The $\eta$-pseudo-hermitic generator in the deformed Woods Saxons potential

TL;DR

This paper introduces a method to solve non-Hermitian, PT-symmetric potentials in quantum mechanics, specifically applied to the deformed Woods-Saxon potential, revealing how real potentials can be expressed as complex, pseudo-Hermitian systems.

Contribution

It develops a general approach using $ ext{ exteta}$-pseudo-Hermiticity and Hamiltonian hierarchy to analyze non-Hermitian potentials, applied to the Dirac equation with the deformed Woods-Saxon potential.

Findings

Real potentials can be decomposed into complex $ ext{ exteta}$-pseudo-Hermitian potentials.

Transmission probabilities relate to potential parameters via $ ext{ exteta}$-pseudo-Hermiticity.

The method applies to solvable potentials like the deformed Woods-Saxon potential.

Abstract

In this paper, we present a general method to solve non-hermetic potentials with PT symmetry using the definition of two $\eta$-pseudo-hermetic and first-order operators. This generator applies to the Dirac equation which consists of two spinor wave functions and non-hermetic potentials, with position that mass is considered a constant value and also Hamiltonian hierarchy method and the shape invariance property are used to perform calculations. Furthermore, we show the correlation between the potential parameters with transmission probabilities that $\eta$-pseudo-hermetic using the change of focal points on Hamiltonian can be formalized based on Schr\"{o}dinger-like equation. By using this method for some solvable potentials such as deformed Woods Saxon's potential, it can be shown that these real potentials can be decomposed into complex potentials consisting of eigenvalues of a class of $\eta$-pseudo-hermitic.