Pseudo-bosons and bi-coherent states out of $\Lc^2(\mathbb{R})$

TL;DR

This paper extends the theory of pseudo-bosons and bi-coherent states beyond the traditional Hilbert space framework, exploring non-square-integrable states and distributional approaches in the context of deformed canonical commutation relations.

Contribution

It introduces a method to define eigenstates and bi-coherent states outside the Hilbert space, broadening the mathematical framework for pseudo-bosons and related states.

Findings

Extended pseudo-boson framework beyond (\u211d)

Constructed non-square-integrable eigenstates

Analyzed distributional bi-coherent states

Abstract

In this paper we continue our analysis on deformed canonical commutation relations and on their related pseudo-bosons and bi-coherent states. In particular, we show how to extend the original approach outside the Hilbert space $\Lc^2(\mathbb{R})$, leaving untouched the possibility of defining eigenstates of certain number-like operators, manifestly non self-adjoint, but opening to the possibility that these states are not square-integrable. We also extend this possibility to bi-coherent states, and we discuss in many details an example based on a couple of superpotentials first introduced in \cite{bag2010jmp}. The results deduced here belong to the same distributional approach to pseudo-bosons first proposed in \cite{bag2020JPA}.