Non-isospectrality of the generalized Swanson Hamiltonian and harmonic oscillator

TL;DR

This paper investigates the spectral properties of a generalized Swanson Hamiltonian, demonstrating that it is not always isospectral to the harmonic oscillator despite having a constant commutator, especially within position-dependent mass frameworks.

Contribution

It reveals that the generalized Swanson Hamiltonian can differ spectrally from the harmonic oscillator even with a constant commutator, challenging previous assumptions about their equivalence.

Findings

The Hamiltonian is not necessarily isospectral to the harmonic oscillator.

Constant commutator does not guarantee spectral equivalence.

Position-dependent mass models explain the spectral anomaly.

Abstract

The generalized Swanson Hamiltonian $H_{GS} = w (\tilde{a}\tilde{a}^\dag+ 1/2) + \alpha \tilde{a}^2 + \beta \tilde{a}^{\dag^2}$ with $\tilde{a} = A(x)d/dx + B(x)$, can be transformed into an equivalent Hermitian Hamiltonian with the help of a similarity transformation. It is shown that the equivalent Hermitian Hamiltonian can be further transformed into the harmonic oscillator Hamiltonian so long as $[\tilde{a},\tilde{a}^\dag]=$ constant. However, the main objective of this paper is to show that though the commutator of $\tilde{a}$ and $\tilde{a}^\dag$ is constant, the generalized Swanson Hamiltonian is not necessarily isospectral to the harmonic oscillator. Reason for this anomaly is discussed in the frame work of position dependent mass models by choosing $A(x)$ as the inverse square root of the mass function.