Quasi-Hermitian Hamiltonians associated with exceptional orthogonal polynomials

TL;DR

This paper constructs exactly solvable non-Hermitian quantum Hamiltonians linked to exceptional orthogonal polynomials, demonstrating their quasi-Hermitian nature and real energy spectra through point canonical transformations and imaginary coordinate shifts.

Contribution

It introduces new rationally extended quantum Hamiltonians associated with exceptional orthogonal polynomials and proves their quasi-Hermitian property with real spectra.

Findings

Hamiltonians are non-Hermitian but have real spectra.

Bound state wave functions relate to Laguerre- or Jacobi-type exceptional polynomials.

Hamiltonians are shown to be quasi and pseudo-Hermitian.

Abstract

Using the method of point canonical transformation, we derive some exactly solvable rationally extended quantum Hamiltonians which are non-Hermitian in nature and whose bound state wave functions are associated with Laguerre- or Jacobi-type $X_1$ exceptional orthogonal polynomials. These Hamiltonians are shown, with the help of imaginary shift of co-ordinate: $ e^{-\alpha p} x e^{\alpha p} = x+ i \alpha $, to be both quasi and pseudo-Hermitian. It turns out that the corresponding energy spectra is entirely real.