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

Abstract

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

9 pages, 1 figure

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