More mathematics for pseudo-bosons

Abstract

We propose an alternative definition for pseudo-bosons. This simplifies the mathematical structure, minimizing the required assumptions. Some physical examples are discussed, as well as some mathematical results related to the biorthogonal sets arising out of our framework.We also briefly extend the results to the so-called non linear pseudo-bosons.

Model pseudofermionic systems: Connections with exceptional points

“…We believe this nice and simple result can be extended to more pseudo-fermionic modes (i.e. to Hilbert spaces with dimension 2 N , for some natural N) and to the much more complicated situation of pseudo-bosons, where (2.1) are replaced by a deformed version of canonical commutation rules, [12]. This will be part of our future analysis.…”

“…In view of what we have just seen, the most general diagonalizable hamiltonian which can be written in terms of a and b is obviously the operator 12) where ω and ρ, in principle, could be complex numbers, α = α 11 α 12 , β = β 11 β 12 , and γ = α 12 β 11 − α 11 β 12 = α 12 β 12 (β − α). Then we can write…”

“…This hamiltonian was introduced quite recently in [8], and can be rewritten as 12) where Γ 1 and Γ 2 are positive quantities, ǫ 1 and ǫ 2 are reals, and ν 0 is complex-valued. It is a simple exercise to show that H GM M can be written as in (2.12) with the following identification: 13) where ∆ǫ = ǫ 2 − ǫ 1 , ∆Γ = Γ 2 − Γ 1 , ǫ = ǫ 2 + ǫ 1 and Γ = Γ 2 + Γ 1 .…”

Appearances of pseudo-bosons from Black-Scholes equation

“…In [18] a weaker version of Assumption D-pb 3 has also been introduced for the first time, particularly interesting for physical applications: let G be a suitable dense subspace of H. Two biorthogonal sets F η = {η n ∈ G, n ≥ 0} and F Φ = {Φ n ∈ G, n ≥ 0} are called G-quasi bases if, for all f, g ∈ G, the following holds:…”

“…Then, using induction on n, one can check that ϕ n , Ψ n = 1, for all n ≥ 0. We refer to [4] and to [18] for more mathematical details on our framework, details which are mostly related to the fact that the operators involved in the game are almost all unbounded. Let us now recall that a basis of H is a set of vectors F = {f n ∈ H, n ≥ 0}, such that each vector g ∈ H can be written as g = n c n f n , with the complex coefficients c n uniquely determined.…”