An alternative construction of the positive inner product for pseudo-Hermitian Hamiltonians: Examples

TL;DR

This paper presents a new method for constructing positive inner products for pseudo-Hermitian Hamiltonians, with examples including harmonic oscillators, finite-dimensional matrices, and perturbative approaches for non-diagonalizable systems.

Contribution

It introduces an alternative construction method for positive inner products applicable to a broad class of pseudo-Hermitian systems, including perturbative cases.

Findings

Method applies to Hermitian and pseudo-Hermitian Hamiltonians.

Constructs positive inner products for 2x2 matrix Hamiltonians.

Perturbative formalism developed for non-diagonalizable systems.

Abstract

This paper builds on our earlier proposal for construction of a positive inner product for pseudo-Hermitian Hamiltonians and we give several examples to clarify our method. We show through the example of the harmonic oscillator how our construction applies equally well to Hermitian Hamiltonians which form a subset of pseudo-Hermitian systems. For finite dimensional pseudo-Hermitian matrix Hamiltonians we construct the positive inner product (in the case of $2\times 2$ matrices for both real as well as complex eigenvalues). When the quantum mechanical system cannot be diagonalized exactly, our construction can be carried out perturbatively and we develop the general formalism for such a perturbative calculation systematically (for real eigenvalues). We illustrate how this general formalism works out in practice by calculating the inner product for a couple of ${\cal PT}$ symmetric quantum mechanical theories.