2013/02/18 by Nurulla Azamov, Azamov, Nurulla
Computer Science · Mathematics · Physics and Astronomy · #47A55 #FOS: Physical sciences #Mathematical Physics (math-ph) #Matrix Theory and Algorithms #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:47A55
paper · pdf · doi:10.48550/arxiv.1302.4142
35 pages
arxiv created 2013/02/18 · openalex publication_date 2013/02/18 · arxiv updated 2013/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we give a new and constructive approach to stationary scattering theory for pairs of self-adjoint operators H0 and H1 on a Hilbert space \mathcal H which satisfy the following conditions: (i) for any open bounded subset Δ of \mathbb R, the operators F EΔH0 and F EΔH1 are Hilbert-Schmidt and (ii) V = H1- H0 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 Schrödinger operator H0 = -Δ+ V0 acting on L2(\mathbb Rν), where V0 is a potential of class Kν (see B. Simon, \it Schrödinger semigroups, Bull. AMS 7, 1982, 447--526) and H1 = H0 + V1, where V1 ∈ L^∞(\mathbb Rν) ∩ L1(\mathbb Rν). Among results of this paper is a new proof of existence and completeness of wave operators W_±(H1,H0) 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_±(λ; H1,H0) acting between fibers \mathfrak hλ(H0,F) and \mathfrak hλ(H1,F) of sheaves \EuScript S(H0,F) and \EuScript S(H1,F) respectively. The wave operators W_±(H1,H0) are then defined as direct integrals of wave matrices and are proved to coincide with classical time-dependent definition of wave operators.