Equivalent Hermitian operator from supersymmetric quantum mechanics

TL;DR

This paper demonstrates how to derive an equivalent Hermitian operator from supersymmetric quantum mechanics, even when the associated metric operator is singular, by using eigenfunctions of the Hermitian counterpart.

Contribution

It introduces a method to find Hermitian equivalents of pseudo-Hermitian Hamiltonians in supersymmetric quantum mechanics despite singular metric operators.

Findings

Eigenfunctions of the Hermitian operator can be obtained using the singular metric operator.

The Hermitian operator can be reconstructed via spectral decomposition.

The approach applies to pseudo-Hermitian Hamiltonians with real, discrete spectra.

Abstract

Diagonalizable pseudo-Hermitian Hamiltonians with real and discrete spectra, which are superpartners of Hermitian Hamiltonians, must be $\eta$-pseudo-Hermitian with Hermitian, positive-definite and non-singular $\eta$ operators. We show that despite the fact that an $\eta$ operator produced by a supersymmetric transformation, corresponding to the exact supersymmetry, is singular, it can be used to find the eigenfunctions of a Hermitian operator equivalent to the given pseudo-Hermitian Hamiltonian. Once the eigenfunctions of the Hermitian operator are found the operator may be reconstructed with the help of the spectral decomposition.