϩ ͒Ϫ1 L͑0 ͑ In ͑ In , R͑0 ͑In ͑ In .

1 nϫn Example 5.5: Suppose Q(x) is even and belongs to L1(R;C ). This implies that f r(0,x) ϭ Ϫ ͑ ͒ ͑ ͒ ͑ ͒ ⌬ ⌬ f l(0, x) and from 2.31 , 2.33 , and 2.34 we conclude that l is self-adjoint. Hence l is diagonalizable and there are no Jordan chains of length greater than 1. We have ␮ϭ␯, n␣ϭ1 for р␣р␬ ␬ϭ ϭ ϭ ⌬ 1 , and n. We also have P1 P2 In . It is possible that l has some nonzero eigenvalues, so ␮Ͻn in general. If ␰෈N, then

͒␰ϭ Ϫ ͒␰ f l͑0,x f r͑0, x , ␰ ␰෈M NϭM which implies that f r(0,x) is bounded. This means and hence . Furthermore, using ͑4.2͒ and ͑4.3͒ we conclude that

͒␹ϭ Ϫ ͒␹ϭ Ϫ ͒␰ ͑ ͒ f l͑0,x f r͑0, x f l͑0, x , 5.3

where ␰෈N and ␹ϭ⌫␰. Letting x→Ϫϱ, we see that ⌫␹ϭ␰, that is

2 ⌫ ϭI␮ . ͑5.4͒

p pϪ1 It follows that ⌫ is diagonalizable because (I␮Ϯ⌫) ϭ2 (I␮Ϯ⌫) for pу1 and has eigenvalues Ϯ ⑀ ⑀ ϩ⑀ ϭ␮ ϭ 1. Let Ϯ denote the corresponding multiplicities ( ϩ Ϫ ). Since n␣ 1, we put u␣1 ϭ ⌬ u␣ for the vectors of the Jordan basis for l and assume that they are normalized and arranged such that

⌫u␣ϭu␣ , ␣ϭ1,...,⑀ϩ,

⌫u␣ϭϪu␣ , ␣ϭ⑀ϩϩ1,...,␮.

† We also set w ϭw , so that w u␣ϭ␦ ␣ for sϭ1,...,n and ␣ϭ1,...,n. Note that as a consequence sns s s s of ͑5.3͒, ⑀ϩ(⑀Ϫ) is the number of linearly independent bounded even ͑odd͒ solutions of ͑2.26͒. Then from ͑4.8͒ and ͑5.4͒ we conclude that

†R† ϭ͑⌫Ϫ1 ͒† ϭ͑⌫ ͒† ϭ †⌫† ͑ ͒ ws u␣ ws u␣ ws u␣ ws u␣ , 5.5

where ⌫† is the adjoint of ⌫ as a mapping from N to itself. Using ͑5.5͒ in ͑4.32͒ we obtain A ϭ †͓⌫ϩ⌫†͔ 1,sj iws u j , and therefore ͑AϪ1͒ ϭϪ †͓⌫ϩ⌫†͔Ϫ1 1 sj iws u j .

As a result, from ͑4.36͒, ͑4.41͒, and ͑4.42͒ we deduce that

͒ϭ ⌫ϩ⌫†͒Ϫ1 Tl͑0 ͓2͑ ͔ 0,

where the direct sum refers to the direct decomposition CnϭN N Ј with

N ϭ Ј Span͕u␮ϩ1 ,...,un͖.

ACKNOWLEDGMENTS This material is based on work supported by the National Science Foundation under Grant No. DMS-9803219, by the Department of Energy under Grant No. DE-FG02-01ER45951, and by INDAM and MURST.

