Quantum mechanics of P T $\mathcal {P}\mathcal {T}$ and non- P T $\mathcal {P}\mathcal {T}$ -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.

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