Complexified von Roos Hamiltonian's η-weak-pseudo-Hermiticity, isospectrality and exact solvability

Abstract

A complexified von Roos Hamiltonian is considered and a Hermitian first-order intertwining differential operator is used to obtain the related position-dependent mass η-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the η-weak-pseudo-Hermitian von Roos Hamiltonians Hx are mapped into the traditional Schrödinger Hamiltonian form Hq, where exact isospectral correspondence between Hx and Hq is obtained. Under a 'user-friendly' position-dependent-mass setting, it is observed that for each exactly solvable η-weak-pseudo-Hermitian reference-Hamiltonian Hq there is a set of exactly solvable η-weak-pseudo-Hermitian isospectral target-Hamiltonians Hx. A non-Hermitian \mathcal{PT} -symmetric Scarf II and a non-Hermitian periodic-type \mathcal{PT} -symmetric Samsonov–Roy potentials are used as reference models and the corresponding η-weak-pseudo-Hermitian isospectral target-Hamiltonians are obtained.