Two-parameters pseudo-bosons

TL;DR

This paper constructs a two-parameter example of pseudo-bosons, demonstrating they are not regular and that associated biorthogonal bases are generally not Riesz bases, highlighting new mathematical properties.

Contribution

Introduces a novel two-parameter class of pseudo-bosons and analyzes their basis properties, expanding understanding of their mathematical structure.

Findings

Pseudo-bosons are not regular in this new example.

Biorthogonal bases constructed are generally not Riesz bases.

The example broadens the class of pseudo-bosonic systems studied.

Abstract

We construct a two-parameters example of {\em pseudo-bosons}, and we show that they are not regular, in the sense previously introduced by the author. In particular, we show that two biorthogonal bases of $\Lc^2(\Bbb R)$ can be constructed, which are not Riesz bases, in general.