LAPLACE EQUATIONS, CONFORMAL SUPERINTEGRABILITY AND BÔCHER CONTRACTIONS

Authors

  • Ernest G. Kalnins University of Waikato
  • Willard Miller, Jr School of Mathematics, University of Minnesota
  • Eyal Subag Pennsylvania State University

DOI:

https://doi.org/10.14311/AP.2016.56.0214

Keywords:

conformal superintegrability, contractions, Laplace equations

Abstract

Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often ``hidden''.

The symmetry generators of 2nd order superintegrable systems in 2 dimensions close under commutation to define quadratic algebras, a generalization of Lie algebras. Distinct systems and their algebras are related by geometric limits, induced by generalized Inönü-Wigner Lie algebra contractions of the symmetry algebras of the underlying spaces. These have physical/geometric implications, such as the Askey scheme for hypergeometric orthogonal polynomials. The systems can be best understood by transforming them to Laplace conformally superintegrable systems and using ideas introduced in the 1894 thesis of Bôcher to study separable solutions of the wave equation. The contractions can be subsumed into contractions of the conformal algebra  so(4,C) to itself. Here we announce main findings, with detailed classifications in papers under preparation.

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Author Biographies

  • Ernest G. Kalnins, University of Waikato
    Department of Mathematics, Professor
  • Willard Miller, Jr, School of Mathematics, University of Minnesota
    School of mathematics, Professor Emeritus
  • Eyal Subag, Pennsylvania State University
    Mathematics Department,  Postdoctoral Fellow

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Published

2016-06-30

Issue

Vol. 56 No. 3 (2016)

Section

Articles

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How to Cite

Kalnins, E. G., Miller, Jr, W., & Subag, E. (2016). LAPLACE EQUATIONS, CONFORMAL SUPERINTEGRABILITY AND BÔCHER CONTRACTIONS. Acta Polytechnica, 56(3), 214-223. https://doi.org/10.14311/AP.2016.56.0214