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

Abstract

Hamiltonian which naturally arised in our formalism.

New Quasi-Exactly Solvable Difference Equation

“…In this section we discuss two examples, the Meixner-Pollaczek polynomials and the continuous Hahn polynomials, in their full generality. In our previous work on discrete quantum mechanics [14,15], only the special case of the Meixner-Pollaczek polynomials with the phase angle φ = π/2 and the special case of the continuous Hahn polynomials with two real parameters a 1 and a 2 are discussed, partly because these special cases of the two polynomials appear in several other dynamical contexts [6,3,2] and, in particular, they appear in the description of the equilibrium positions [19,13,14] of the classical Ruijsenaars-Schneider van Diejen systems [20,27].…”

“…For a ≤ 0, however, there appear other singularities of φ 0 (x; a), which break the hermiticity. The special case discussed in [14,6,2,3] is β = 0 or φ = π/2. The groundstate wavefunction φ 0 , as annihilated by the operator A, Aφ 0 = 0, is given by…”

“…The special case discussed in [14,6,2,3] is β = 0 or φ = π/2. The groundstate wavefunction φ 0 , as annihilated by the operator A, Aφ 0 = 0, is given by…”

Hermite polynomials and Fibonacci oscillators

“…One should mention that there are several similar studies in the literature [19][20][21][22][23][24][25][26] with multi-parameters deformed oscillators that do not necessarily obey the Fibonacci properties in the sense of the seminal paper by Arik [18] where the spectrum is given by a generalized Fibonacci sequence. They provide a unification of quantum oscillators with quantum groups [27][28][29][30][31], keeping the degeneracy property of the spectrum invariant under the symmetries of the quantum group. The quantum algebra with two deformation parameters may have a greater flexibility when it comes to applications in realistic phenomenological physical models [32,33].…”

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

“…In the last years the interest to studying of the oscillator-like systems (so-called "generalized oscillators") and to using such systems in various areas of quantum mechanics ( [1]- [3]) has significantly increased. In works [4]- [12] authors suggested a new approach to definition of generalized oscillators, connected with the given system of orthogonal polynomials and to construction of coherent states for these oscillators. Within the framework of this approach we investigated generalized oscillators, connected with classical orthogonal polynomials of a continuous argument (such as Laguerre, Legendre, Chebyshev, and Gegenbauer polynomials), orthogonal polynomials of a discrete argument (such as Meixner and Charlier polynomials) as well as systems connected with q-analogues of Hermite polynomials.…”