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arXiv:2506.22933

Zernike polynomials from the tridiagonalization of the radial harmonic oscillator in displaced Fock states

Hashim A. Yamani, Zouhaı̈r Mouayn

Published Jun 28, 2025• 0 citations
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Abstract

We revisit the J-matrix method for the one dimensional radial harmonic oscillator (RHO) and construct its tridiagonal matrix representation within an orthonormal basis phi(z)n of L2 (R+);parametrized by a fixed z in the complex unit disc D and n = 0,1,2,.... Remarkably, for fixed n,and varying z in D, the system phi(z)n forms a family of Perelomov-type coherent states associated with the RHO. For each fixed n, the expansion of phi(z)n over the basis (fs) of eigenfunctions of the RHO yields coefficients cn,s(z; z) precisely given by two-dimensional complex Zernike polynomials. The key insight is that the algebraic tridiagonal structure of RHO contains the complete information about the bound state solutions of the two-dimensional Schrödinger operator describing a charged particle in a magnetic field (of strength proportional to B > 1/2) on the Poincaré disc D.

DOI
10.48550/arxiv.2506.22933

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We revisit the J-matrix method for the one dimensional radial harmonic oscillator (RHO) and construct its tridiagonal matrix representation within an orthonormal basis phi(z)n of L2 (R+);parametrized by a fixed z in the complex unit disc D and n = 0,1,2,.... Remarkably, for fixed n,and varying z in D, the system phi(z)n forms a family of Perelomov-type coherent states associated with the RHO. For each fixed n, the expansion of phi(z)n over the basis (fs) of eigenfunctions of the RHO yields coefficients cn,s(z; z) precisely given by two-dimensional complex Zernike polynomials. The key insight is that the algebraic tridiagonal structure of RHO contains the complete information about the bound state solutions of the two-dimensional Schrödinger operator describing a charged particle in a magnetic field (of strength proportional to B > 1/2) on the Poincaré disc D.
While dealing with the J-Matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L2(R); we rederive a set of generalized coherent states (GCS) of Perelomov type labeled by points z of the complex plane C and depending on a positive integer number m The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian associated with the mth Landau level. We extend this procedure to the Morse oscillator by constructing a new set of GCS of Glauber type.
While dealing with the J -matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L 2 ( R ) , we rederive a set of generalized coherent states (GCS) of Perelomov type labelled by points z of the complex plane C and depending on an integer number m ∈ Z + . The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials and whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian, associated with the m th Landau level. We further generalize this construction to the Morse oscillator by introducing a new set of Glauber-type GCS.
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We develop an effective radial framework for analyzing new field-modified potential function of a charged spinless particle in collinear electric and magnetic fields. Using the weak- magnetic field approximation, we neglect the diamagnetic term and other angular projections under an explicit smallness condition ≪ΔE\ll ΔE≪ΔE, where EEE is the electric field, and the replacement of the angular factor by C=cos⁡θC=\cosθC=cosθ in the spherical coordinates. The resulting equation is therefore a field-aligned radial model whose validity is limited to angularly localized states. Within this setting, we apply the indirect mode (or search mode) of Tridiagonal representation approach in the Laguerre and Jacobi bases. In the Laguerre basis, the tridiagonal constraint generates linear-plus-inverse and linear-plus-quadratic radial potentials, and the coefficient recurrences are connected with Meixner--Pollaczek and Meixner polynomial type structures, therefore yielding the analytic spectra and expansion wavefunctions. In the Jacobi basis, the mapping of the radial half-line to a finite interval produces a singular confining effective potential and a three-term recurrence that is treated through a truncated generalized eigenvalue problem. The formulation provides analytically and numerically tractable benchmark models for field-modified potential functions, while keeping the assumptions, parameter restrictions, endpoint behavior, and convergence tests explicit. In the Laguerre basis the resultant spectra are obtained in closed form and are independent of the orientation and of the electric field, which enter through the shape of the effective central potential, whereas in the Jacobi basis the spectrum is obtained numerically from a new three-term recurrence; in all cases the magnetic field enters through a linear Zeeman shift.
We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with T∗R2=C2T^*{\mathbb R}^{2}={\mathbb C}^2T∗R2=C2 as classical phase space. We show that the eigenfunctions ψnψ_nψn​ of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space C2/Zn⊂C2{\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2C2/Zn​⊂C2. They describe Zn{\mathbb Z}_nZn​-invariant motion of particle along a circle S1S^1S1 in lens space S3/Zn⊂C2/ZnS^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_nS3/Zn​⊂C2/Zn​, where Zn{\mathbb Z}_nZn​ is the cyclic group of rotation by an angle 2π/n2π/n2π/n on the circle S1S^1S1, n=1,2,... n=1,2,...\,n=1,2,.... Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states ψnψ_nψn​ that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.
We study the problem of a charged particle in a uniform magnetic field with two different gauges, known as Landau and symmetric gauges. By using a similarity transformation in terms of the displacement operator we show that, for the Landau gauge, the eigenfunctions for this problem are the harmonic oscillator number coherent states. In the symmetric gauge, we calculate the SU(1; 1) Perelomov number coherent states for this problem in cylindrical coordinates in a closed form. Finally, we show that these Perelomov number coherent states are related to the harmonic oscillator number coherent states by the contraction of the SU(1; 1) group to the Heisenberg-Weyl group.