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

Abstract

We study the stationary scattering theory for the matrix Schrödinger equation on the half line, with the most general boundary condition at the origin, and with integrable selfadjoint matrix potentials. We prove the limiting absorption principle, we construct the generalized Fourier maps, and we prove that they are partially isometric with initial space the subspace of absolute continuity of the matrix Schrödinger operator and final space L 2 ((0, ∞)). We prove the existence and the completeness of the wave operators and we establish that they are given by the stationary formulae. We also construct the spectral shift function and we give its high-energy asymptotics. Furthermore, assuming that the potential also has a finite first moment, we prove a Levinson's theorem for the spectral shift function.

The number of eigenvalues of the matrix Schrödinger operator on the half line with general boundary conditions

“…It is proven in [9] (see also Section 3 below) that the formal differential operator (1.1) has a selfadjoint, bounded below, realization in L 2 , defined by quadratic forms, with the boundary condition (1.2). We also denote this selfadjoint realization by H A,B .…”

“…We also denote this selfadjoint realization by H A,B . For this purpose, we only need that (1.5) holds and that the potential matrix is integrable, Furthermore, it is proven in [9] that H A,B has no singular continuous spectrum and its absolutely continuos spectrum is [0, ∞). Moreover, H A,B has no positive or zero eigenvalues and its negative eigenvalues are of finite multiplicity and can only accumulate at zero.…”

“…Moreover, H A,B has no positive or zero eigenvalues and its negative eigenvalues are of finite multiplicity and can only accumulate at zero. If V is in the Faddeev class (1.6) the number of eigenvalues of H A,B is finite [7], [9].…”

“…In Section 2 we introduce notations and definitions and we state preliminary results that we need. In Section 3, following [9], we define the matrix Schrödinger operator as a selfadjoint operator by means of quadratic form techniques and we estimate the difference in the number of bound states of two matrix Schrödinger operators with the same potential and with different boundary conditions. In Section 4 we state results from [9] on the integral kernel of the resolvent in the case where the potential matrix is identically zero.…”

Inverse Scattering on the Half Line for the Matrix Schrodinger Equation

“…This definition is motivated by the theory of quantum graphs, where the Neumann boundary condition is usually used for the unperturbed problem. We refer the reader to [15,17,18,22] for further details.…”