Generic nature of asymptotic completeness in dissipative scattering theory

Abstract

We review recent results obtained in the scattering theory of dissipative quantum systems representing the long-time evolution of a system $S$ interacting with another system $S^{\prime }$ and susceptible of being absorbed by $S^{\prime }$. The effective dynamics of $S$ is generated by an operator of the form $H=H_0+V-\mathrm{i}{C}^{\ast }C$ on the Hilbert space of the pure states of $S$, where $H_0$ is the self-adjoint generator of the free dynamics of $S$, $V$ is symmetric and $C$ is bounded. The main example is a neutron interacting with a nucleus in the nuclear optical model. We recall the basic objects of the scattering theory for the pair $(H,H_0)$, as well as the results, proven in arXiv:1703.09018 and arXiv:1808.09179, on the spectral singularities of $H$ and the asymptotic completeness of the wave operators. Next, for the nuclear optical model, we show that asymptotic completeness generically holds.

Contribution to the proceedings of QMATH 14: Mathematical Results in Quantum Physics. Contains an overview of arXiv:1703.09018, arXiv:1808.09179 and the proof of a new result. 18 pages, 1 figure