Forward scattering on the line with a transfer condition

Only point transfer matrices at the origin will be considered, and henceforth we will refer to them as transfer matrices. In [1] conditions for self-adjointness and limit point/limit circle criteria are considered for a more general problem.

In the physical context, the transfer matrix represents a change of medium which affects the incident wave as represented by components of the matrix. Livšic in [[2], p.7] gives a concrete physical example of a scattering problem of the form given above. He considers a uniform, infinite string, attached at the point $x=0$ to a transverse spring. The behaviour at the point of attachment is described by what we have called the transfer matrix M.

Our transfer matrices will be real constant transfer matrices, i.e., all components will be constants. The theory could be extended to eigenparameter-dependent transfer matrices, this will be considered in future studies. Gordon and Pearson, in [3], characterized the self-adjoint constant point transfer matrices as well as eigenparameter-dependent ones.

The Jost solutions of (1.1) represent oscillations moving left or right on ℝ. The scattering data is defined in terms of these solutions. The classical Jost solutions correspond to the case where the transfer matrix M is the identity. In particular, asymptotic approximations to the classical Jost solutions and their derivatives, as well as some basic relations regarding them, are given. The Jost solutions for the scattering problem with a transfer condition ($M\ne I$) are then expressed in terms of the classical Jost solutions. Consequently, we consider functional analytic aspects of the operator L such as location of eigenvalues and continuous spectrum, multiplicity of eigenvalues, properties of the scattering coefficients/scattering matrix and the reflection coefficient. We show that the operator L has finitely many eigenvalues, they are negative and simple, and that the positive real axis $[0,\mathrm{\infty })$ is the continuous spectrum of L.

This paper is structured as follows. The Jost solutions of the scattering problem (1.1) and (1.2) are defined in Section 2. In addition, some basic properties and asymptotic approximations of the classical Jost solutions and their derivatives are recalled. In Section 3, the scattering problem is reformulated as a system spectral problem, and we prove that L is self-adjoint if and only if $detM=1$. Moreover, it is shown that all the eigenvalues are negative and that the entire non-negative real half-line is the continuous spectrum of L. In Section 4, since the scattering problem (1.1) and (1.2) can be considered as two half-line problems, the Jost solutions are expressed in terms of the classical Jost solutions. Asymptotics for the scattering coefficients are determined, and we prove that the problem has finitely many eigenvalues all of which are simple. In the final section, Section 5, we give a relationship between the norming constants and the derivative of the scattering coefficient.

The results obtained in this paper provide the foundation needed in order to consider the associated inverse problem, this will be the topic of a subsequent paper.

In this section, asymptotic solutions $y(x,\zeta ),(x,\zeta )\in \mathbb{R}\times {\overline{\mathbb{C}}}^{+}$, will be developed for equation (1.1) for large values of $|x|+|\zeta |$, where ${\overline{\mathbb{C}}}^{+}=\{\xi +i\eta \mid \xi ,\eta \in \mathbb{R},\eta \ge 0\}$. The focus of this section will be on the so-called Jost solutions of (1.1) and their derivatives. The Jost solutions $f_{+,M}$ and $f_{-,M}$ of (1.1) and (1.2) are the solutions of (1.1) and (1.2) defined by their asymptotic behaviour at ±∞ as follows.

Definition 2.1 [[4], p.297]

where $C(x)$ is non-increasing and ρ and σ are as defined earlier.

Note that the Wronskian, $W[f_{+},{\overline{f}}_{+}]$, of $f_{+}$ and ${\overline{f}}_{+}$ for $\xi \in \mathbb{R}$ and $x\in \mathbb{R}$ is given by $W[f_{+},{\overline{f}}_{+}](x,\xi )=-2i\xi$, see [7]. Thus $f_{+}(x,\xi )$ and ${\overline{f}}_{+}(x,\xi )=f_{+}(x,-\xi )$ for $\xi =\zeta \in \mathbb{R}\setminus \{0\}$ are independent.

Here we consider the scattering problem on the line with transfer condition (1.2) at $x=0$. In order for (1.1) and (1.2) to be self-adjoint in $L^2((-\mathrm{\infty },0)\cup (0,\mathrm{\infty }))=L^2(\mathbb{R})$, the transfer matrix M will have to be of a certain form. Here, we restrict our attention to the most interesting case of M invertible, the case of M non-invertible will be considered elsewhere. The scattering problem can then be treated as two classical half-line problems joined at the origin by matrix condition (1.2).

It is easily verified that Ψ is a bijective isometry between $L^2(\mathbb{R})$ and ${(L^2(0,\mathrm{\infty }))}^2$ with the additional properties that $\mathrm{\Psi }(D(L))=D(T)$ and ${\mathrm{\Psi }}^{-1}T\mathrm{\Psi }=L$.

We now obtain conditions on the transfer matrix which ensure that T is a self-adjoint operator. We begin by defining the minimal and maximal operators associated with T.

Note that $T_0$ has deficiency indices $(2,2)$.

Theorem 3.1 If $detM\ne 0$, then the operator L is a self-adjoint operator if and only if $detM=1$.

