Scattering matrix, phase shift, spectral shift and trace formula for one-dimensional dissipative Schrödinger-type operators (Q2470878)

This paper is devoted to real-valued potential Schrödinger operators with (possibly non-self-adjoint) dissipative boundary conditions on a compact interval. Continuing the analysis initiated in [\textit{H.--C.\thinspace Kaiser, H.\,Neidhardt} and \textit{J.\,Rehberg}, Math.\ Nachr.\ 252, 51--69 (2003; Zbl 1028.47023)], the authors study the phase shift and the high energy asymptotic behaviour of the associated Lax--Phillips scattering theory. Two results are to be highlighted. On the one hand, they show that the phase shift of the scattering matrix has the same asymptotic behaviour as the eigenvalue distribution function of the Dirichlet problem. On the other hand, they establish a precise connection between spectral shift, phase shift and eigenvalue distribution. Using these connections, they are able to recover direct proofs of the trace and the Birman--Krein formulae