A constructive approach to stationary scattering theory

Abstract

In this paper we give a new and constructive approach to stationary scattering theory for pairs of self-adjoint operators $H_0$ and $H_1$ on a Hilbert space $\mathcal H$ which satisfy the following conditions: (i) for any open bounded subset $\Delta$ of $\mathbb R,$ the operators $F E_\Delta^{H_0}$ and $F E_\Delta^{H_1}$ are Hilbert-Schmidt and (ii) $V = H_1- H_0$ is bounded and admits decomposition $V = F^*JF,$ where $F$ is a bounded operator with trivial kernel from $\mathcal H$ to another Hilbert space $\mathcal K$ and $J$ is a bounded self-adjoint operator on $\mathcal K.$ An example of a pair of operators which satisfy these conditions is the Schr\"odinger operator $H_0 = -\Delta + V_0$ acting on $L^2(\mathbb R^\nu),$ where $V_0$ is a potential of class $K_\nu$ (see B.\,Simon, {\it Schr\"odinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 \in L^\infty(\mathbb R^\nu) \cap L^1(\mathbb R^\nu).$ Among results of this paper is a new proof of existence and completeness of wave operators $W_\pm(H_1,H_0)$ and a new constructive proof of stationary formula for the scattering matrix. This approach to scattering theory is based on explicit diagonalization of a self-adjoint operator $H$ on a sheaf of Hilbert spaces $\EuScript S(H,F)$ associated with the pair $(H,F)$ and with subsequent construction and study of properties of wave matrices $w_\pm(\lambda; H_1,H_0)$ acting between fibers $\mathfrak h_\lambda(H_0,F)$ and $\mathfrak h_\lambda(H_1,F)$ of sheaves $\EuScript S(H_0,F)$ and $\EuScript S(H_1,F)$ respectively. The wave operators $W_\pm(H_1,H_0)$ are then defined as direct integrals of wave matrices and are proved to coincide with classical time-dependent definition of wave operators.