Scattering theory with localized non-Hermiticities

TL;DR

This paper explores how choosing different metrics in non-Hermitian Hamiltonian models can resolve paradoxes in scattering theory, demonstrated through an exactly solvable toy model.

Contribution

It introduces a method to select metrics that clarify physical interpretations in non-Hermitian scattering models, resolving existing paradoxes.

Findings

A suitable metric restores a consistent physical scattering picture.

The approach is demonstrated on an exactly solvable toy model.

The variability of the metric can resolve apparent paradoxes in non-Hermitian scattering.

Abstract

In the context of the recent interest in solvable models of scattering mediated by non-Hermitian Hamiltonians (cf. H. F. Jones, Phys. Rev. D 76, 125003 (2007)) we show that and how the well known variability of our ad hoc choice of the metric $\Theta$ which defines the physical Hilbert space of states can help us to clarify several apparent paradoxes. We argue that with a suitable $\Theta$ a fully plausible physical picture of the scattering is recovered. Quantitatively, our new recipe is illustrated on an exactly solvable toy model.