APPENDIX: PROOF OF PROPOSITION 4.2 Proof: Since the assertions of Proposition 4.2 concern the small-k asymptotics, we assume that k lies in a fixed interval ͓Ϫ␦, ␦͔ with ␦Ͼ0. In the following C is used to denote various constants that may depend on the choice of ␦ but not on k or x. The solution ␸(k,x)of͑1.1͒ defined by the initial conditions ͑4.10͒ satisfies the integral equation

sin kx 1 x ␸͑ ͒ϭ ͑ ͒ ϩ Ј͑ ͒ͩ ͪ ϩ ͵ ͓ ͑ Ϫ ͔͒ ͑ ͒␸͑ ͒ ͑ ͒ k,x f l 0,0 cos kx f l 0,0 dy sin k x y Q y k,y , A1 k k 0

which can be solved by iteration. A standard Gronwall inequality shows that

ʈ␸͑k,x͒ʈрC͑1ϩ͉x͉͒, x෈R. ͑A2͒ we have 0͖͕گTherefore, by using ͑A1͒ and ͑A2͒, it follows that for each k෈R

ikx Ϫikx ␸͑k,x͒ϭ␣Ϯ͑k͒e ϩ␤Ϯ͑k͒e ϩ⑀Ϯ͑k,x͒, ͑A3͒

where ⑀Ϯ(k,x) and ⑀ϮЈ (k,x) are both o(1) as x→Ϯϱ, and where

1 1 1 Ϯϱ ␣ ͑ ͒ϭ ͑ ͒ϩ Ј͑ ͒ϩ ͵ Ϫiky ͑ ͒␸͑ ͒ ͑ ͒ Ϯ k f l 0,0 f l 0,0 dy e Q y k,y , A4 2 2ik 2ik 0

1 1 1 Ϯϱ ␤ ͑ ͒ϭ ͑ ͒Ϫ Ј͑ ͒Ϫ ͵ iky ͑ ͒␸͑ ͒ Ϯ k f l 0,0 f l 0,0 dy e Q y k,y . 2 2ik 2ik 0

From ͑A3͒ and ͑A4͒, together with ͑1.2͒ and ͑1.3͒, it follows that:

ϱ ͓␸͑ ͒† ͑ ͔͒ϭ ␣ ͑ ͒†ϭ ͑ ͒†Ϫ Ј͑ ͒†Ϫ ͵ ikz␸͑ ͒† ͑ ͒ ͑ ͒ k,x ; f l k,x 2ik ϩ k ikfl 0,0 f l 0,0 dz e k,z Q z , A5 0

0 ͓ ͑Ϫ ͒† ␸͑ ͔͒ϭ ␣ ͑ ͒ϭ ͑ ͒ϩ Ј͑ ͒Ϫ ͵ Ϫikz ͑ ͒␸͑ ͒ ͑ ͒ f r k,x ; k,x 2ik Ϫ k ikfl 0,0 f l 0,0 dz e Q z k,z . A6 Ϫϱ

In order to control the remainder terms in the subsequent asymptotic expansions, we will need the estimates

kx 2 ʈ␸͑k,x͒Ϫ␸͑0,x͒ʈрC͑1ϩmax͕0,Ϫx͖͒ͩ ͪ , ͑A7͒ 1ϩ͉k͉͉x͉

kx 2 ʈ͓␸͑k,x͒Ϫ␸͑0,x͔͒␰ʈрCͩ ͪ ʈ␰ʈ, ␰෈N. ͑A8͒ 1ϩ͉k͉͉x͉

The term max͕0,Ϫx͖ in ͑A7͒ accounts for the fact that ␸(0,x) is in general unbounded and O(x) as x→Ϫϱ.In͑A8͒, this term is absent because ␸(0,x)␰ is bounded when ␰෈N. We omit the

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proofs of ͑A7͒ and ͑A8͒ here because ͑A7͒ follows from ͑A1͒ by some standard estimates and ͑A8͒ can be proved by mimicking the proof in the scalar case ͑see Lemma 2.2 in Ref. 12͒. Now consider the integral on the right-hand side of ͑A5͒ and write it as

