A non-trivial PT-symmetric continuum Hamiltonian and its Eigenstates and Eigenvalues

TL;DR

This paper analyzes a non-trivial PT-symmetric continuum Hamiltonian, revealing its spectral equivalence to the harmonic oscillator, deriving its eigenfunctions, and identifying its hidden symmetry operator through exact analytical methods.

Contribution

It introduces a new continuum PT-symmetric Hamiltonian, demonstrates its iso-spectrality with the harmonic oscillator, and analytically finds its eigenfunctions and symmetry operator.

Findings

Hamiltonian is iso-spectral to the harmonic oscillator

Eigenfunctions form an orthonormal set along a complex path

Hidden symmetry operator ${\ m C}$ is explicitly derived

Abstract

In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, ${\cal C}$, for this system. All calculations are performed analytically and without approximation.