Real Spectra in PT Symmetry Hamiltonians using Tridiagonal Representation Approach

TL;DR

This paper introduces a tridiagonal representation approach (TRA) to accurately compute real spectra of PT symmetry Hamiltonians, providing improved eigenvalues and wavefunctions, including for non-integer parameters, using semi-analytic methods.

Contribution

The paper presents a novel application of TRA to PT symmetry Hamiltonians, enabling precise eigenvalue and wavefunction calculations, including for non-integer parameter values.

Findings

TRA yields more accurate energy levels and wavefunctions.

The method effectively handles integer and non-integer parameter N.

Semi-analytic application enhances computational efficiency.

Abstract

We consider the solution of PT symmetry Hamiltonians using the technique of tridiagonal representation approach. This methodology provides more accurate results and proper depiction of the Hamiltonian energy level and wavefunctions. It is well know that PT symmetry condition of a Hamiltonian ensure that its spectra are real and positive even if the Hamiltonian is non Hermitian. Here, we introduce the method of TRA to get the eigenvalues and wavefunction of this Hamiltonian for integers values of $N$ and show an approximation solution for non-integer value of $N$. Due to the nature of the Hamiltonian, the TRA was applied in a semi-Analytic manner in the paper.