Time behaviour near to spectral singularities

TL;DR

This paper investigates the time evolution of wave functions near spectral singularities, revealing unique decay behaviors and linear time dependence at exceptional points in a simple quantum model.

Contribution

It demonstrates how initial conditions influence wave function decay near spectral singularities and characterizes the linear time dependence at exceptional points.

Findings

Wave functions exhibit very fast decay or long lifetime near exceptional points.

At the exceptional point, wave functions include a linear time term.

The study provides insights into time behavior in non-Hermitian quantum systems.

Abstract

Spectral singularities such as exceptional points invoke specific physical effects. The present paper focuses upon the time dependent solutions of the Schr\"odinger equation. In a simple model it is demonstrated that - depending on initial conditions - within close proximity of exceptional points the time behaviour of the wave function displays characteristic features such as very fast decay or the opposite, i.e. very long life time. At the exceptional point the wave function typically has a linear term in time besides the usual exponential behaviour.