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

Багарелло, Фабио; Gargano, Francesco; Volpe, D. della

doi:10.1007/s10773-014-2487-9

Cited by 14 publications

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“…This approach is particularly interesting since it is heavily connected with the general settings proposed in recent years for deformed canonical commutation and anti-commutation relations, see [9,11,10] for recent applications, and since makes no use of equality (2.12), which is not always satisfied even in simple cases, as in the first example discussed in Section III.1.…”

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confidence: 99%

Coordinate representation for non-Hermitian position and momentum operators

et al. 2017

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In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q^q^ and p^p^ similar, in a suitable sense, to the self-adjoint position and momentum operators q^0 q^0 and p^0 p^0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions, and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q^ q^ and p^ p^, based on the so-called quasi *-algebras

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Section: )mentioning

confidence: 99%

Coordinate representation for non-Hermitian position and momentum operators

et al. 2017

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“…In [11] the commutation rule (1.1) have been used to consider a truncated version of the harmonic oscillator, living in the finite dimensional Hilbert space H N , and then considering its limit for diverging N. In [17] it has been discussed that the self-adjoint Hamiltonian of the oscillator produces, using similarity transformations, several non self-adjoint quadratic Hamiltonians with known spectra and eigenstates which may, or may not, form bases for the infinite-dimensional Hilbert space where the model is defined. Among these Hamiltonians, one can recover the ones for the shifted harmonic oscillator and for the Swanson model.…”

Section: Iv2 a Truncated Swanson Modelmentioning

confidence: 99%

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

Багарелло

2018

Physics Letters A

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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 ...

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“…Therefore, with this choice, H ef f becomes a non selfadjoint shifted harmonic oscillator which appears, in slightly different ways, in several models in pseudo-hermitian quantum mechanics. We refer to [5], and references therein, and to [21], for some examples of similar systems. Fixing W (x) as in (3.11) the operators A and B look like…”

Section: How D-pbs Appearmentioning

confidence: 99%

Appearances of pseudo-bosons from Black-Scholes equation

Багарелло

2016

Journal of Mathematical Physics

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It is a well known fact that the Black-Scholes equation admits an alternative representation as a Schrödinger equation expressed in terms of a non self-adjoint hamiltonian. We show how pseudo-bosons, linear or not, naturally arise in this context, and how they can be used in the computation of the pricing kernel. IntroductionPseudo-Hermitian quantum mechanics, together with its many relatives, received an increasing interest by the physicists community and, more recently, by the mathematicians too, because of its possible applications in concrete systems and also in connection with several interesting mathematical properties appearing in systems driven by non-hermitian Hamiltonians. We refer to [1]-[4] for some recent and not so recent reviews or volumes dedicated to this subject.Most examples of non hermitian Hamiltonians come from physics, see [5]-[8] just to cite a few, or from mathematics, see [9], where the interest is more focused to a rigorous treatment of the models under consideration, and to their main constituents.But, as shown in [10] and, recently, in [11, 12], more examples of this kind of operators come from an unexpected field of research, i.e. from economics. In fact, with a suitable change of variable, the Black-Scholes equation can be rewritten as a Schrödinger equation, but with a non self-adjoint hamiltonian. This fact was used in [10]-[12] to shown how quantum mechanical techniques can be useful to compute the pricing kernel for various situations, and how in some cases the Black-Scholes Hamiltonian can be factorized and used also as an interesting example in supersymmetric quantum mechanics, [13]. In this paper we show that similar (or identical) models also provide examples of D-pseudo bosons (D-PBs) and non linear D-PBs (D-NLPBs), which are excitations recently introduced by us, considering suitable deformations of the canonical commutation relations (CCRs). We also discuss how pricing kernels can be defined and computed in these cases.The paper is organized as follows: in the next section we give a brief mathematical introduction to D-PBs and D-NLPBs, useful to keep the paper self-contained. In Section 3 we show how D-PBs arise naturally out of the Black-Scholes equation, while in Section 4 we discuss the appearance of D-NLPBs in the same context. Section 5 contains our conclusions. Mathematical preliminariesIn view of their use in Sections 3 and 4, we devote this section to some preliminary definitions and results on D-PBs and D-NLPBs. We refer to [5] and to [14]-[16] for more details. Something about D-PBsLet H be a given Hilbert space with scalar product ., . and related norm . . Let further a and b be two operators on H, with domains D(a) and D(b) respectively, a † and b † their adjoints,

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

Cited by 14 publications

References 20 publications

Coordinate representation for non-Hermitian position and momentum operators

Coordinate representation for non-Hermitian position and momentum operators

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

Appearances of pseudo-bosons from Black-Scholes equation

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