Quantum mechanics of $\mathcal {P}\mathcal {T}$ and non-$\mathcal {P}\mathcal {T}$-symmetric potentials in three dimensions

Abstract

With a view of exploring new vistas with regard to the nature of complex eigenspectra of a non-Hermitian Hamiltonian, the quasi-exact solutions of the Schrödinger equation are investigated for a shifted harmonic potential under the framework of extended complex phase-space approach. Analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of the eigenvalues and eigenfunctions of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as excited state including their $\mathcal {P}\mathcal {T}$-symmetric version are worked out.