Bi-squeezed states arising from pseudo-bosons

Abstract

Extending our previous analysis on bi-coherent states, we introduce here a new class of quantum mechanical vectors, the bi-squeezed states, and we deduce their main mathematical properties. We relate bi-squeezed states to the so-called regular and non regular pseudo-bosons. We show that these two cases are different, from a mathematical point of view. Some physical examples are considered.

Susy for Non-Hermitian Hamiltonians, with a View to Coherent States

“…Bϕ (2) (J, γ ; x) = i d dγ ϕ (1) for all n 0, with N 1N2 = e −iθ √ π in order to have ϕ 0 , 0 = 1. In [10] we have shown that the bounds in (4.3) (for j = 1) are satisfied and that J min = ∞. Therefore, ϕ (1) (J, γ ; x) and ψ (1) (J, γ ; x) are well defined for all (J, γ ) ∈ R 2 .…”

“…The commutator between A and B is the difference between the two Hamiltonians: [5,8,9] and references therein, and [10] for a more recent results. We will consider this particular case in Section 5.…”

“…In recent years, with the growing interest for non-Hermitian Hamiltonians, many attempts to define coherent states also in this case have been carried out, see the recent paper [22] for instance. In particular, we have proposed one of these extentions, the so-called bi-coherent states, see [10] and references therein, which have a lot of nice properties. Here, extending what we did in [16], we propose a different kind of bi-coherent states which we call of the Gazeau-Klauder type, [6], which are rather different from those proposed in [23][24][25][26][27].…”

Some results on the rotated infinitely deep potential and its coherent states

“…If we restrict to H 0 , rather than to H ϕ , we get the following result, which relates K ϕ with K −ϕ : 16) for all f (x), g(x) ∈ D(H 0 ). The proof is based on a double integration by part and on the use of the boundary conditions for functions belonging to D(H 0 ), see (2.2).…”

Tridiagonality, supersymmetry and non self-adjoint Hamiltonians

“…Of course, if h = 1 we return to the situation considered in Section II. Hence, to make the situation interesting, in this section we assume h > 1 we consider an Hamiltonian from which a (bi)-squeezed state can be obtained by applying our recurrence procedure, [10].…”