A concise review of pseudobosons, pseudofermions, and their relatives

Abstract

We review some basic definitions and few facts recently established for D-pseudo bosons and for pseudo-fermions. We also discuss an extended version of these latter, based on biorthogonal bases, which lives in a finite dimensional Hilbert space. Some examples are described in details.

Coherent Gamow states for the hyperbolic Pöschl–Teller potential

“…This shows the analogy of our formalism with that in [19]. Nevertheless, there are some minor differences that should not affect the essential idea.…”

“…We have also shown that the two families of Gamow states behave as pseudo-bosons in the sense given by Bagarello in [19]. We have constructed respective topologies on the spaces spanned by both families of Gamow states and shown that the corresponding ladder operators are continuous under these topologies.…”

“…This is a slight generalization of the formalism for pseudo-bosons introduced by Bagarello [19]. In fact, if we identify a ≡ B − , b ≡ B + , ϕ 0 ≡ ϕ D 0 , Ψ 0 ≡ ϕ G 0 , so that a † ≡ C + , b † ≡ C − , we have the following expressions: The vectors…”

“…It seems more natural to introduce in the linear space spanned by ϕ D n and ϕ G n a Krein pseudometrics, such that along (35), one has ϕ D n |ϕ D m = ϕ G n |ϕ G m for all values of m and n [24,25], see also [26]. In any case, the pair (z D , |z G ), z ∈ C may be looked as bi-coherent states in the sense of [19]. So far, we have assumed that D D is the space of linear combinations of decaying Gamow vectors.…”

Coupled Susy, pseudo-bosons and a deformed $\mathfrak{su}(1,1)$ Lie algebra

“…These equalities imply that F Φ and F ξ are biorthonormal, and, [31], that they are D-quasi bases if and only if F ϕ and F ψ are D-quasi bases. This property, useful to deduce several mathematical properties of the system, as already stated, is always true in all the physical systems where D-PBs have been shown to appear so far, [31,32]. We end this section with Tables 1 and 2 which contain several useful formulas involving all the vectors introduced in this section.…”