A Swanson-like Hamiltonian and the inverted harmonic oscillator

TL;DR

This paper analyzes a parameter-dependent Hamiltonian related to the Swanson Hamiltonian, deriving its eigenvalues and eigenvectors, constructing bi-coherent states, and connecting its eigensystem to that of the inverted harmonic oscillator without needing a modified scalar product.

Contribution

It provides a novel approach to understanding the inverted harmonic oscillator's eigensystem through a distributional method, avoiding the need for ad hoc metric operators.

Findings

Eigenvalues and eigenvectors of the Hamiltonian $H_ heta$ are derived.

Bi-coherent states are constructed for the Hamiltonian.

The eigensystem of the inverted harmonic oscillator is obtained without a modified scalar product.

Abstract

We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian $H_\theta$ which is closely related to the Swanson Hamiltonian, and we construct bi-coherent states for it. After that, we show how and in which sense the eigensystem of the Hamiltonian $H$ of the inverted quantum harmonic oscillator can be deduced from that of $H_\theta$. We show that there is no need to introduce a different scalar product using some ad hoc metric operator, as suggested by other authors. Indeed we prove that a distributional approach is sufficient to deal with the Hamiltonian $H$ of the inverted oscillator.