Families of superintegrable Hamiltonians constructed from exceptional polynomials

We introduce a family of exactly-solvable two-dimensional Hamiltonians whose wave functions are given in terms of Laguerre and exceptional Jacobi polynomials. The Hamiltonians contain purely quantum terms which vanish in the classical limit leaving only a previously known family of superintegrable systems. Additional, higher-order integrals of motion are constructed from ladder operators for the considered orthogonal polynomials proving the quantum system to be superintegrable.

💡 Research Summary

The paper presents a new family of exactly‑solvable two‑dimensional quantum Hamiltonians whose eigenfunctions are built from Laguerre polynomials in the radial coordinate and exceptional Jacobi polynomials in the angular coordinate. The construction begins by separating the Schrödinger equation in polar coordinates, writing the wavefunction as (\Psi(r,\theta)=R(r)\Theta(\theta)). The radial part (R(r)) is proportional to a Gaussian factor times a Laguerre polynomial (L_n^{(\alpha)}(r^2)); this reproduces the spectrum of the two‑dimensional isotropic oscillator. The angular part (\Theta(\theta)) is expressed through an exceptional Jacobi polynomial (X_m^{(\beta,\gamma)}(\cos 2\theta)), which differs from the ordinary Jacobi basis by the presence of a “missing degree’’ (the exceptional index) and additional parameters.

The Hamiltonian takes the form

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