Laplace equations, conformal superintegrability and bôcher contractions

Ernest Kalnins, Willard Miller, Eyal Subag

Research output: Contribution to journalArticlepeer-review

4 Scopus citations


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 generalizedInönü-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 submitted and under preparation.

Original languageEnglish (US)
Pages (from-to)214-223
Number of pages10
JournalActa Polytechnica
Issue number3
StatePublished - 2016
Externally publishedYes

Bibliographical note

Publisher Copyright:
© Czech Technical University in Prague, 2016.


  • Conformal superintegrability
  • Contractions
  • Laplace equations


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