Conceptual Problems in Scattering from Localized non-Hermitian Potentials

TL;DR

This paper examines conceptual issues in applying quasi-Hermitian frameworks to scattering from localized non-Hermitian potentials, emphasizing the non-locality of the metric and its implications for probability conservation.

Contribution

It clarifies the conceptual problems and proposes a detailed analysis of the non-local metric in non-Hermitian scattering models, especially for delta-function potentials.

Findings

Probability conservation requires a non-local metric.

Standard probability differs from the new one at large distances.

Bound states unaffected by distant non-Hermitian potentials.

Abstract

We highlight the conceptual issues that arise when one applies the quasi-Hermitian framework to analyze scattering from localized non-Hermitian potentials, in particular complex square-wells or delta-functions. When treated in the framework of conventional quantum mechanics, these potentials are generally considered as effective theories, in which probability is not conserved because of processes that have been ignored. However, if they are treated as fundamental theories, the Hilbert-space metric must be changed. In order for the newly-defined probability to be conserved, it must differ from the standard one, even at asymptotically large distances from the scattering centre, and the mechanism for this is the non-locality of the new metric, as we show in detail in the model of a single complex delta function. However, properties of distant bound-state systems, which do not interact physically with the non-Hermitian scattering potential, should not be affected. We analyze a model Hamiltonian that supports this contention.