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On the eigenvectors of the 5D discrete Fourier transform number operator in Newtonian basis

Natig Atakishiyev

TL;DR

This work develops an analytic framework to compute the eigenvalues and eigenvectors of the 5D discrete number operator $\mathcal{N}_5 = A_5^{\intercal} A_5$ by exploiting the $P_d$-symmetry of the intertwining operators with respect to the 5-point DFT $\Phi_5$. It constructs an explicit ladder of eigenvectors $f_0,\dots,f_4$ with corresponding eigenvalues and shows $\mathcal{N}_5$ commutes with $\Phi_5$, thus classifying $\Phi_5$'s eigenvectors and resolving degeneracies. A sparsealization procedure isolates symmetric/antisymmetric annihilators, enabling a discrete analog of the continuous harmonic-oscillator formulas, notably expressing $f_n$ as $f_n = d_n^{-1}\mathcal{P}_n(X_5) f_0$ where $\mathcal{P}_n$ are Newtonian-basis polynomials. The paper links the discrete framework to an Askey-Wilson/Cubic-algebra structure and provides a concrete, algebraic method for spectral analysis of the 5-point DFT with potential numerical applications.

Abstract

A simple analytic approach to the evaluation of the eigenvalues and eigenvectors f_n of the 5D discrete number operator N_5 is formulated. This approach is essentially based on the symmetry of the intertwining operators with respect to the discrete reflection operator. A procedure for the sparsealization of the intertwining operators has been developed, which made it possible to establish a discrete analog of the well-known continuous case formula. A discrete analog for the eigenvectors f_n of another continuous case formula is constructed in the Newtonian basis polynomials, times the lowest eigenvector f_0.

On the eigenvectors of the 5D discrete Fourier transform number operator in Newtonian basis

TL;DR

This work develops an analytic framework to compute the eigenvalues and eigenvectors of the 5D discrete number operator by exploiting the -symmetry of the intertwining operators with respect to the 5-point DFT . It constructs an explicit ladder of eigenvectors with corresponding eigenvalues and shows commutes with , thus classifying 's eigenvectors and resolving degeneracies. A sparsealization procedure isolates symmetric/antisymmetric annihilators, enabling a discrete analog of the continuous harmonic-oscillator formulas, notably expressing as where are Newtonian-basis polynomials. The paper links the discrete framework to an Askey-Wilson/Cubic-algebra structure and provides a concrete, algebraic method for spectral analysis of the 5-point DFT with potential numerical applications.

Abstract

A simple analytic approach to the evaluation of the eigenvalues and eigenvectors f_n of the 5D discrete number operator N_5 is formulated. This approach is essentially based on the symmetry of the intertwining operators with respect to the discrete reflection operator. A procedure for the sparsealization of the intertwining operators has been developed, which made it possible to establish a discrete analog of the well-known continuous case formula. A discrete analog for the eigenvectors f_n of another continuous case formula is constructed in the Newtonian basis polynomials, times the lowest eigenvector f_0.
Paper Structure (5 sections, 1 theorem, 88 equations)

This paper contains 5 sections, 1 theorem, 88 equations.

Table of Contents

  1. Introduction
  2. 5D intertwining operators $A_5$ and $A^{\intercal}_5$
  3. Eigenvectors and eigenvalues of the discrete number operator $\mathcal{N}_5$
  4. Explicit forms of discrete analogs
  5. Conclusions

Key Result

Proposition 1

The product $\Phi_5 X_5$ can be represented as either where ${\mathcal{A}}^{(s)}$ is a symmetric annihilator operator that annuls every $P_d$-symmetric vector $f^{(s)}:=(a,b,c,c,b)^{\intercal}$, and ${\mathcal{B}}^{(s)}$ is a sparse matrix, or where ${\mathcal{A}}^{(a)}$ is an antisymmetric annihilator operator that annuls every $P_d$-antisymmetric vector $f^{(a)}:=(0,b,c,- c,- b)^{\intercal}$,

Theorems & Definitions (3)

  • Proposition 1
  • proof
  • Remark 3.1