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

Asiri Nanayakkara, Thilagarajah Mathanaranjan

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 $$H$$.

http://arxiv.org/abs/1407.4633
Mathematical Physics (math-ph)