# Heisenberg uncertainty relations for the non-Hermitian resonance state solutions to the Schr\"odinger equation

## TL;DR

This paper extends the Heisenberg uncertainty relations to non-Hermitian resonance states in quantum mechanics, deriving analytical expressions and validating the relations through model calculations.

## Contribution

It introduces a new definition of expectation values for resonance states and demonstrates the validity of uncertainty relations for these non-Hermitian solutions.

## Key findings

- Heisenberg uncertainty relations hold for a broad range of potential parameters.
- Derived analytical expressions for expectation values of position and momentum.
- Comparison with Zel'dovich's regularization method shows similar results except near the energy threshold.

## Abstract

Resonance (quasinormal) states correspond to non-Hermitian solutions to the Schr\"odinger equation obeying outgoing boundary conditions which lead to complex energy eigenvalues and momenta. Following the normalization rule for resonance states obtained from the residue at a complex pole of the outgoing Green's function to the problem, we propose a definition of expectation value for these states and use it to investigate the extent of validity of the Heisenberg uncertainty relations for potentials that vanish after a distance. We derive analytical expressions for the expectation values involving the momentum and the position for a given resonance state and find in model calculations that the Heisenberg uncertainty relations are satisfied for a broad range of potential parameters. A comparison of our approach with that based on the regularization method by Zel'dovich yields very similar results except for resonance energies very close to the energy threshold. Our work shows that the validity of the Heisenberg uncertainty relations may be extended to the non-Hermitian resonance state solutions to the Schr\"odinger equation.