Beyond conventional factorization: Non-Hermitian Hamiltonians with radial oscillator spectrum

TL;DR

This paper explores a non-Hermitian radial oscillator Hamiltonian derived from factorization methods, revealing that it shares the same spectrum as the conventional oscillator but features complex eigenfunctions related to Laguerre polynomials.

Contribution

It introduces a novel non-Hermitian Hamiltonian with the same spectrum as the radial oscillator, constructed via non-mutually adjoint factorization operators.

Findings

Non-Hermitian Hamiltonian shares the radial oscillator spectrum.

Eigenfunctions are complex Darboux-deformations of Laguerre polynomials.

Provides a new perspective on oscillator spectra using non-Hermitian operators.

Abstract

The eigenvalue problem of the spherically symmetric oscillator Hamiltonian is revisited in the context of canonical raising and lowering operators. The Hamiltonian is then factorized in terms of two not mutually adjoint factorizing operators which, in turn, give rise to a non-Hermitian radial Hamiltonian. The set of eigenvalues of this new Hamiltonian is exactly the same as the energy spectrum of the radial oscillator and the new square-integrable eigenfunctions are complex Darboux-deformations of the associated Laguerre polynomials.