Quantum superintegrable Zernike system

Abstract

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account the quantum Zernike system, where that differential equation is seen as a Schrödinger equation with a potential, is special in that it has a potential and boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates in two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schrödinger-type equations with Pöschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.

Submitted to J Math Phys

References in corpus (2)

• Superintegrable classical Zernike system
• Separation of variables and Lie algebra contractions. Applications to special functions

Cited by in corpus (10)

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• Higher-order superintegrable momentum-dependent Hamiltonians on curved spaces from the classical Zernike system
• An infinite family of Dunkl type superintegrable curved Hamiltonians through coalgebra symmetry: Oscillator and Kepler-Coulomb models
• Generalized quantum Zernike Hamiltonians: Polynomial Higgs-type algebras and algebraic derivation of the spectrum
• Quantum Aberrations: Entangling Photons with Zernike Polynomials