Superintegrable systems, polynomial algebra structures and exact derivations of spectra

Superintegrable systems are a class of physical systems which possess more conserved quantities than their degrees of freedom. The study of these systems has a long history and continues to attract significant international attention. This thesis investigates finite dimensional quantum superintegrable systems with scalar potentials as well as vector potentials with monopole type interactions. We introduce new families of $N$-dimensional superintegrable Kepler-Coulomb systems with non-central terms and double singular harmonic oscillators in the Euclidean space, and new families of superintegrable Kepler, MIC-harmonic oscillator and deformed Kepler systems interacting with Yang-Coulomb monopoles in the flat and curved Taub-NUT spaces. We show their multiseparability and obtain their Schr\"{o}dinger wave functions in different coordinate systems. We show that the wave functions are given by (exceptional) orthogonal polynomials and Painlev\'{e} transcendents (of hypergeometric type). We construct higher-order algebraically independent integrals of motion of the systems via the direct and constructive approaches. These integrals form (higher-rank) polynomial algebras with structure constants involving Casimir operators of certain Lie algebras. We obtain finite dimensional unitary representations of the polynomial algebras and present the algebraic derivations for degenerate energy spectra of these systems. Finally, we present a generalized superintegrable Kepler-Coulomb model from exceptional orthogonal polynomials and obtain its energy spectrum using both the separation of variable and the algebraic methods.Comment: PhD Thesis, School of Mathematics and Physics, The University of Queensland, Australia, January 2018, 175 page

Quadratic symmetry algebras and spectrum of the 3D nondegenerate quantum superintegrable system

In this paper, we present the quadratic associative symmetry algebra of the 3D nondegenerate maximally quantum superintegrable system. This is the complete symmetry algebra of the system. It is demonstrated that the symmetry algebra contains suitable quadratic subalgebras, each of which is generated by three generators with relevant structure constants, which may depend on central elements. We construct corresponding Casimir operators and present finite-dimensional unirreps and structure functions via the realizations of these subalgebras in the context of deformed oscillators. By imposing constraints on the structure functions, we obtain the spectrum of the 3D nondegenerate superintegrable system. We also show that this model is multiseparable and admits separation of variables in cylindrical polar and paraboloidal coordinates. We derive the physical spectrum by solving the Schr\"{o}dinger equation of the system and compare the result with those obtained from algebraic derivations.Comment: 21 page

Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

In this paper we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of Theorem 1 in [A Marchesiello and L \v{S}nobl 2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203]. We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of [A Marchesiello and L \v{S}nobl 2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203].Comment: 18 page