Generalized Coherent States for the q-Oscillator Associated with Discrete q-Hermite Polynomials

We continue studying generalized coherent states of the Barut-Girardello type for oscillator-like systems related to a given set of orthogonal polynomials. In this paper we construct a family of coherent states associated with discrete q-Hermite polynomials of the II-type and prove the overcompleteness of this family by constructing the measure in the unity decomposition for this family of coherent states. Bibliography: 49 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 308, 2004, pp. 48–66.     

Generalized coherent states for oscillators connected with Meixner and Meixner—Pollaczek polynomials

The authors continue to study generalized coherent states for oscillator-like systems connected with a given family of orthogonal polynomials. In this work, we consider oscillators connected with Meixner and Meixner— Pollaczek polynomials and define generalized coherent states for these oscillators. A completeness condition for these states is proved by solution of a related classical moment problem. The results are compared with the other authors ones. In particular, we show that the Hamiltonian of the relativistic model of a linear harmonic oscillator can be treated as the linearization of a quadratic Hamiltonian, which arises naturally in our formalism. Bibliography: 56 titles. The authors dedicate this work to their friend and colleague P. P. Kulish on the occasion of his 60th birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 317, 2004, pp. 66–93.     

Compound model of the generalized oscillator. I

We study the problem of realization of a given generalized oscillator by a system of N generalized oscillators of a different type. We consider a generalized oscillator related to a fixed system of orthogonal polynomials that are determined by three-term recurrent relations and the corresponding three-diagonal Jacobi matrix J. The case N =2 was considered in a previous paper. It was shown that in this case the orthogonality measure is symmetric with respect to rotation at angle π. In this paper, we consider the case N =3. We prove that such a problem has a solution only in two cases. In the first case, the Jacobi matrix related to the given “composite” generalized oscillator has block-diagonal form and consists of similar 3×3 blocks. In the second (more interesting) possible case, the Jacobi matrix is not block-diagonal. For this matrix, we construct the corresponding system of orthogonal polynomials. This system decomposes into three series which are related to Chebyshev polynomials of the second kind. The main result of the paper is a solution of the moment problem for the corresponding Jacobi matrix. In this case, the constructed measure is symmetric with respect to rotation at angle 2π/3. Bibliography: 6 titles.     

Chebyshev-Koornwinder oscillator

We construct a generalized oscillator related to bivariate Chebyshev-Koornwinder polynomials associated with the Lie algebra sl(3) root system.     

Generalized coherent states for oscillators associated with the Charlier q-polynomials

We continue studying the generalized coherent states for oscillator-like systems associated with a given family of orthogonal polynomials. We consider the case of generalized oscillators generated by the Charlier q-polynomials. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 39–46, April, 2008.     

Quaternionic vector coherent states and the supersymmetric harmonic oscillator

The quaternionic vector coherent states are realized as coherent states of the supersymmetric harmonic oscillator with broken symmetry in analogy with the standard canonical coherent states of the ordinary harmonic oscillator. We study the nonclassical properties of the oscillator, such as the photon number distribution and signal-to-quantum-noise ratio in terms of these states and discuss the squeezing properties and the temporal stability of the coherent states. We obtain the orthogonal polynomials associated with the quaternionic vector coherent states. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 80–98, October, 2006.     

Generalized Coherent States: A Novel Approach

We define generalized coherent states for oscillator-like systems connected with orthogonal polynomials (classical, q-deformed, etc.). In considered cases such polynomials play the same role as the Hermite polynomials in the case of the usual boson oscillator. Bibliography: 26 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 65–71.     

Barut–Girardello Coherent States for the Gegenbauer Oscillator

We define the Gegenbauer oscillator and introduce a family of Barut–Girardello coherent states (eigenstates of a relevant annihilation operator) for this oscillator. Gegenbauer (ultraspherical) polynomials play the same role for this oscillator as Hermite polynomials do in the case of the usual boson oscillator. We establish the validity of unity decomposition for the introduced states and evaluate their overlap. These results reproduce similar results obtained earlier by the authors for Legendre and Chebyshev polynomials. Bibliography: 38 titles.     

A generating function for hermite polynomials associated with euclidean Landau levels

We construct a generating function for the Hermite polynomials by comparing two expressions for the same coherent states associated with Landau levels in the planar problem. The first expression is found using a group theory construction, and the second expression is obtained using generalized canonical coherent states expanded as series in the basis of number states.     

