Inverse scattering for the matrix Schrödinger operator and Schrödinger operator on graphs with general self-adjoint boundary conditions

Abstract

Using a parameterisation of general self-adjoint boundary conditions in terms of Lagrange planes we propose a scheme for factorising the matrix Schrodinger operator and hence construct a Darboux transformation, an interesting feature of which is that the matrix potential and boundary conditions are altered under the transformation. We present a solution of the inverse problem in the case of general boundary conditions using a Marchenko equation and discuss the specialisation to the case of a graph with trivial compact part, that is, with diagonal matrix potential. The matrix Schrodinger operator on the semi-axisWe consider here the matrix Schrodinger operator on the semi-axis, that is, . The matrix Schrodinger operator, J2o, defined on smooth functions vanishing at the origin and with compact support, is a symmetric operator with deficiency indices (n, n). Using von Neumann extension theory [2] we may parameterise all self-adjoint extensions of this operator by unitary mappings between the deficiency subspaces, that is, U e U(n). For practical purposes, however, it is more convenient to describe the self-adjoint extensions in terms of self-adjoint boundary conditions at the origin.

Hermitian symplectic geometry and extension theory

“…It is convenient to consider the Schrödinger operator on the graph with n rays as a matrix operator, with diagonal potential, see [8,9]. Let us consider the matrix of n solutions of Schrödinger equation LΞ = λΞ on the graph satisfying the following boundary conditions at the origin…”

Spectral problems and scattering on noncompact star-shaped graphs containing finite rays

“…Results of [1] are extended by Harmer [9] to arbitrary selfadjoint boundary conditions at origin, and then it becomes possible to treat the operator on noncompact star graphs as a specialization to the case of operator with diagonal matrix potential.…”

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.…”