Isospectral Hermitian counterpart of complex non Hermitian Hamiltonian $p^{2}-gx^{4}+a/x^{2}$

TL;DR

This paper demonstrates that certain non-Hermitian Hamiltonians are isospectral to Hermitian ones, revealing a new class of such pairs and extending previous results to cases with complex parameters.

Contribution

It establishes conditions under which non-Hermitian Hamiltonians are isospectral to Hermitian counterparts, including cases with complex parameters, expanding the understanding of spectral equivalences.

Findings

Non-Hermitian Hamiltonians $H=p^{2}-gx^{4}+a/x^{2}$ are isospectral to Hermitian $h=p^2+4gx^{4}+bx$ under specific parameter relations.

The class includes previously known pairs as special cases.

For certain parameters, the Hermitian counterpart becomes non-Hermitian with complex $b$, yet the spectra remain identical.

Abstract

In this paper we show that the non-Hermitian Hamiltonians $H=p^{2}-gx^{4}+a/x^2$ and the conventional Hermitian Hamiltonians $h=p^2+4gx^{4}+bx$ ($a,b\in \mathbb{R}$) are isospectral if $a=(b^2-4g\hbar^2)/16g$ and $a\geq -\hbar^2/4$. This new class includes the equivalent non-Hermitian - Hermitian Hamiltonian pair, $p^{2}-gx^{4}$ and $p^{2}+4gx^{4}-2\hbar \sqrt{g}x,$ found by Jones and Mateo six years ago as a special case. When $a=\left(b^{2}-4g\hbar ^{2}\right) /16g$ and $a<-\hbar^2/4,$ although $h$ and $H$ are still isospectral, $b$ is complex and $h$ is no longer the Hermitian counterpart of $H$.