Decay Semigroups for the Resonances of Quantum Mechanical Scattering Systems

TL;DR

This paper constructs decay semigroups for certain quantum Hamiltonians, linking their spectra directly to resonances, using an adapted Lax-Phillips theory under specific analyticity and multiplicity conditions.

Contribution

It introduces a canonical method to associate decay semigroups with quantum Hamiltonians, connecting their spectra to resonances through a novel adaptation of scattering theory.

Findings

Spectrum of the generator matches all resonances.

Applicable to trace class perturbations with analyticity.

Extended to potential scattering with symmetric, compact support potentials.

Abstract

For selected classes of quantum mechanical Hamiltonians a canonical association of a decay semigroup is presented. The spectrum of the generator of this semigroup is a pure eigenvalue spectrum and it coincides with the set of all resonances. The essential condition for the results is the meromorphic continuability of the scattering matrix onto $\Bbb{C}\setminus(-\infty,0]$ and the rims $\Bbb{R}_{-}\pm i0$. Further finite multiplicity is assumed. The approach is based on an adaption of the Lax-Phillips scattering theory to semi-bounded Hamiltonians. It is applied to trace class perturbations with analyticity conditions. A further example is the potential scattering for central-symmetric potentials with compact support and angular momentum 0.