A description of pseudo-bosons in terms of nilpotent Lie algebras

Abstract

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.

On the presence of families of pseudo-bosons in nilpotent Lie algebras of arbitrary corank

“…In particular, one needs to have clear the notions of -bounded operators in Hilbert spaces (see [6, Definition 3.1]) -Riesz basis (see [22]), -G-quasi basis in [6, Definition 3.3], -canonical commuting relations (see [1]), -distribution (see [6,Eq. (3.2)]), -Assumption D-pb 1, Assumption D-pb 2, Assumption D-pb 3, Assumption D-pbw 3, Assumption D-pbs 3, reported in [6, Section 3].…”

“…Proof. About the notations and the terminology for pseudo-bosonic operators, we will use the same in [6,Section 3]. We begin to consider the nilpotent Lie algebra…”

“…We are going to prove that both of them may be described in terms of pseudo-bosonic operators. About the notations and the terminology for pseudo-bosonic operators, we will use again that in [6,Section 3]. Let c be a bosonic operator such that [c, c † ] = cc † − c † c = I.…”

“…We will refer to the recent works [19,20,21], which generalize [8,9,10], but, at the same time, the present contribution continues the efforts we made in [6], where we looked at dynamical systems involving pseudo-bosons in terms of finite dimensional nilpotent Lie algebras for values of the dimension smaller than five.…”

“…They can be found in [12,24,26] and are integrated by some recent material in [19,20,21]. We refer to [6,Section 3] for the terminology and notations which will be involved in the proofs of our main results, where a framework of functional analysis and of theory of the operators is applied. Finally, Section 3 illustrates the main results with corresponding proofs.…”