# Remarks to the Resonance-Decay Problem in Quantum Mechanics from a mathematical point of view

## TL;DR

This paper discusses the mathematical formulation of the resonance-decay problem in quantum mechanics, highlighting the Lax-Phillips approach, spectral theory, and the limitations of existing solutions, with a focus on scattering matrices and decaying states.

## Contribution

It provides a detailed analysis of the resonance-decay problem using spectral theory and the Lax-Phillips approach, including a new solution with a meromorphic scattering matrix.

## Key findings

- A solution for scattering systems with semi-bounded Hamiltonians and holomorphic scattering matrices.
- A comparison of different approaches to the spectral-theoretical component.
- A No-Go theorem indicating the limitations of non-relativistic quantum mechanics in solving the problem.

## Abstract

The description of bumps in scattering cross-sections by Breit-Wigner amplitudes led in the framework of the mathematical Physics to its formulation as the so-called Resonance-Decay Problem. It consists of a spectraltheoretical component and the connection of this component with the construction of decaying states. First the note quotes a solution for scattering systems, where the absolutely continuous parts of the Hamiltonians are semi-bounded and the scattering matrix is holomorphic in the upper half plane. This result uses the approach developed by Lax and Phillips, where the energy scale is extended to the whole real axis. The relationship of the spectraltheoretical part of its solution and corresponding solutions obtained by other approaches is explained in the case of the Friedrichs model. A No-Go theorem shows the impossibility of the total solution within the specific framework of non-relativistic quantum mechanics. This points to the importance of the Lax-Phillips approach. At last, a solution is presented, where the scattering matrix is meromorphic in the upper half plane.