Scattering Matrix, Phase Shift, Spectral Shift and Trace Formula for One-dimensional Dissipative Schrödinger-type Operators

Abstract

The paper is devoted to Schrödinger operators with dissipative boundary conditions on bounded intervals. In the framework of the LaxPhillips scattering theory the asymptotic behaviour of the phase shift is investigated in detail and its relation to the spectral shift is discussed. In particular, the trace formula and the Birman-Krein formula are verified directly. The results are exploited for dissipative Schrödinger-Poisson systems. Mathematics Subject Classification (2000). Primary 47A20; Secondary 47B44, 47A40.

Uniqueness for Dissipative Schrödinger-Poisson Systems

“…The phase shift ω[V ] is defined by where H D [V ] denotes the Schrödinger-type operator with Dirichlet boundary conditions. Theorem 3.6 [26,Theorem 4.7] Let the Schrödinger assumption Q 1 and Q 2 be satisfied.…”

“…Let us verify that the projection P K H is K[V ]-smooth. To this end we need the following lemma which was proved in [26].…”