Deformed quons and bi-coherent states

Багарелло, Фабио

doi:10.1098/rspa.2017.0049

Cited by 25 publications

(23 citation statements)

References 25 publications

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“…Other approaches could be used to study this kind of systems. For instance, we could use Riesz basis like in [5,6,7,8,9], or the property of pseudo-hermiticity [25].…”

Section: Discussionmentioning

confidence: 99%

“…The idea of defining two families of coherent states can be seen as a generalization (in the sense of Gilmore-Perelomov coherent states) of bi-coherent states for the standard bosonic operators [4,5,6,7,8,9].…”

Section: Non-hermitian Gilmore-perelomov Coherent Statesmentioning

confidence: 99%

“…Non-hermitian Hamiltonians have been used for a long time as effective Hamiltonians (think, for instance, of the optical potential in nuclear physics [1]). With the introduction of P T -symmetric Hamiltonians [2] in Quantum Mechanics they became very popular, however only a few papers have been devoted to coherent states (CS) for non-Hermitian systems (see [3], where Gazeau-Klauder CS are constructed using the definition of scalar product in terms of the CPT norm, [4,5,6,7,8,9] where the notion of pseudo-bosons and bi-coherent states are introduced, or [10,11]) as compared with the huge amount of papers devoted to usual coherent states.…”

Section: Introduction and Physical Motivationmentioning

confidence: 99%

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Non-Hermitian Coherent States for Finite-Dimensional Systems

Guerrero

2018

Springer Proceedings in Physics

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We introduce Gilmore-Perelomov coherent states for non-unitary representations of non-compact groups, and discuss the main similarities and differences with respect to ordinary unitary Gilmore-Perelomov coherent states. The example of coherent states for the non-unitary finite dimensional representations of SU (1, 1) is considered and they are used to describe the propagation of light in coupled PT-symmetric optical devices.1 The first theoretical proposal of P T -symmetry in optics was given in [12].

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“…Other approaches could be used to study this kind of systems. For instance, we could use Riesz basis like in [5,6,7,8,9], or the property of pseudo-hermiticity [25].…”

Section: Discussionmentioning

confidence: 99%

Section: Non-hermitian Gilmore-perelomov Coherent Statesmentioning

confidence: 99%

Section: Introduction and Physical Motivationmentioning

confidence: 99%

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Non-Hermitian Coherent States for Finite-Dimensional Systems

Guerrero

2018

Springer Proceedings in Physics

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“…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, .…”

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confidence: 99%

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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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Bosons and Pseudo-Bosons

Багарелло

2022

Mathematical Physics Studies

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Deformed quons and bi-coherent states

Cited by 25 publications

References 25 publications

Non-Hermitian Coherent States for Finite-Dimensional Systems

Non-Hermitian Coherent States for Finite-Dimensional Systems

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

Bosons and Pseudo-Bosons

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