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Quantum Systems Associated with the Hahn and Continuous Hahn Polynomials

Abdulaziz D Alhaidari, Y.-T. Li

arXiv:1711.05943Reports on Mathematical PhysicsDec 1, 201810 citations

Integrable multi-boson systems and orthogonal polynomials

Anatol Odzijewicz, Maciej Horowski, Agnieszka Tereszkiewicz

The strict relation between a certain class of multi-boson Hamiltonian systems and the corresponding class of orthogonal polynomials is established. The correspondence is effectively used to integrate the systems. As an explicit example we integrate the class of multi-boson systems corresponding to q -Hahn class polynomials.

arXiv:math-ph/0101013Journal of Physics A Mathematical and GeneralMay 9, 200134 citations

Quantum Integrable Systems on a Classical Integrable Background

Andrii Nikolaevich Liashyk, Nicolai Reshetikhin, Ivan Andreevich Sechin

arXiv:2405.17865Communications in Mathematical PhysicsDec 8, 20254 citations

Representation of the quantum mechanical wavefunction by orthogonal polynomials in the energy and physical parameters

Abdulaziz D Alhaidari

Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.Abstract We present a formulation of quantum mechanics based on the theory of orthogonal polynomials. The wavefunction is expanded over a complete set of square integrable basis where the expansion coefficients are orthogonal polynomials in the energy and physical parameters. Information about the corresponding physical systems (both structural and dynamical) are derived from the properties of these polynomials. We demonstrate that an advantage of this formulation is that the class of exactly solvable quantum mechanical problems becomes larger than in the conventional formulation (see, for example, table 3 in the text). We limit our investigation in this work to the Askey classification scheme of hypergeometric orthogonal polynomials and focus on the Wilson polynomial and two of its limiting cases (the Meixner–Pollaczek and continuous dual Hahn polynomials). Nonetheless, the formulation is amenable to other classes of orthogonal polynomials.

Communications in Theoretical PhysicsDec 30, 201919 citations

QUANTUM GROUP AND QUANTUM INTEGRABLE SYSTEMS

Mo‐Lin Ge

Jul 1, 199245 citations

Explicitly solvable models of a two-mode coupler in Kerr media

Agnieszka Tereszkiewicz, Anatol Odzijewicz, Maciej Horowski +2 more

We present exact solutions for two nonlinear models each of which describes parametric down conversion of photons as well as the Kerr effect. The Hamiltonians for these models are related to the dual Hahn finite orthogonal polynomials. Explicit expressions are obtained for the spectra and for the eigenvectors of the Hamiltonians. A discussion of the physical characteristics of the systems is presented.

Physical Review AJun 19, 20077 citations

Example of the two–mode bosonic system integrable by the dual Hahn polynomials

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Example of the two–mode bosonic system integrable by the dual Hahn polynomials

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