Exact solution of Schr\"odinger equation for the complex Morse potential to investigate physical systems with position-dependent complex mass

TL;DR

This paper derives exact solutions for a quantum system with position-dependent complex mass under a Morse potential, revealing conditions for real spectra and stable bound states, with implications for exotic physical systems.

Contribution

It provides the first exact ground state solutions for such systems without assuming a specific mass profile, introducing a novel normalization method for non-Hermitian Hamiltonians.

Findings

Real energy spectra can occur under certain parameters.

Normalized eigenfunctions describe stable, localized states.

Potential applications in high-energy and cosmological physics.

Abstract

This paper presents the exact ground state solution for a diatomic particle system with position-dependent complex mass under action of a complex Morse potential in the quantum domain. By solving the position-dependent Schr\"odinger equation in extended complex phase space without assuming a specific mass profile, we derive both the eigenfunctions and corresponding eigenenergies using the analyticity conditions of the eigenfunctions. A key focus is placed on addressing the challenge of normalization inherent in non-Hermitian Hamiltonians. To overcome the limitations of conventional normalization methods in systems with complex potentials and spatially varying mass, we propose a modified normalization approach based on a two-dimensional integral over phase space. The results reveal that, under certain parameter constraints, real energy spectra can arise in non Hermitian settings, supported by normalized and physically meaningful eigenfunctions. Probability density plots validate the existence of stable, localized bound states, maintaining essential characteristics of the traditional Morse potential. Moreover, the model offers potential applications in high-energy and cosmological physics, particularly in the quantum description of exotic systems like dark matter.