Model pseudofermionic systems: Connections with exceptional points

Abstract

We discuss the role of pseudo-fermions in the analysis of some two-dimensional models, recently introduced in connection with non self-adjoint hamiltonians. Among other aspects, we discuss the appearance of exceptional points in connection with the validity of the extended anti-commutation rules which define the pseudo-fermionic structure.

𝒟 $\mathcal {D}$ -Deformed Harmonic Oscillators

“…In this paper we set up a systematic analysis of several two-dimensional models which can be described completely in terms of D-PBs, some of them already considered in the past and others, in our knowledge, new. Doing so, we propose a list of models which are simply D-deformed two-dimensional harmonic oscillators, for which the approach discussed in, say, [5] and reviewed in [6] can be adopted (see also [7] for models involving anti-commutation relations).…”

Intertwining operators for non-self-adjoint Hamiltonians and bicoherent states

“…Of course, this has useful consequences, as discussed in [13,16], since the general framework proposed there for the deformed anti commutation relations can now be used for Θ 2 . Viceversa, of course, one could also look now for explicit connections of Θ 2 with some quantum mechanical system, for instance in the context of PT-quantum mechanics.…”

Multiply Degenerate Exceptional Points and Quantum Phase Transitions

“…In a suitable basis they acquire a Jordan-blocks-containing canonical form [1,5,6]. Thus, even if one of values of λ (EP ) j is real, the limiting transition λ → λ (EP ) j must still be perceived as a process during which certain eigenvectors of H (λ) "parallelize" and become linearly dependent, resulting in the loss of the usual probabilistic tractability of the quantum system in question [7][8][9][10][11][12][13].…”