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 $V=F^{\ast }JF,$ 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 Schrödinger 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 Schrödinger 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.

35 pages

References in corpus (1)

• Absolutely continuous and singular spectral shift functions