$e$-product of distributions, with applications

TL;DR

This paper revisits a recent approach to multiplying distributions, applying it to demonstrate biorthonormality of certain distributions linked to eigenvalues of a non self-adjoint operator related to weak pseudo-bosons, with detailed examples.

Contribution

It reformulates a recent definition of distribution multiplication and applies it to analyze eigenvalues in the context of weak pseudo-bosons, providing new insights and examples.

Findings

Established biorthonormality of distributions related to eigenvalues.

Applied the distribution multiplication to non self-adjoint operators.

Provided detailed examples illustrating the theory.

Abstract

We consider and reformulate a recent definition of multiplication between distributions. We show that this definition can be adopted, in particular, to prove biorthonormality of some distributions arising when looking to the (generalized) eigenvalues of a specific non self-adjoint number-like operator, considered in connection with the recently introduced {\em weak pseudo-bosons}. Several examples are discussed in details.