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Intertwining symmetry algebras of quantum superintegrable systems. (English) Zbl 1161.37049

Summary: We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwining operators, that span pairs of Lie algebras like \((\text{su}(n),\text{so}(2n))\) or \((\text{su}(p,q),\text{su}(2p,2q))\). The eigenstates of the associated Hamiltonian hierarchies belong to unitary representations of these algebras. It is shown that these intertwining operators, related with separable coordinates for the system, are very useful to determine eigenvalues and eigenfunctions of the Hamiltonians in the hierarchy. A study of the corresponding superintegrable classical systems is also included for the sake of completeness.

MSC:

37K30 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
17B80 Applications of Lie algebras and superalgebras to integrable systems
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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