Finite-dimensional pseudo-bosons: A non-Hermitian version of the truncated harmonic oscillator

Abstract

We propose a deformed version of the commutation rule introduced in 1967 by Buchdahl to describe a particular model of the truncated harmonic oscillator. The rule we consider is defined on a N -dimensional Hilbert space H N , and produces two biorhogonal bases of H N which are eigenstates of the Hamiltonians h = 1 2 (q 2 +p 2 ), and of its adjoint h † . Here q and p are non-Hermitian operators obeying [q, p] = i(1 1−N k), where k is a suitable orthogonal projection operator. These eigenstates are connected by ladder operators constructed out of q, p, q † and p † . Some examples are discussed.Quantum mechanics is often thought to be naturally associated to self-adjoint (or Hermitian 1 ) operators. In particular, the dynamics is deduced out of a self-adjoint Hamiltonian, and the observables of the system are almost always assumed to be self-adjoint as well.In recent years, mainly since the seminal work by Bender and Boettcher, [1], it was understood that self-adjointness is not an essential requirement, since other operators exist, not self-adjoint, having purely real (and discrete) spectra. We refer to [2,3,4] for some reviews on this alternative approach. What is interesting, from a mathematical point of view, is that orthonormal (o.n.) bases of eigenstates are replaced by biorthonormal sets which can be, or not, bases of the Hilbert space where the physical system lives. Also, different scalar products can play a role, and this different products produce different adjoints of the same operators. Moreover, the role of pseudospectra in connection with unbounded operators becomes relevant, [5]. Then, in a sense, loosing self-adjointness makes the mathematical structure reacher. Not only that: from a physical point of view the situation is also rather interesting since, for instance, some so-called PT-symmetric Hamiltonians can be naturally used to describe quantum systems with gain and loss phenomena, see [6,7] and references therein.In recent years, in connection with this kind of operators, we have developed a rather general formalism based on some suitable deformations of the canonical commutation and anticommutation relations (CCR and CAR). These deformations produce what we have called D-pseudo bosons and pseudo-fermions. A rather complete review on both these topics can be found in [8], to which we refer for several details and for some physical applications. Later on a similar framework was proposed for quons and for generalized Heisenberg algebra, [9,10].Here we consider a deformation of a different commutation rule, originally considered in [11], and later analyzed in [12], in connection with a truncated version of the harmonic oscillator. The operator c considered in these papers obeys the following rule 1) in which N = 2, 3, 4, . . . is a fixed natural number, while K is a self-adjoint projection operator, K = K 2 = K † , satisfying the equality Kc = 0. The presence of the term N K in (1.1) makes it possible to find a representation of K and c in terms of N × N matrices. In fact, in absence of this term ...

A concise review of pseudobosons, pseudofermions, and their relatives

“…The price to pay is that, at a first sight, these new operators do not obey any interesting commutation relation. However, this point is currently under analysis, [26], since it is known, [27], that o.n. bases in finite dimensional Hilbert spaces (different from H 1 ) can be deduced by suitable modifications of the CCR.…”

“…Also, this does not exclude that a different commutation or anticommutation rule can be found which returns the same results. The analysis of this particular aspect will be postponed to a future paper, [26]. The starting point is, as before, a basis F We refer to [25] for more details, and to the role of M in our construction.…”

Eigenvalues of non‐Hermitian matrices: A dynamical and an iterative approach—Application to a truncated Swanson model

“…Vice versa, if ( (t), Ψ (t)) = ( (0), Ψ (0)), with ⟨ Ψ (0), (0)⟩ ≠ 0, and if ( (t), Ψ (t)) satisfy (11), then ( (t), Ψ (t)) is an equilibrium pair of (10).…”

“…and k is a projection operator, which annihilates the vector e N of the canonical orthonormal basis of C N and satisfies k = k 2 = k † together with kA = B k = 0. In Bagarello, 11 it is shown that H is similar to the truncated quantum harmonic oscillator Hamiltonian h,…”

Краткий Обзор Исследований Псевдобозонов, Псевдофермионов И Родственных Им Частиц

“…На первый взгляд, ценой за это является то, что эти новые операторы не подчиняются каким-либо интересным коммутационным соотношениям. Однако это утверждение в настоящее время анализируется [26], поскольку известно [27], что ортонормированный базис в конечномерном гильбертовом…”