ϱ ͵ ikz␸͑ ͒† ͑ ͒ϭ ͑ ͒ϩ ͑ ͒ ͑ ͒ dz e k,z Q z A1 k A2 k , A9 0

where

ϱ ͑ ͒ϭ ͵ ikz␸͑ ͒† ͑ ͒ ͑ ͒ A1 k dz e 0,z Q z , A10 0

ϱ ͑ ͒ϭ ͵ ikz͓␸͑ ͒†Ϫ␸͑ ͒†͔ ͑ ͒ ͑ ͒ A2 k dz e k,z 0,z Q z . A11 0

When mϭ1, from ͑A.10͒ we get

ϱ ϱ ͑ ͒ϭ ͵ ␸͑ ͒† ͑ ͒ϩ ͵ ␸͑ ͒† ͑ ͒ϩF͑ ͒ A1 k dz 0,z Q z ik dz z 0,z Q z k , ͭ 0 0 ͑A12͒ ϭϪ Ј͑ ͒†ϩ ͓ ͑ ͒†Ϫ ͔ϩF͑ ͒ f l 0,0 ik f l 0,0 In k ,

where

ϱ F͑k͒ϭ ͵ dz ͑eikzϪ1Ϫikz͒␸͑0,z͒†Q͑z͒. ͑A13͒ 0

F ͑ ͒ ϩϱ Note that (k)iso(k)by 4.11 , the boundedness of f l(0,z)on͓0, ), and the estimate

Cz2 ͉eikzϪ1Ϫikz͉р , zу0. 1ϩz

In deriving ͑A12͒ we have also used the relations

ϱ ͵ ␸͑ ͒† ͑ ͒ϭϪ Ј͑ ͒† dz 0,z Q z f l 0,0 , 0 ͑ ͒ ͭ ϱ A14 ͵ ␸͑ ͒† ͑ ͒ϭ ͑ ͒†Ϫ dz z 0,z Q z f l 0,0 In, 0

which follow from ͑2.28͒. Using ͑A7͒ in ͑A11͒ we see that

͒ϭ ͒ ͑ ͒ A2͑k o͑k . A15

Combining ͑A5͒, ͑A9͒, ͑A12͒, and ͑A15͒ we obtain

␸ ͒† ͒ ϭ ϩ ͒ ͓ ͑k,x ; f l͑k,x ͔ ikIn o͑k ,

which agrees with ͑4.13͒ for mϭ1. ͑ ͒ ϭ ෈ 1 nϫn Now consider A5 for m 2, that is, Q L2(R;C ). In this case we can expand the remainder F(k)in͑A12͒ as

ϱ ͑ikz͒2 F͑k͒ϭ ͵ dz ␸͑0,z͒†Q͑z͒ϩo͑k2͒, ͑A16͒ 0 2

where we have used ͑A7͒, ͑A13͒, and the estimate

k2z2 C͉͑k͉z͒3 ͯeikzϪ1Ϫikzϩ ͯр , zу0. 2 1ϩ͉k͉z

The integral in ͑A16͒ can be expressed in a form that does not involve Q explicitly. To see this, substitute ␸Љ(0,z)† for ␸(0,z)†Q(z) and replace the upper limit of integration by N. Then inte- grate by parts twice and let N→ϩϱ. This gives

k2 ϱ † k2 ϱ Ϫ ͫ ͵ dz z2␸Љ͑0,z͒ͬ ϭϪ lim G† ϭϪk2 ͵ dz ͓ f ͑0,z͒†ϪI ͔, N l n 2 0 2 N→ϩϱ 0

where we have deﬁned

N G ϭ 2␸Ј͑ ͒Ϫ ͓␸͑ ͒Ϫ ͔ϩ ͵ ͓␸͑ ͒Ϫ ͔ N N 0,N 2N 0,N In 2 dz 0,z In . 0

Thus

ϱ F͑ ͒ϭϪ 2 ͵ ͓ ͑ ͒†Ϫ ͔ϩ ͑ 2͒ ͑ ͒ k k dz f l 0,z In o k . A17 0

In the derivation of ͑A17͒ we have also used

␸ ͒ϭ 2͒ ␸ ͒Ϫ ϭ ͒ ␸ ͒Ϫ ෈ 1 ϩ nϫn͒ ͑ ͒ Ј͑0,N o͑1/N , ͑0,N In o͑1/N , ͑0,N In L ͑R ;C . A18

