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.     

Construction of a Universal Twist Element from an R-Matrix

We propose a method for construction of a universal twist element based on a constant quasi-classical unifary matrix solution of the Yang–Baxter equation. The method is applied to few known R-matrices corresponding to Lie (super) algebras of rank one. Bibliography: 13 titles.     

Differential equations for the elementary 3-symmetric Chebyshev polynomials

We continue the study of a “compound model of a generalized oscillator” and related elementary 3-symmetric Chebyshev polynomials. For these polynomials, we obtain second-order differential equations which are of Fuchs type and have 13 singular points. In the considered simplest case, the obtained results give us an answer to a more general question: What changes in the differential equations for polynomials of the Askey–Wilson scheme when the Jacobi matrix related to these polynomials is perturbed by a diagonal matrix with a complex diagonal? Bibliography: 8 titles.     

Local Perturbation of the Discrete Schrödinger Operator and a Generalized Chebyshev Oscillator

Theoretical and Mathematical Physics - We discuss the conditions under which a special linear transformation of the classical Chebyshev polynomials (of the second kind) generate a class of...     

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.     

Quantum Heisenberg group and reflection equation. I

The quantum group of upper triangular matrices and the Hopf algebra associated with the reflection equation are constructed for the R-matrix corresponding to the quantum Heisenberg group. Bibliography: 12 titles. Dedicated to P. P. Kulish on the occasion of his 50th birthday Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 209, 1994, pp. 28–36. Translated by E. V. Damanskinsky.     

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.     

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.     

Chebyshev-Koornwinder oscillator

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