Generalized Bogoliubov transformations versus ${\mathcal{D}}$% -pseudo-bosons

TL;DR

This paper explores the conditions under which generalized Bogoliubov transformations produce ${\\mathcal{D}}$-pseudo-bosons, revealing that specific parameter constraints are necessary for their correspondence and analyzing the basis properties of related eigenvector sets.

Contribution

It establishes the precise parameter constraints needed for generalized Bogoliubov transformations to yield ${\mathcal{D}}$-pseudo-bosons and analyzes the basis properties of the resulting eigenvector sets.

Findings

Not all generalized Bogoliubov transformations produce ${\mathcal{D}}$-pseudo-bosons.

Specific parameter constraints are required for the correspondence.

Eigenvector sets form bases but not Riesz bases, and become ${\mathcal{D}}$-quasi bases when constraints are relaxed.

Abstract

We demonstrate that not all generalized Bogoliubov transformations lead to $\cal D$-pseudo-bosons and prove that a correspondence between the two can only be achieved with the imposition of specific constraints on the parameters defining the transformation. For certain values of the parameters we find that the norms of the vectors in sets of eigenvectors of two related apparently non self-adjoint number-like operators possess different types of asymptotic behavior. We use this result to deduce further that they constitute bases for a Hilbert space, albeit neither of them can form a Riesz base. When the constraints are relaxed they cease to be Hilbert space bases, but remain $\cal D$-quasi bases.