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.

In geometric optics the Maxwell fish-eye is a medium where light rays follow circles, while in scalar wave optics this medium can only 'trap' fields of certain discrete frequencies. In the monochromatic case characterized by a positive integer ℓ, there are + ℓ 2 1independent fields. We identify two bases of functions: one, known as the Sherman-Volobuyev functions, is characterized as of 'most definite' momenta; the other is new and composed of 'most definite' positions and normal derivatives for the fish-eye scalar wavefields. Their construction uses the stereographic projection of the sphere, and their identification is corroborated in the → ∞ ℓ contraction limit to the homogeneous Helmholtz medium.

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Abstract. The sphere is well understood manifold, as is the basis of spherical harmonics for functions thereof. A stereographic projection of the sphere realizes the Maxwell fish-eye, an optical medium whose refractive index distribution is such that light rays travel in circles. Definite angular momentum on the sphere corresponds with monochromatic light in the fish-eye. A contraction of these manifolds (as the radius of the sphere grows without bound) results in a plane medium of wave functions subject to the Helmholtz equation. In the latter, there is the continuous basis of 'momenta' (plane waves) and a denumerably infinite basis of 'positions' and 'normal derivatives' (Bessel ∼ J0(x) and ∼ J1(x)/x functions) for the Hilbert space of these wavefields. We present the pre-contraction of these bases to the finite-dimensional systems of fish-eye medium and the sphere. The 'momentum' basis is a subset of Sherman-Volobuyev functions; in this paper we propose new 'position' and 'normal derivative' bases for this finite system. The bases are not orthogonal, so their measure is non-local, and here we find their dual bases.1. The harmonic basis on the sphere Phase space is a concept created in classical mechanics that applies as well in geometric optics [1]. In quantum mechanics and in wave optics, there are several approaches to phase space through the definition of the Wigner [2] and Wigner-like [3] quasidistribution functions. Those that follow Wigner's original formulation hinge on the Heisenberg-Weyl algebra of noncommuting operators of position and momentum; other formulations also rely on a supporting Lie algebra [4].In this work we start on a different path, based on the monochromatic Maxwell fish-eye wave-optical system, presenting what appear to be the natural wavefields of definite momentum and position. Since this system is a stereographic projection of the sphere on a plane or higherdimensional flat manifold [5,6,7,8,9,10,11],[12, Sect. 6.4], the analysis remits us to the sphere, treated as in angular momentum theory through the well-known spherical harmonics, whose relevant properties are recalled in Sect. 2, while the link to the fish-eye model is summarized in Sect. 3. In our case we have not a continuous, but a finite system, i.e., the Hilbert space of wavefields for which we provide the bases is finite-dimensional.The basis of functions of 'most definite momentum' are known as the Sherman-Volobuyev functions, of which we need only a finite number for fixed angular momentum in Sect. 4, because they correspond to monochromatic wavefields in the optical model. The basis of 'most definite position' presented in Sect. 5 actually involves two sub-bases, of positions and of normal