Intertwining Symmetry Algebras of Quantum Superintegrable Systems

We present an algebraic study of a kind of quantum systems belonging to a family of superintegrable Hamiltonian systems in terms of shape-invariant intertwinig operators, that span pairs of Lie algebras like $(su(n),so(2n))$ or $(su(p,q),so(2p,2q))$. 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. An study of the corresponding superintegrable classical systems is also included for the sake of completness.

💡 Research Summary

The paper investigates a broad class of quantum superintegrable Hamiltonian systems by exploiting shape‑invariant intertwining operators. Superintegrability means that, in addition to the Hamiltonian, the system possesses more independent constants of motion than the number of degrees of freedom, which leads to highly degenerate spectra and separability in several coordinate systems. The authors start from a basic Hamiltonian defined on an n‑dimensional sphere (or its non‑compact analogues) and construct first‑order differential operators that act as ladder (raising and lowering) operators between Hamiltonians whose parameters differ by a shift. These operators satisfy intertwining relations of the form

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