Inverse scattering on matrices with boundary conditions

Abstract

We describe inverse scattering for the matrix Schrödinger operator with general selfadjoint boundary conditions at the origin using the Marchenko equation. Our approach allows the recovery of the potential as well as the boundary conditions. It is easily specialised to inverse scattering on star-shaped graphs with boundary conditions at the node.

Scattering theory for the matrix Schrödinger operator on the half line with general boundary conditions

“…Matrix Schrödinger operators on the half line are important in quantum mechanical scattering of particles with internal structure, in quantum graphs and in quantum wires. See, for example, [1]- [5] and [9]- [21], and the references quoted there.The matrix Schrödinger operator on the half line (1.1) corresponds to a star graph. It describes the behavior of n connected very thin quantum wires that form a star-graph, i.e.…”

Schrödinger operators with nonlocal point interactions

“…Indeed, since χ L 2 2 √ , the determinant of system (15) approaches 1 as → 0, thus it is not zero if √ ( f 1 + f 2 ) < 1 4 . This means that if B 1 and B 2 in definition (14) of the function u are chosen to be solutions of system (15), then Γ 0 u = (C 1 , C 2 ) and (7) and Theorem 1, we get the known result that (8) is a boundary-value triple for the one-dimensional Schrödinger operator with a one-point interaction in the point x = 0. In particular, this gives a description of all self-adjoint one-dimensional Schrödinger operators on L 2 (R 1 ) with a one-point interaction in terms of one of the following boundary-value conditions for ψ ∈ W 2 2 (R 1 \ {0}) [4,8,12,14,15,17,18,25]:…”

Some inverse scattering problems on star-shaped graphs

“…The article [17] considers a star-shaped graph consisting of N infinite branches and solves the inverse scattering problem assuming the measurement of N − 1 reflection coefficients. Next, in [18], Harmer provides an extension of the previous result with general self-adjoint boundary conditions at the central node. This however necessitates the knowledge of N reflection coefficients.…”