Non-Hermitian von Roos Hamiltonian's $\eta$-weak-pseudo-Hermiticity, isospectrality and exact solvability

TL;DR

This paper explores complexified von Roos Hamiltonians, establishing their $ ext{eta}$-weak-pseudo-Hermiticity and isospectrality with solvable Schrödinger models through variable transformations and specific PT-symmetric potentials.

Contribution

It introduces a method to construct exactly solvable $ ext{eta}$-weak-pseudo-Hermitian Hamiltonians from known models via variable change and intertwining operators.

Findings

Exact isospectral correspondence between complexified von Roos and Schrödinger Hamiltonians.

Construction of new exactly solvable $ ext{eta}$-weak-pseudo-Hermitian Hamiltonians.

Application to PT-symmetric Scarf II and Samsonov-Roy potentials.

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 $\eta$-weak-pseudo-Hermitian Hamiltonians. Using a Liouvillean-type change of variables, the $\eta$-weak-pseudo-Hermitian von Roos Hamiltonians H(x) are mapped into the traditional Schrodinger Hamiltonian form H(q), where exact isospectral correspondence between H(x) and H(q) is obtained. Under a user-friendly position dependent mass settings, it is observed that for each exactly-solvable $\eta$-weak-pseudo-Hermitian reference-Hamiltonian H(q)there is a set of exactly-solvable $\eta$-weak-pseudo-Hermitian isospectral target-Hamiltonians H(x). A non-Hermitian PT-symmetric Scarf II and a non-Hermitian periodic-type PT-symmetric Samsonov-Roy potentials are used as reference models and the corresponding $\eta$-weak-pseudo-Hermitian isospectral target-Hamiltonians are obtained.