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$e$-product of distributions, with applications

Fabio Bagarello

TL;DR

This work presents the $e$-product, a basis-dependent extension of the scalar product to distributions, aimed at handling generalized eigenstates of non-self-adjoint operators arising in weak pseudo-boson frameworks. By anchoring the product to a fixed orthonormal basis ${\cal F}_e$ and defining ${\cal M}_e$, the author derives structural properties and applies the framework to both abstract settings and concrete examples, including delta distributions, exponentials, and the harmonic-oscillator basis. A central result demonstrates biorthonormality for a weak pseudo-boson pair via $\langle\varphi_n,\psi_m\rangle_e=\delta_{nm}$, using Abel-type summation to manage otherwise divergent series. The work also analyzes when the $e$-product fails to exist, and discusses adjoint-like constructions $\ddagger$, highlighting both mathematical and physical implications for distributions in quantum-like systems. Overall, the paper provides a practical, technically grounded approach to extending distribution multiplication with potential applications to non-Hermitian quantum mechanics and rigged Hilbert space formalisms, while outlining clear avenues for future development and alternative basis choices.

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.

$e$-product of distributions, with applications

TL;DR

This work presents the -product, a basis-dependent extension of the scalar product to distributions, aimed at handling generalized eigenstates of non-self-adjoint operators arising in weak pseudo-boson frameworks. By anchoring the product to a fixed orthonormal basis and defining , the author derives structural properties and applies the framework to both abstract settings and concrete examples, including delta distributions, exponentials, and the harmonic-oscillator basis. A central result demonstrates biorthonormality for a weak pseudo-boson pair via , using Abel-type summation to manage otherwise divergent series. The work also analyzes when the -product fails to exist, and discusses adjoint-like constructions , highlighting both mathematical and physical implications for distributions in quantum-like systems. Overall, the paper provides a practical, technically grounded approach to extending distribution multiplication with potential applications to non-Hermitian quantum mechanics and rigged Hilbert space formalisms, while outlining clear avenues for future development and alternative basis choices.

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.
Paper Structure (13 sections, 2 theorems, 82 equations)

This paper contains 13 sections, 2 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $e$-product
  4. Something about the adjoint
  5. Examples
  6. Example nr. 1
  7. Example nr. 2
  8. Example nr. 3
  9. Example nr. 4
  10. An introductory case: $n=m=0$
  11. Larger $n$ and $m$
  12. Two variations on the same theme
  13. Conclusions

Key Result

Proposition 4

If ${\cal F}_e$ is ${\cal F}_e$-separating, then ${\cal V}$ is $e$-separating. Viceversa, if ${\cal V}_0$ is $e$-separating, then ${\cal F}_e$ is ${\cal F}_e$-separating.

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Corollary 5