Pseudo-Supersymmetric Quantum Mechanics and Isospectral Pseudo-Hermi tian Hamiltonians

TL;DR

This paper explores pseudo-supersymmetry in quantum systems with pseudo-Hermitian Hamiltonians, showing how it relates to isospectrality, Hamiltonian factorization, and the generation of real spectra in non-Hermitian systems.

Contribution

It establishes pseudo-supersymmetry as a fundamental framework for understanding and constructing isospectral pseudo-Hermitian Hamiltonians with real spectra.

Findings

Pseudo-supersymmetry ensures isospectrality of Hamiltonian pairs with real or complex-conjugate eigenvalues.

Every diagonalizable Hamiltonian with discrete spectrum can be factored as H=L^# L, linking to pseudo-supersymmetry.

Application to two-level systems demonstrates the practical implications of the theoretical results.

Abstract

We examine the properties and consequences of pseudo-supersymmetry for quantum systems admitting a pseudo-Hermitian Hamiltonian. We explore the Witten index of pseudo-supersymmetry and show that every pair of diagonalizable (not necessarily Hermitian) Hamiltonians with discrete spectra and real or complex-conjugate pairs of eigenvalues are isospectral and have identical degeneracy structure except perhaps for the zero eigenvalue if and only if they are pseudo-supersymmetric partners. This implies that pseudo-supersymmetry is the basic framework for generating non-Hermitian PT-symmetric and non-PT-symmetric Hamiltonians with a real spectrum via a Darboux transformation, and shows that every diagonalizable Hamiltonian H with a discrete spectrum and real or complex-conjugate pairs of eigenvalues may be factored as H=L^# L where L is a linear operator with pseudo-adjoint L^#. In particular, this factorization applies to PT-symmetric and Hermitian Hamiltonians. The nondegenerate two-level systems provide a class of Hamiltonians that are pseudo-Hermitian. We demonstrate the implications of our general results for this class in some detail.