# Bi-Orthogonal Approach to Non-Hermitian Hamiltonians with the Oscillator Spectrum: Generalized Coherent States for Nonlinear Algebras

## TL;DR

This paper develops a bi-orthogonal framework for non-Hermitian Hamiltonians with harmonic oscillator spectra, introducing generalized coherent states for nonlinear algebras, enabling analysis similar to Hermitian systems.

## Contribution

It introduces a bi-orthogonal approach for non-Hermitian oscillators, constructs nonlinear algebras, and derives generalized coherent states, extending the analysis of non-Hermitian quantum systems.

## Key findings

- Bi-orthogonal eigenvector systems for non-Hermitian oscillators.
- Construction of nonlinear algebras with ladder operators.
- Generalized coherent states bridging non-Hermitian and Hermitian systems.

## Abstract

A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a mathematical procedure to satisfy the superposition principle. In this form the non-Hermitian oscillators can be studied in much the same way as in the Hermitian approaches. Two different nonlinear algebras generated by properly constructed ladder operators are found and the corresponding generalized coherent states are obtained. The non-Hermitian oscillators can be steered to the conventional one by the appropriate selection of parameters. In such limit, the generators of the nonlinear algebras converge to generalized ladder operators that would represent either intensity-dependent interactions or multi-photon processes if the oscillator is associated with single mode photon fields in nonlinear media.