Interplay between Riccati, Ermakov and Schroedinger equations to produce complex-valued potentials with real energy spectrum

TL;DR

This paper explores how nonlinear Riccati and Ermakov equations can be combined with the Darboux method to generate complex-valued quantum potentials that retain real energy spectra, including additional eigenvalues, without singularities.

Contribution

It introduces a novel approach linking Riccati, Ermakov, and Schrödinger equations to produce complex potentials with real spectra, extending the Darboux method to non-Hermitian systems.

Findings

Complex potentials inherit spectra of Hermitian systems.

Additional real eigenvalues can be incorporated without singularities.

Examples include Morse and Poeschl-Teller potentials.

Abstract

Nonlinear Riccati and Ermakov equations are combined to pair the energy spectrum of two different quantum systems via the Darboux method. One of the systems is assumed Hermitian, exactly solvable, with discrete energies in its spectrum. The other system is characterized by a complex-valued potential that inherits all the energies of the former one, and includes an additional real eigenvalue in its discrete spectrum. If such eigenvalue coincides with any discrete energy (or it is located between two discrete energies) of the initial system, its presence produces no singularities in the complex-valued potential. Non-Hermitian systems with spectrum that includes all the energies of either Morse or trigonometric Poeschl-Teller potentials are introduced as concrete examples.