Abstract
Quantum superintegrable systems are solvable eigenvalue problems. Their solvability is due to symmetry, but the symmetry is often ``hidden''.<p style="-qt-block-indent: 0; text-indent: 0px; margin: 0px;">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 <em>so</em>(4,C) to itself. Here we announce main findings, with detailed classifications in papers under preparation.</p>
Highlights
A quantum superintegrable system is an integrable Hamiltonian system on an n-dimensional Riemannian/pseudo-Riemannian manifold with potential: H = ∆n + V that admits 2n − 1 algebraically independent partial differential operators Lj commuting with H, the maximum possible. [H, Lj] = 0, j = 1, 2, · · ·, 2n − 1
Just as for Lie algebras we can define a contraction of a quadratic algebra in terms of 1-parameter families of basis changes in the algebra: As → 0 the 1-parameter family of basis transformations becomes singular but the structure constants go to a finite limit [15]
Acta Polytechnica form Ψ = R(u)Πnj=1ψj(uj) where R is a fixed gauge function and ψj depends only on the variable uj and the separation constants.) We show that his limit procedure can be interpreted as constructing generalized Inönü-Wigner Lie algebra contractions of so(4, C) to itself
Summary
V (x) = a1V(1)(x) + a2V(2)(x) + a3V(3)(x) + a4, where {V(1)(x), V(2)(x), V(3)(x), 1} is a linearly independent set For these the symmetry algebra generated by H, L1, L2 always closes under commutation and gives the following quadratic algebra structure: Define 3rd order commutator R by R = [L1, L2]. All 2nd order 2D superintegrable systems with potential and their quadratic algebras are known. There is a bijection between quadratic algebras generated by 2nd order elements in the enveloping algebra of o(3, C), called free, and 2nd order nondegenerate superintegrable systems on the complex 2-sphere. There is a bijection between quadratic algebras generated by 2nd order elements in the enveloping algebra of e(2, C) and 2nd order nondegenerate superintegrable systems on the 2D complex flat space. Most of the classical special functions in the Digital Library of Mathematical Functions, as well as Wilson polynomials, appear in these ways [21]
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