分野別セミナー

Superintegrable quantum systems with second order integrals display very interesting properties which make them interesting from perspective of mathematics and physics. These models are multiseparable, exactly solvable and with a degenerate energy spectrum. Their spectrum can be described via representations of quadratic algebras. Their wavefunctions have been related to the full Askey scheme of orthogonal polynomials. Until recently, much less was know for Hamiltonians with integrals of degree 3 or higher, even on two dimensional spaces. I will review results on classification of quantum superintegrable systems on two-dimensional Euclidean space with integrals of higher order. They have been connected with the Chazy class of differential equations and the six Painlevé transcendents. I will discuss how this search has been extended to Riemannian manifolds. An Hamiltonian involving the sixth Painlevé transcendent have been discovered. I will briefly discuss how for these Hamiltonians the symmetry algebra can take the form of polynomial algebras.