An explicit analytic construction of the eigenvectors of the number operator N5 = A5-transpose A5 associated with the 5x5 discrete Fourier transform (DFT), obtained by treating the intertwining operators A5 and A5-transpose as discrete analogs of the harmonic-oscillator lowering and raising operators. The construction exploits two structural facts: the intertwining relations A5 Phi5 = i Phi5 A5 (which make A5, A5-transpose step between DFT eigenvectors), and the symmetry of these operators under the discrete reflection operator Pd, which commutes with the DFT and splits its eigenvectors into symmetric and antisymmetric classes. A 'sparsealization' procedure simplifies the intertwining operators enough to build a discrete counterpart of the continuous formula psi_n = (a-dagger)^n psi_0 / sqrt(n!), generating each eigenvector by repeated raising from the lowest one, and a discrete Hermite-type formula expressing the eigenvectors as Newtonian basis polynomials P_n(X5) times the ground eigenvector f_0. The contribution is the elementary, symmetry-driven route to closed-form DFT-number-operator eigenvectors, tying the discrete problem to the harmonic-oscillator algebra and to cubic (Askey-Wilson-type) algebras.
This math-physics paper constructs, by an elementary analytic method, the eigenvalues and eigenvectors of the number operator N5 = A5-transpose A5 built from the 5-point discrete Fourier transform (D…