Willard Miller

Classical and quantum superintegrability with applications

A superintegrable system is, roughly speaking, a system that allows more integrals of motion than degrees of freedom. This review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta. We present a classification of second-order superintegrable systems in two-dimensional Riemannian and pseudo-Riemannian spaces. It is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis. The determining equations for the existence of integrals of motion of arbitrary order in real Euclidean space E2 are presented and partially solved for the case of third-order integrals. A systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order. The algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally. The relation between superintegrability and the classification of orthogonal polynomials is analyzed.

Completeness of superintegrability in two-dimensional constant-curvature spaces

We classify the Hamiltonians H = px2 + py2 + V(x,y) of all classical superintegrable systems in two-dimensional complex Euclidean space with two additional second-order constants of the motion. We similarly classify the superintegrable Hamiltonians H = J12 + J22 + J32 + V(x, y, z) on the complex two-sphere where x2 + y2 + z2 = 1. This is achieved in all generality using properties of the complex Euclidean group and the complex orthogonal group.

Superintegrable systems in Darboux spaces

Almost all research on superintegrable potentials concerns spaces of constant curvature. In this paper we find by exhaustive calculation, all superintegrable potentials in the four Darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta, in addition to the Hamiltonian. These are two-dimensional spaces of nonconstant curvature. It turns out that all of these potentials are equivalent to superintegrable potentials in complex Euclidean 2-space or on the complex 2-sphere, via “coupling constant metamorphosis” (or equivalently, via Stackel multiplier transformations). We present a table of the results.

The Group

The representation theory of the group

O(4)

O(4)

, Separation of Variables and the Hydrogen Atom

is considered systematically in different bases, corresponding to the reduction of

Superintegrability and associated polynomial solutions: Euclidean space and the sphere in two dimensions

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Killing Tensors and Variable Separation for Hamilton-Jacobi and Helmholtz Equations

to various continuous or discrete subgroups. The results are applied to the hydrogen atom and we investigate the six different bases corresponding to separation of variables in the Schrodinger equation in momentum space and the four different bases corresponding to separation in coordinate space. It is shown that a classification of different bases (and of complete sets of commuting operators determining the bases) corresponds to a classification of interactions, breaking the original symmetry, while preserving certain aspects of it. Vector and scalar potentials providing such breaking of the

Superintegrability in Classical and Quantum Systems

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Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory

symmetry of the hydrogen atom are constructed explicitly. The relationship between