On a Discrete Number Operator Associated with the 5D Discrete Fourier Transform

AIP Conference Proceedings

2019

In the present work, we discuss some additional findings concerning algebraic properties of the N -dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81-92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N -dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N (N ) , that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.

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Section: Eigenvectors For the Even Cases N=2lmentioning

confidence: 99%

“…which represents at the same time the eigenvector of the DFT operator Φ (6) , associated with the eigenvalue i 0 = 1.…”

mentioning

confidence: 99%

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More on algebraic properties of the discrete Fourier transform raising and lowering operators

AIP Conference Proceedings

2019

show abstract

“…with the DFT operator Φ (N ) . Then it is straightforward to verify, by using definition (6), that a discrete number operator, defined as a difference operator of the form…”

Section: Introductionmentioning

confidence: 99%

“…In order to test the consistency of this approach to finding eigenfunctions and eigenvalues of the DFT operator Φ (N ) , the particular case of the 5-dimensional DFT was studied in detail in [6]. It was confirmed that the eigenvalues of the discrete number operator N (5) are represented by distinct non-negative numbers and the corresponding eigenvectors f k , 0 ≤ k ≤ 4, can be successively defined with the aid of the raising operator b † 5 and the lowest eigenvector f 0 , which is found as a solution of the difference equation b 5 f 0 = 0.…”

Section: Introductionmentioning

confidence: 99%

More on algebraic properties of the discrete Fourier transform raising and lowering operators

2019

4open

In the present work, we discuss some additional findings concerning algebraic properties of the N-dimensional discrete Fourier transform (DFT) raising and lowering difference operators, recently introduced in [Atakishiyeva MK, Atakishiyev NM (2015), J Phys: Conf Ser 597, 012012; Atakishiyeva MK, Atakishiyev NM (2016), Adv Dyn Syst Appl 11, 81–92]. In particular, we argue that the most authentic symmetrical form of discretization of the integral Fourier transform may be constructed as the discrete Fourier transforms based on the odd points N only, while in the discrete Fourier transforms on the even points N this symmetry is spontaneously broken. This heretofore undetected distinction between odd and even dimensions is shown to be intimately related with the newly revealed algebraic properties of the above-mentioned DFT raising and lowering difference operators and, of course, is very consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional DFT. In addition, we propose a general approach to deriving the eigenvectors of the discrete number operators N(N), that avoids the above-mentioned pitfalls in the structure of each even-dimensional case N = 2L.

show abstract

On the Discrete Fourier Transform Eigenvectors and Spontaneous Symmetry Breaking

Springer Proceedings in Mathematics &Amp; Statistics

2020