Remarks to the Resonance-Decay Problem in Quantum Mechanics from a Mathematical Point of View

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 spectral theoretical 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 semibounded 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 spectral theoretic 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.