Proof We will prove the result for the operator T and, consequently, it will be true for the operator L.

As the deficiency indices of $T_0$ are $(2,2)$, we need to restrict the domain of $T_{\max }$ by two boundary conditions to ensure self-adjointness, see Weidmann [[8], p.53]. For the operator to be self-adjoint with two linear domain conditions, they must ensure that $(f_1{\overline{g}}_1^{\prime }-f_1^{\prime }{\overline{g}}_1+f_2{\overline{g}}_2^{\prime }-f_2^{\prime }{\overline{g}}_2)(0)=0$.

Denote $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$, then we have that $J^T=-J$ and $J^2=-I$.

Thus ${(detM)}^2=-m_{12}{m}_{21}+m_{11}{m}_{22}=detM$ giving that $detM=1$ since M is invertible. □

It should be noted that in [1] a condition on the transfer matrix M is given for self-adjointness; however, for the case of $detM\ne 0$, where the entries of M are real, the proof presented above is substantially simpler.

Throughout the remainder of the paper, we will now assume that $detM=1$.

Theorem 3.2 Let $detM=1$. All eigenvalues (if any) of L as defined in (1.4), (1.5), and consequently of T, are negative.

if $|c_1|+|c_2|\ne 0$ and L has no λ positive eigenvalues.

is a solution of (1.1) which is asymptotic to x for $x\to \mathrm{\infty }$, [[7], p.173], and therefore linearly independent of $f_{+}(x,0)$. But $a{f}_{+}(x,0)+bg(x)$ is asymptotic to $a+bx$ as $x\to \mathrm{\infty }$, and thus not in $L^2(0,\mathrm{\infty })$ unless $|a|+|b|=0$. Hence ${\xi }^2=0$ is not an eigenvalue of L. □

We now study the continuous spectrum.

Theorem 3.3 The continuous spectrum of T (and thus of L) is $[0,\mathrm{\infty })$.

Proof From [[9], p.66] or [[10], p.251], as T is self-adjoint, the spectrum of T, $\sigma (T)$, is comprised of continuous and point spectrum only, i.e., T has no residual spectrum. In addition, the continuous spectrum of T is real. By [6] the essential spectrum of the minimal operator generated by T is $[0,\mathrm{\infty })$ and this is the same as the essential spectrum of T, see [10]. Thus, as T has no residual spectrum, we have that the continuous spectrum of T is $[0,\mathrm{\infty })$. □

We now consider the Jost solutions to problem (3.3). Since this problem can be considered as two half-line problems, solutions to (3.3) can be given in terms of the classical Jost solutions $f_{+}(x,\zeta )$ and $f_{-}(x,\zeta )$, i.e., when $M=I$.

Here, it should be noted that the existence of an extension of $f_{+}$ from $(0,\mathrm{\infty })$ to the solution $f_{+,M}$ on $\mathbb{R}\setminus \{0\}$ relies on M being non-singular.

where the transfer condition holds at $x=0$.

Note that (4.4) also holds on $(-\mathrm{\infty },0)$ as $detM=1$.

 □

Conversely, if $A(\zeta )=0$, then $f_{-,M}(x,\zeta )$ and $f_{+,M}(x,\zeta )$ are linearly dependent making $f_{+,M},f_{-,M}\in L^2(-\mathrm{\infty },\mathrm{\infty })$. Thus ${\zeta }^2$ is an eigenvalue and $\zeta =i\eta$ for some $\eta \in {\mathbb{R}}^{+}$.

Theorem 4.2 The function $A(\zeta )$ has a finite set of zeros.

Proof We know, from the above reasoning, that if $A(\zeta )=0$ then $\zeta =i\eta$ for some $\eta \in {\mathbb{R}}^{+}$. It should be noted that $A(\zeta )$ is analytic on the upper half-plane and continuous there and up to the boundary.

Here, we observe that $|cos(-\zeta \tau )e^{i\zeta |\tau |}|\le 1$ and $q\in L^1(\mathbb{R})$, thus the integral term in (4.11) is uniformly bounded in the upper half-plane. In addition, as $detM=1$ and M has real entries, not both of $m_{12}$ and $m_{11}+m_{22}$ can simultaneously be zero. Hence $|A(\zeta )|$ is bounded away from zero for large ζ in the upper half-plane giving that $A(\zeta )$ does not have an unbounded sequence of zeros.

Using the same approach as in [6], by Lemma 4.1, $\frac{1}{|A(\zeta )|}$ is bounded in a neighbourhood of the real axis, so the zeros of $A(\zeta )$ have no finite accumulation point and are thus a finite set. □

From Theorem 4.2 and the reasoning preceding Theorem 4.2, we obtain the following corollary.

Corollary 4.3 The scattering problem with a transfer condition has a finite number of eigenvalues, all of which are simple.

We note for reference the following properties of $B(\xi )$.

where $f_{-,M}(x,i{\eta }_k)=d_k{f}_{+,M}(x,i{\eta }_k)$ for all $x\in \mathbb{R}\setminus \{0\}$ and $d_k\ne 0$.