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Quadratic Algebra and Spectrum of Superintegrable System

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Geometric Methods in Physics XXXIX (WGMP 2022)

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Abstract

Algebraic methods are powerful tools in classical and quantum mechanics. Superintegrable systems are an important class of classical and quantum systems that can be solved using algebraic approaches. In this chapter, we overview higher-rank quadratic algebra of the N-dimensional quantum Smorodinsky–Winternitz system, which is a maximally superintegrable and exactly solvable system. It is shown that the system is multiseparable and the wave function can be expressed in terms of Laguerre and Jacobi polynomials. We present a complete symmetry algebra SW(N) of the system, which is a higher-rank quadratic one containing Racah algebra R(N) as subalgebra. The distinct quadratic Q(3) algebras involving Racah algebra R(N) and their related cubic Casimir invariants are also studied. The energy spectrum of the N-dimensional Smorodinsky–Winternitz system is obtained algebraically via the different sets of subalgebras.

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Authors and Affiliations

  1. Faculty of Nuclear Sciences and Physical Engineering, Department of Physics, Czech Technical University in Prague, Prague, Czech Republic

    Md Fazlul Hoque

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  1. Md Fazlul Hoque
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Correspondence to Md Fazlul Hoque .

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Editors and Affiliations

  1. Departamento de Física, CINVESTAV, Ciudad de México, Mexico

    Piotr Kielanowski

  2. Faculty of Mathematics, University of Białystok, Białystok, Poland

    Alina Dobrogowska

  3. Departments of Mathematics, Physics, Learning & Teaching, Rutgers University, New Brunswick, NJ, USA

    Gerald A. Goldin

  4. Faculty of Mathematics, University of Białystok, Białystok, Poland

    Tomasz Goliński

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Hoque, M.F. (2023). Quadratic Algebra and Spectrum of Superintegrable System. In: Kielanowski, P., Dobrogowska, A., Goldin, G.A., Goliński, T. (eds) Geometric Methods in Physics XXXIX. WGMP 2022. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-30284-8_18

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