These properties follow directly from ͑2.28͒ and ͑4.11͒. The expression ͑A17͒ for F(k) has the advantage that it allows us to combine F(k) with another term that arises from the expansion of ͑ ͒ ␸ Ϫ␸ A2(k). To see this we return to A11 . In order to expand the difference (k,x) (0,x), we use the variation of parameters formula in the form

x x ␸͑ ͒ϭ␸͑ ͒ϩ 2 ͑ ͒ ͵ ˙ ͑ ͒†␸͑ ͒ϩ 2˙ ͑ ͒ ͵ ͑ ͒†␸͑ ͒ k,x 0,x ik f l 0,x dz fl 0,z k,z ik f l 0,x dz fl 0,z k,z . 0 0 ͑A19͒

We brieﬂy mention some details of the derivation of ͑A19͒ because there is a useful identity that falls out in the process. We write ͑2.26͒ as a ﬁrst-order system with 2n components and note that a fundamental matrix ⌿(x) for this system and its inverse ⌿(x)Ϫ1 are given by

f ͑0,x͒ ˙ ͑ ͒ ˙Ј͑ ͒† Ϫ˙ ͑ ͒† l f l 0,x f l 0,x f l 0,x ⌿͑x͒ϭͫ ͬ, ⌿͑x͒Ϫ1ϭiͫ ͬ. Ј͑ ͒ ˙Ј͑ ͒ Ј͑ ͒† Ϫ ͑ ͒† f l 0,x f l 0,x f l 0,x f l 0,x

By taking x→ϩϱ and using ͑2.4͒, one can prove that det ⌿(x)ϭi. For this and also later we need ˙ ˙Ј to use certain asymptotic information about the functions f l(0,x) and f l (0,x). It sufﬁces to men- ˙ tion that f l(0,x) is the unique solution of the integral equation

ϱ ˙ ͑ ͒ϭ ϩ ͵ ͑ Ϫ ͒ ͑ ͒˙ ͑ ͒ ͑ ͒ f l 0,x ixIn dy y x Q y f l 0,y , A20 x

˙ ͑ ͒ which, incidentally, shows that f l(0,x) is also a matrix solution of 2.26 . Moreover, a Gronwall inequality gives

ʈ˙ ͒ʈр ϩ͉ ͉͒ ෈ ͑ ͒ f l͑0,x C͑1 x , x R. A21 ⌿ Ϫ1⌿ ϭ The identity (x) (x) I2n is easily veriﬁed by using the Wronskian relations

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͒† ͒ ϭ ˙ ͒† ˙ ͒ ϭ ͓ f l͑0,x ; f l͑0,x ͔ ͓ f l͑0,x ; f l͑0,x ͔ 0,

͒† ˙ ͒ ϭ ˙ ͒† ͒ ϭ ͓ f l͑0,x ; f l͑0,x ͔ ͓ f l͑0,x ; f l͑0,x ͔ iIn , ͑ ͒ ͑ ͒ ͑ ͒ Ј which follow from 2.28 , A20 , and the first formula in A18 which indicates that f l (0,x) ϭo(1/x2)asx→ϩϱ. Then ͑A19͒ is an easy consequence of the variation of parameters formula for first-order systems. The useful identity alluded to above appears when we write out the identity ⌿ ⌿ Ϫ1ϭ ͑ ͒ (x) (x) I2n in this order! in terms of the entries of the matrices involved. Among the resulting identities we find

Ј͑ ͒˙ ͑ ͒†ϩ˙Ј͑ ͒ ͑ ͒†ϭ f l 0,x f l 0,x f l 0,x f l 0,x iIn ,

which will be useful later. By iterating ͑A19͒ once and using ͑4.11͒ we obtain

x x ␸͑ ͒ϭ␸͑ ͒ϩ 2 ͑ ͒ ͵ ˙ ͑ ͒† ͑ ͒ϩ 2˙ ͑ ͒ ͵ ͑ ͒† ͑ ͒ϩ␳͑ ͒ k,x 0,x ik f l 0,x dz fl 0,z f l 0,z ik f l 0,x dz fl 0,z f l 0,z k,x , 0 0 ͑A22͒

where ␳(k,x) obeys

kx 2 ʈ␳͑k,x͒ʈрCk2͑1ϩ͉x͉͒2ͩ ͪ . ͑A23͒ 1ϩ͉k͉x͉

͑ ͒ ͑ ͒ ͑ ͒ This estimate follows by using A7 and A21 . Taking the adjoint of A2(k) given in A11 and expanding the exponential function there we get

