Structure, Time Propagation and Dissipative Terms for Resonances

TL;DR

This paper reviews and extends complex scaling methods for resonances, introducing a unified approach for structure, time evolution, and dissipation in quantum systems with unstable resonances.

Contribution

It generalizes complex scaling to include time propagation and dissipative effects, unifying resonance structure and dynamics in quantum systems.

Findings

Time propagation results match existing quantum dynamics solvers

Method incorporates dissipative processes naturally

Provides a unified framework for resonance structure and dynamics

Abstract

For odd anharmonic oscillators, it is well known that complex scaling can be used to determine resonance energy eigenvalues and the corresponding eigenvectors in complex rotated space. We briefly review and discuss various methods for the numerical determination of such eigenvalues, and also discuss the connection to the case of purely imaginary coupling, which is PT-symmetric. Moreover, we show that a suitable generalization of the complex scaling method leads to an algorithm for the time propagation of wave packets in potentials which give rise to unstable resonances. This leads to a certain unification of the structure and the dynamics. Our time propagation results agree with known quantum dynamics solvers and allow for a natural incorporation of structural perturbations (e.g., due to dissipative processes) into the quantum dynamics.