Pseudobosons, Riesz bases, and coherent states

Abstract

In a recent paper, Trifonov suggested a possible explicit model of a PT-symmetric system based on a modification of the canonical commutation relation. Although being rather intriguing, in his treatment many mathematical aspects of the model have just been neglected, making most of the results of that paper purely formal. For this reason we are reconsidering the same model and we repeat and extend the same construction paying particular attention to all the subtle mathematical points. From our analysis the crucial role of Riesz bases clearly emerges. We also consider coherent states associated with the model.

More mathematics for pseudo-bosons

“…Let us call D ∞ (X) := ∩ p≥0 D(X p ) the common domain of all the powers of the operator X. In [1] we have considered the following working assumptions:…”

“…We should recall, in fact, that completeness of a set F is equivalent to F being a basis if F is an orthonormal (o.n.) set, but not in general, at least if H is infinite dimensional, which is the only situation we are interested here in this paper 1 . Actually, there exist intriguingly simple examples of non o.n.…”

“…In a series of papers, [1]- [8], we have considered two operators a and b, with b = a † , acting on a Hilbert space H, and satisfying, in some suitable sense, the commutation rule [a, b] = 1 1. A nice functional structure has been deduced under suitable assumptions, and some connections with physics, and in particular with quasi-hermitian quantum mechanics and with the technique of intertwining operators, have been established.…”

Weak commutation relations of unbounded operators and applications

“…However, in many concrete applications, like in Quantum Theories, they play a so relevant role to deserve a full-fledged mathematical consideration. In recent papers by one of us [3]- [5], generalizing the commutation relations for bosons [a, a † ] = 1 1, the more general case [a, b] = 1 1, where b is not the adjoint of a, has been considered and several interesting results on these pseudo-bosons have been derived, in particular for what concerns the existence and the behavior of bases of eigenvectors of two non self-adjoint operators.…”

Quantization and Quantum-Like Phenomena: A Number Amplitude Approach

“…A number of other generalization procedures to the usual form of canonical commutation rules have also been considered. These include so-called qcalculus versions of the canonical quantization rule [15] that are consistent with q-derivatives and the pseudo-boson formalism [16] in which operators for di¤erent …elds do not commute.…”