Preprint : H. A. Yamani & Z. Mouayn 28-06-2025

[Zouhair Mouayn]

Dear All,

Comments about Preprint are welcome.
Attached file.
Best regards
Zouhaïr Mouayn
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[Submitted on 28 Jun 2025]
Zernike polynomials from the tridiagonalization of the radial harmonic oscillator in displaced Fock states
Hashim A. Yamani, Zouhaïr Mouayn
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(IR+);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 (f_s) of eigenfunctions of the RHO yields coefficients C_n,s(z; zbar) 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.