ϱ ͑ ͒†ϭ ͵ ͑ ͓͒␸͑ ͒Ϫ␸͑ ͔͒ϩ ͑ 2͒ ͑ ͒ A2 k dz Q z k,z 0,z o k , A24 0

where we have used ͑A7͒ to determine the order of the error term. Now we insert ͑A22͒ into ͑A24͒ ͑ ͒ ˙Љ ϭ ˙ and proceed as in the derivation of A17 , using f l (0,x) Q(x) f l(0,x) and two integrations ͑ ͒ ͑ ͒ ˙Ј Ϫ ϭ →ϩϱ by parts. We also use A21 , A23 , and the property f l (0,N) iIn o(1/N)asN , which follows from ͑A20͒. The result is

ϱ ϱ ϱ ͵ ͑ ͓͒␸͑ ͒Ϫ␸͑ ͔͒ϭ 2 ͵ ͓ ͑ ͒Ϫ ͔Ϫ 2 ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔ϩ ͑ 2͒ dz Q z k,z 0,z k dz f l 0,z In k dz f l 0,z f l 0,z In o k . 0 0 0 ͑A25͒

Combining ͑A9͒, ͑A12͒, ͑A17͒, ͑A24͒, and ͑A25͒ we obtain

ϱ ͓␸͑ ͒† ͑ ͔͒ϭ ϩ 2 ͵ ͓ ͑ ͒† ͑ ͒Ϫ ͔ϩ ͑ 2͒ k,x ; f l k,x ikIn k dz f l 0,z f l 0,z In o k , 0

which is the desired result in ͑4.13͒ for mϭ2. To prove ͑4.14͒ we return to the Wronskian in ͑A6͒.Ifmϭ1, we have

0 0 ͵ dz eϪikzQ͑z͒␸͑k,z͒ϭ ͵ dz Q͑z͒␸͑0,z͒ϩo͑1͒, ͭ Ϫϱ Ϫϱ ͑A26͒ ϭϪ⌬ ϩ Ј͑ ͒ϩ ͑ ͒ l f l 0,0 o 1 ,

where we have used ͑2.28͒ and ͑A14͒. The order of the error term is again a consequence of ͑A7͒. Substituting ͑A26͒ in ͑A6͒ we get ͑4.14͒ for mϭ1. If mϭ2, we have

0 0 ͵ Ϫikz ͑ ͒␸͑ ͒ϭϪ⌬ ϩ Ј͑ ͒Ϫ ͵ ͑ ͒␸͑ ͒ϩ ͑ ͒ ͑ ͒ dz e Q z k,z l f l 0,0 ik dz zQ z 0,z o k , A27 Ϫϱ Ϫϱ

and, using ͑3.1͒, ͑A14͒, and ͑A27͒ we obtain

0 ͵ ͑ ͒␸͑ ͒ϭ ϩ Ϫ ͑ ͒ ͑ ͒ dz zQ z 0,z In El f l 0,0 . A28 Ϫϱ

Substituting this in ͑A6͒ we get

Ϫ ͒† ␸ ͒ ϭ⌬ ϩ ϩ ͒ϩ ͒ ͓ f r͑ k,x ; ͑k,x ͔ l ik͑In El o͑k ,

proving ͑4.14͒ when mϭ2. It remains to prove ͑4.15͒. So pick ␰෈N and assume mϭ1. Then ␸(0,x)␰ stays bounded as x→Ϫϱ, which has the same effect on the integral in ͑A6͒, when it acts on ␰,asifm were 2. In particular, ͑A28͒ now becomes

0 ͵ ͑ ͓͒␸͑ ͒␰͔ϭ⌫␰Ϫ ͑ ͒␰ dz zQ z 0,z f l 0,0 , Ϫϱ

͑ ͒ ͑ ͒ ⌬ ␰ϭ ͑ ͒ where we have used 2.28 and 4.5 . Since l 0, from A27 we obtain

0 ͵ Ϫikz ͑ ͓͒␸͑ ͒␰͔ϭ Ј͑ ͒␰Ϫ ⌫␰ϩ ͑ ͒␰ϩ ͑ ͒ dz e Q z k,z f l 0,0 ik ikfl 0,0 o k . Ϫϱ

