On a Discrete Number Operator Associated with the 5D Discrete Fourier Transform
We construct an explicit form of a difference analogue of the quantum number operator in terms of the raising and lowering operators that govern eigenvectors of the 5D discrete (finite) Fourier transform. Eigenvalues of this difference operator are represented by distinct non-negative numbers so that it can be used to systematically classify, in complete analogy with the case of the continuous classical Fourier transform, eigenvectors of the 5D discrete Fourier transform, thus resolving the ambiguity caused by the well-known degeneracy of the eigenvalues of the discrete Fourier transform.
KeywordsDiscrete Fourier transform Raising and lowering operators 5D eigenvectors
We are grateful to Vladimir Matveev, Timothy Gendron and Ívan Area for illuminating discussions and thank Fernando González for graphical support of our results and the computation of the eigenvalues (24)–(27) and eigenvectors (28)–(32) with the aid of Mathematica. The participation of MKA in this work has been partially supported by the SEP-CONACyT project 168104 ‘Operadores integrales y pseudodiferenciales en problemas de fisíca matemática’, and NMA has been partially supported by the UNAM–DGAPA project IN101115 ‘ Óptica matemática’.
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