Orthogonal Polynomials and Generalized Oscillator Algebras

For any orthogonal polynomials system on real line we construct an appropriate oscillator algebra such that the polynomials make up the eigenfunctions system of the oscillator hamiltonian. The general scheme is divided into two types: a symmetric scheme and a non-symmetric scheme. The general approach is illustrated by the examples of the classical orthogonal polynomials: Hermite, Jacobi and Laguerre polynomials. For these polynomials we obtain the explicit form of the hamiltonians, the energy levels and the explicit form of the impulse operators.Comment: 23 pages, no figures, submitted to Integral Transforms and Special Function

Coherent states and Chebyshev polynomials

We define the coherent states for the oscillator-like systems, connected with the Chebyshev polynomials $T_n(x)$ and $U_n(x)$ of the 1-st and 2-nd kind.Comment: 5 pages, no figures, Latex2e, Based on the talk given on the Chebyshev Workshop, Obninsk, 200

Comment on "On the dimensions of the oscillator algebras induced by orthogonal polynomials" [J. Math. Phys. {\bf 55}, 093511 (2014)]

In the interesting paper G. Honnouvo and K. Thirulogasanthar [J. Math. Phys. {\bf 55} , 093511 (2014)] the authors obtained the necessary and sufficient conditions under which the oscillator algebra connected with orthogonal polynomials on real line is finite-dimensional (and in this case the dimension of the algebra is always equal four). In the cited article, only the case when polynomials are orthogonal with respect to a symmetric measure on the real axis was considered. Unfortunately, the sufficient condition from this paper is incomplete. Here we clarify the sufficient part of the corresponding theorem from that paper and extend the results to the case when measure is not symmetric.Comment: Submitte

Orthogonal polynomials and deformed oscillators

We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.Comment: 12 pages Submited to TMF based on the talk on conference "In search of fundamental symmetries" dedicated to 90-anniversary of Yu.V Novozilo

The discrete spectrum of Jacobi matrix related to recurrence relations with periodic coefficients

In this note we investigate the discrete spectrum of Jacobi matrix corresponding to polynomials defined by recurrence relations with periodic coefficients. As examples we consider a)the case when period $N$ of coefficients of recurrence relations equals three (as a particular case we consider "parametric" Chebyshev polynomials introduced by authors early); b)the elementary $N$-symmetrical Chebyshev polynomials ($N=3,4,5$), that was introduced by authors in the study of the "composite model of generalized oscillator".Comment: 16 pages, 1 figure, Submited to ZNS POM

The generalized coherent states for oscillators, Connected with Meixner and Meixner-Pollaczek polynomials % %

The investigation of the generalized coherent states for oscillator-like systems connected with given family of orthogonal polynomials is continued. In this work we consider oscillators connected with Meixner and Meixner-Pollaczek polynomials and define generalized coherent states for these oscillators. The completeness condition for these states is proved by the solution of the 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 linear harmonic oscillator can be thought of as the linearization of the quadratic Hamiltonian which naturally arised in our formalism

Generalized Coherent States for Classical Orthogonal Polynomials

For the oscillator-like systems, connected with the Laguerre, Legendre and Chebyshev polynomials coherent states of Glauber-Barut-Girardello type are defined. The suggested construction can be applied to each system of orthogonal polynomials including classical ones as well as deformed ones.Comment: LaTeX2e, 8 pages, no figures, submitted to Proc. of conference "Day of Difraction 2002" Sankt-Petersburg, Russia, 200

Invariance of the generalized oscillator under linear transformation of the related system of orthogonal polynomials

We consider two families of polynomials \mathbb{P}=\polP and \mathbb{Q}=\polQ\footnote{Here and below we consider only monic polynomials.} orthogonal on the real line with respect to probability measures $\mu$ and $\nu$ respectively. Let \polQ and \polP connected by the linear relations $$Q_n(x)=P_n(x)+a_1P_{n-1}(x)+...+a_kP_{n-k}(x).$$ Let us denote $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ generalized oscillator algebras associated with the sequences $\mathbb{P}$ and $\mathbb{Q}$. In the case $k=2$ we describe all pairs ($\mathbb{P}$,$\mathbb{Q}$), for which the algebras $\mathfrak{A}_P$ and $\mathfrak{A}_Q$ are equal. In addition, we construct corresponding algebras of generalized oscillators for arbitrary $k\geq1$.Comment: 10 pages, 0 figures, The work is based on the report presented at the international conference MQFT 201

Coherent States for generalized oscillator with finite-dimensional Hilbert space

The construction of oscillator-like systems connected with the given set of orthogonal polynomials and coherent states for such systems developed by authors is extended to the case of the systems with finite-dimensional state space. As example we consider the generalized oscillator connected with Krawtchouk polynomials.Comment: English translation of the article published in Russia

Coherent states for the Legendre oscillator

A new oscillator-like system called by the Legendre oscillator is introduced in this note. The two families of coherent states (coherent states as eigenvectors of the annihilation operator and the Klauder-Gazeau temporally stable coherent states) are defined and investigated for this oscillator.Comment: No figures, Latex2