Substituting this expression in ͑A6͒ we get

Ϫ ͒† ␸ ͒ ␰ϭ ⌫␰ϩ ͒ ͓ f r͑ k,x ; ͑k,x ͔ ik o͑k ,

which agrees with ͑4.15͒ for mϭ1. If mϭ2 and ␰෈N, then we can carry the expansion in ͑A27͒ further as in the case of ͑A9͒ and ͑A11͒. To obtain the corresponding coefﬁcients in the expansion ˙ we could proceed by using variation of parameters in terms of the solutions f r(0,x) and f l(0,x). However, there is a simpler approach that exploits the connection between the left and right Jost ,(Q#(xۋ(Ϫx, that is, under the transformation Q(xۋsolutions for ͑1.1͒ under the substitution x where Q#(x)ϭQ(Ϫx). We use the superscript # to indicate that a given quantity pertains to ͑1.1͒ with potential Q#. It is straightforward to show that

͑ ͒ϭ #͑ Ϫ ͒ ͑ ͒ϭ #͑ Ϫ ͒ ͑ ͒ f r k,x f l k, x , f l k,x f r k, x . A29

We now introduce a solution ␻(k,x)of͑1.1͒ satisfying the initial conditions

␻͑ ͒ϭ ͑ ͒ ␻Ј͑ ͒ϭ Ј͑ ͒ k,0 f r 0,0 , k,0 f r 0,0 .

Then it follows from ͑4.3͒ and ͑4.10͒ that for ␰෈N we have

␸͑k,x͒␰ϭ␻͑k,x͒␹, ͑A30͒

where ␹ϭ⌫␰. Since, by ͑4.10͒ and ͑A29͒

␸#͑ ͒ϭ #͑ ͒ϭ ͑ ͒ ␸#Ј͑ ͒ϭ #Ј͑ ͒ϭϪ Ј͑ ͒ k,0 f 1 k,0 f r k,0 , k,0 f 1 k,0 f r k,0 ,

we get ␸#(k,x)ϭ␻(k,Ϫx), which, together with ͑A30͒, yields ␸#(k,Ϫx)␹ϭ␸(k,x)␰. In the following argument we use the more elaborate notation. ͓G(k,x);H(k,x)͔ to denote the (x0) ϭ Wronskian of two matrix functions G(k,x) and H(k,x) evaluated at x x0 . Then we have

Downloaded 07 Oct 2001 to 130.18.24.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp 4652 J. Math. Phys., Vol. 42, No. 10, October 2001 Aktosun, Klaus, and van der Mee

͓ ͑Ϫ ͒† ␸͑ ͔͒ ␰ϭ͓ #͑Ϫ Ϫ ͒† ␸#͑ Ϫ ͔͒ ␹ f r k,x ; k,x ͑x͒ f 1 k, x ; k, x ͑x͒ # † # ϭϪ͓ f ͑Ϫk,x͒ ;␸ ͑k,x͔͒ Ϫ ͒␹ 1 ͑ x ͑ ͒ ͭ ϭ͓␸#͑ ͒† #͑Ϫ ͔͒† ␹ A31 k,x ; f 1 k,x ͑Ϫx͒ ϭ͓␸#͑Ϫ ͒† #͑Ϫ ͔͒† ␹ k,x ; f 1 k,x ͑x͒ , where in the the last step we have used the fact that the Wronskian is constant and that ␸(k,x)is an even function of k. The latter follows from the fact that the initial conditions in ͑4.10͒ are independent of k. Now the Wronskian on the right-hand side of ͑A31͒ is of the same form as that ͑ ͒ in 4.13 . We can, therefore, apply the expansion given there. Then the integrand of Y 2 involves # ͑ ͒ ͑ ͒ f 1(0,z) which can be rewritten in terms of f r(0,z) by means of A29 . Using also 4.2 , we obtain ͑4.15͒. The proof of Proposition 4.2 is now complete. ᭿

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Downloaded 07 Oct 2001 to 130.18.24.42. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/jmp/jmpcr.jsp