Anisotropic oscillator with pseudo-hermiticity and its noncommutative extension

TL;DR

This paper constructs a PT pseudo-hermitian anisotropic oscillator with a real spectrum, extending it to noncommutative space, highlighting the robustness of spectral reality under such modifications.

Contribution

It introduces a new PT pseudo-hermitian anisotropic oscillator model and extends it to noncommutative space, demonstrating the preservation of real spectra.

Findings

The constructed Hamiltonian has a real spectrum.

The real spectrum persists in noncommutative space.

Extension to noncommutative space does not break spectral reality.

Abstract

When a non-hermitian hamiltonian has a certain symmetry, such as the PT pseudo-hermiticity, it is still possible that the hamiltonian has a real spectrum. In this note, by adding an imaginary potential proportional to ip_1p_2 to the hamiltonian of an anisotropic planar oscillator, we construct a PT pseudo-hermitian hamiltonian and obtain its real spectrum. In addition, we find that the reality of the energy spectrum remains when the model is extended to the canonical noncommutative space.