Coherent States Quantization for Generalized Bargmann Spaces with Formulae for Their Attached Berezin Transforms in Terms of the Laplacian on ? N

While dealing with a class of generalized Bargmann spaces, we rederive their reproducing kernels from the knowledge of an orthonormal basis by using an addition formula for Laguerre polynomials involving the disk polynomials. We construct for each of these spaces a set of coherent states to apply a coherent states quantization method. This provides us with another way to recover the Berezin transforms attached to these spaces. Finally, two new formulae representing these transforms as functions of the Euclidean Laplacian are established and a possible physics direction for the application of such formulae is discussed.     

Nonlinear Commutation Relations: Representations by Point-Supported Operators

We present a class of non-Lie commutation relations admitting representations by point-supported operators (i.e., by operators whose integral kernels are generalized point-supported functions). For such relations we construct all operator-irreducible representations (up to equivalence). Each representation is realized by point-supported operators in the Hilbert space of antiholomorphic functions. We show that the reproducing kernels of these spaces can be represented via hypergeometric series and the theta function, as well as via their modifications. We construct coherent states that intertwine abstract representations with irreducible representations.     

Coherent states for a generalized oscillator with a finite-dimensional Hilbert space

A construction of oscillator-like systems connected with a given set of orthogonal polynomials and coherent states for such systems developed by the authors is extended to the case of systems with a finite-dimensional state space. As an example, we consider a generalized oscillator connected with Krawtchouk polynomials. Bibliography: 24 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 75–99.     

非谐振子Gazeau-Klauder与Klauder-Perelomov相干态

利用Kinani-Daoud方法构造了非谐振子势的Gazeau-Klauder(GK)相干态和Klauder-Perelomov(KP)相干态，表明两种相干态在非线性谐振子势下具有完全不同的形式，并对两种相干态的完备性以及各自构成的Hilbert空间进行了讨论. 对相干态的Mandel Q参数的研究表明：GK相干态服从亚Poisson统计分布，KP相干态服从超Poisson统计分布.     

Calculation of Berry's phase in squeezed states

A simple method based on generalized coherent states is proposed for calculation of Berry's phase. In this paper we calculate Berry's phase for a translated oscillator in standard coherent states as well as Berry's phase in squeezed states and spin coherent states, i.e., coherent states for the SU(1, 1) and SU(2) groups, respectively.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 169, pp. 51–59, 1988.     

Coherent states of a time-dependent forced harmonic oscillator and their Aharonov-Anandan phase

A new method to construct coherent states of a time-dependent forced harmonic oscillator was given. The close relation to the classical forced oscillator and the minimum uncertainty relation were investigated. The applied periodic force (off-resonance case), in general, will attenuate the AA phase.     

The q-deformed harmonic oscillator, coherent states, and the uncertainty relation

For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the “coordinate-momentum” uncertainties in q-oscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the q-deformation results in a violation of the standard uncertainty relation; the product of the coordinate-and momentum-operator uncertainties is always less than 1/2. States with the minimum uncertainty, which tends to zero, correspond to the values of λ near the convergence radius of the q-exponential. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 315–322, May, 2006.     

Charlier Polynomials and Charlier Oscillator as Discrete Realization of the Harmonic Oscillator

The Charlier polynomials are used for constructing oscillator-like systems. We introduce coherent states of the system are defined and study the main properties (in particular, the (over)completeness property). We show that the coherent states on the associated uncertainty relation take the minimum value. In the case under consideration, the Mandel parameter vanishes, which corresponds to the Poisson statistics of quasiexcitation spectrum for the considered oscillator. Bibliography: 30 titles.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 3–15.     

Using the Evolution Operator Method to Describe a Particle in a Homogeneous Alternating Field

Using the evolution operator method, we construct coherent states of a nonrelativistic free particle with a variable mass M(t) and a nonrelativistic particle with a variable mass M(t) in a homogeneous alternating field. Under certain physical conditions, they can be regarded as semiclassical states of particles. We discuss the properties (in particular, the completeness relation, the minimization of the uncertainty relations, and the time evolution of the corresponding probability density) of the found coherent states in detail. We also construct exact wave functions of the oscillator type and of the plane-wave type for the considered systems and compute the quantum Wigner distribution functions for the wave functions of coherent and oscillator states. We establish the unitary equivalence of the problems of a free particle and a particle in a homogeneous alternating field.     

Darboux integrability of a Mathieu-van der Pol-Duffing oscillator

In this paper, we characterize all the Darboux polynomials of a Mathieu-van der Pol-Duffing oscillator by transforming from the original system into a three dimensional system. We also provide a complete classification of the rational first integrals and of the Darboux first integrals through the analysis of its Darboux polynomials and its exponential factors.