quantum groups
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1: 17.17 Physical Applications
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►See Berkovich and McCoy (1998) and Bethuel (1998) for recent surveys.
►Quantum groups also apply -series extensively.
Quantum groups are really not groups at all but certain Hopf algebras.
They were given this name because they play a role in quantum physics analogous to the role of Lie groups and special functions in classical mechanics.
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2: Tom H. Koornwinder
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►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC.
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3: Bibliography V
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Representation of Lie Groups and Special Functions. Volume 3: Classical and Quantum Groups and Special Functions.
Mathematics and its Applications (Soviet Series), Vol. 75, Kluwer Academic Publishers Group, Dordrecht.
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4: Bibliography K
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Quantum Groups.
Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York.
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5: Bibliography W
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Group Theory and its Application to the Quantum Mechanics of Atomic Spectra.
Pure and Applied Physics. Vol. 5, Academic Press, New York.
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6: 12.17 Physical Applications
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►Miller (1974) treats separation of variables by group theoretic methods.
Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator.
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7: Bibliography D
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The constrained quantum mechanical harmonic oscillator.
Proc. Cambridge Philos. Soc. 62, pp. 277–286.
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On the computation of Mathieu functions.
J. Engrg. Math. 7, pp. 39–61.
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Quantum Fields and Strings: A Course for Mathematicians. Vol. 1, 2.
American Mathematical Society, Providence, RI.
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Monodromy of certain Painlevé-VI transcendents and reflection groups.
Invent. Math. 141 (1), pp. 55–147.
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8: Bibliography G
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The GAP Group, Centre for Interdisciplinary Research in Computational Algebra,
University of St. Andrews, United Kingdom.
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Linear Differential Equations and Group Theory from Riemann to Poincaré.
2nd edition, Birkhäuser Boston Inc., Boston, MA.
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General form of the quantum-defect theory.
Phys. Rev. A 19 (4), pp. 1485–1509.
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Quantum Electrodynamics of Strong Fields: With an Introduction into Modern Relativistic Quantum Mechanics.
Texts and Monographs in Physics, Springer.
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9: Bibliography
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The Riemann Hypothesis and the Hamiltonian of a Quantum Mechanical System.
In Number Theory and Dynamical Systems (York, 1987), M. M. Dodson and J. A. G. Vickers (Eds.),
London Math. Soc. Lecture Note Ser., Vol. 134, pp. 153–172.
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Normal forms of functions near degenerate critical points, the Weyl groups
and Lagrangian singularities.
Funkcional. Anal. i Priložen. 6 (4), pp. 3–25 (Russian).
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Special Functions: Group Theoretical Aspects and Applications.
D. Reidel Publishing Co., Dordrecht.
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