Examples of Pseudo-bosons in quantum mechanics

TL;DR

This paper explores two quantum mechanical models, the extended harmonic oscillator and Swanson model, demonstrating their compatibility with pseudo-bosonic frameworks and analyzing the nature of their biorthogonal bases.

Contribution

It provides the first detailed analysis of these models within the pseudo-bosonic framework, showing their bases are not Riesz bases.

Findings

Both models satisfy pseudo-bosonic assumptions.

The biorthogonal bases are not Riesz bases.

Supports pseudo-hermitian quantum mechanics applications.

Abstract

We discuss two physical examples of the so-called {\em pseudo-bosons}, recently introduced in connection with pseudo-hermitian quantum mechanics. In particular, we show that the so-called {\em extended harmonic oscillator} and the {\em Swanson model} satisfy all the assumptions of the pseudo-bosonic framework introduced by the author. We also prove that the biorthogonal bases they produce are not Riesz bases.