Scattering matrices for dissipative quantum systems

TL;DR

This paper develops a scattering theory framework for dissipative quantum systems modeled by pseudo-Hamiltonians, providing a formula for scattering matrices and conditions for their invertibility, with applications to nuclear optical models.

Contribution

It introduces a representation formula for scattering matrices in dissipative quantum systems and characterizes spectral singularities affecting their invertibility.

Findings

Derived a formula for scattering matrices in dissipative systems.

Identified spectral singularities as a key factor for matrix invertibility.

Applied results to nuclear optical models with real resonances.

Abstract

We consider a quantum system S interacting with another system S and susceptible of being absorbed by S. The effective, dissipative dynamics of S is supposed to be generated by an abstract pseudo-Hamiltonian of the form H = H0 + V -- iC * C. The generator of the free dynamics, H0, is self-adjoint, V is symmetric and C is bounded. We study the scattering theory for the pair of operators (H, H0). We establish a representation formula for the scattering matrices and identify a necessary and sufficient condition to their invertibility. This condition rests on a suitable notion of spectral singularity. Our main application is the nuclear optical model, where H is a dissipative Schr{\"o}dinger operator and spectral singularities correspond to real resonances.