An alternative construction of the positive inner product in non-Hermitian quantum mechanics

TL;DR

This paper introduces a new method for constructing a positive inner product in non-Hermitian quantum mechanics using eigenvector generators, extending previous approaches to a broader class of operators.

Contribution

It presents an alternative construction of the positive inner product applicable to pseudo-Hermitian operators, expanding the theoretical framework of non-Hermitian quantum mechanics.

Findings

Constructs a unitary inner product space using eigenvector generators.

Extends the construction to pseudo-Hermitian operators beyond PT symmetry.

Provides a detailed example illustrating the new approach.

Abstract

Within the context of non-Hermitian quantum mechanics, we use the generators of eigenvectors of the Hamiltonian to construct a unitary inner product space. Such models have been of interest in recent years, for instance, in the context of ${\cal PT}$ symmetry, although our construction extends to the larger class of so-called pseudo-Hermitian Operators. We provide a detailed example to illustrate the concept and compare with known results.