Inverse Scattering on the Line with a Transfer Condition

Abstract

The inverse scattering problem for Sturm-Liouville operators on the line with a matrix transfer condition at the origin is considered. We show that the transfer matrix can be reconstructed from the eigenvalues and reflection coefficient. In addition, for potentials with compact essential support, we show that the potential can be uniquely reconstructed. *

Inverse problems with a general transfer condition

“…Secondly there is the scattering problem (1.1), (1.2) on (−∞, 0) ∪ (0, ∞) where the potential q has compact essential support in [−S, S]. In [9] we show that the scattering data uniquely determines the m-function, m(λ), and indeed the converse holds i.e. given m(λ) we can uniquely find the scattering data.…”

“…The proofs follow in exactly the same manner, we will point out the main differences and provide the necessary asymptotics. Therefore as in [9] we can find w 1 (S, ξ) and w 2 (S, ξ) and since w 1 and w 2 are entire, by analyticity we can extend them to w 1 (S, ζ) and w 2 (S, ζ). 2) and m, the Titchmarsh-Weyl m-function for the same problem but with the potential q replaced by q.…”

“…Proof: The proof follows identically to that given in [9,Theorem 4.2]. It should be noted that only the asymptotics for v and w 2 used in Theorems 5.3 and 5.4 change.…”

“…the norming constants for the Neumann-Neumann problem on [−S, S], given in (?? ), can be uniquely determined.To conclude we combine the above theorem together with the results from Section 2 as well as those given in[9, Sections 3 and 4] to obtain the following Corollary.Corollary 4.7 Given the scattering data i.e. the reflection coefficient and eigenvalues of (4.1), (4.2) it is possible to obtain the spectra of the Neumann-Neumann and Neumann-Dirichlet problems on [−S, S] or the spectrum and norming constants of the Neumann-Neumann problem on [−S, S].…”

“…Firstly there is the finite interval problem (1.1) and (1.2). For this problem we summarize, from [9], how the Titchmarsh-Weyl m-function, m(λ), of (1.1) on [−S, S] obeying the transfer condition (1.2) with separated boundary conditions at the end points is defined. An asymptotic approximation for m(λ) is then obtained and consequently we show that m(λ) can be uniquely determined from two spectra.…”