ON A FAMILY OF DISCRETE ND LADDER-TYPE OPERATORS CONSTRUCTED IN TERMS OF THE HERMITIAN TOEPLITZ COMMUTATOR OPERATOR Z_N

Year 2024, Volume: 6 Issue: 2, 24 - 49, 08.11.2024

Miguel Angel Ortiz Natig Atakishiyev

https://doi.org/10.47087/mjm.1467436

Abstract

A development of an algebraic system with N-dimensional ladder-type
operators associated with the discrete Fourier transform is described,
following an analogy with the canonical commutation relations of the continuous
case. It is found that a Hermitian Toeplitz matrix Z_N, which plays the
role of the identity, is sufficient to satisfy the Jacobi identity and, by solving
some compatibility relations, a family of ladder operators with corresponding
Hamiltonians can be constructed. The behaviour of the matrix Z_N for large
N is elaborated. It is shown that this system can be also realized in terms
of the Heun operator W, associated with the discrete Fourier transform, thus
providing deeper insight on the underlying algebraic structure.

Keywords

ladder operators, Hermitian Toeplitz operator, discrete Fourier transform

References

• I. Bengtsson, K. Zyczkowsky, Geometry of Quantum States: An Introduction to Quantum Entaglement, Cambridge University Press, Cambridge, 2017.
• J. H. McClellan and T.W. Parks, Eigenvalue and Eigenvector Decomposition of the Discrete Fourier Transform, IEEE Transactions on Audio and Electroacoustics, AU-20 (1972) 66-74.
• M. K. Atakishiyeva and N. M. Atakishiyev, On the raising and lowering difference operators for eigenvectors of the finite Fourier transform, J. Phys: Conf. Ser., 597 (2015) 012012.
• M. K. Atakishiyeva and N. M. Atakishiyev, On algebraic properties of the discrete raising and lowering operators, associated with the N-dimensional discrete Fourier transform, Adv. Dyn. Syst. Appl., 11 (2016) 81-92.
• N. M. Atakishiyev, The eigenvalues and eigenvectors of the 5D discrete Fourier transform number operator revisited, Maltepe J. Math., 4 (2022) 55-65.
• A. Kuznetsov and M. Kwásnicki, Minimal Hermite-type eigenbasis of the discrete Fourier transform, Journal of Fourier Anal. Appl., 25 (2019) 1053-1079.
• A. N. Karkishchenko and V. B. Mnukhin, On a system of eigenvectors of the Fourier trans- form over finite Gaussian fields, J. Phys.: Conf. Ser., 1096 (2018) 012040.
• B. W. Dickinson and K. Steiglitz, Eigenvectors and functions of the discrete Fourier trans- form, IEEE Trans. Acoust. Speech and Signal Proc., 30 (1982) 25-31.
• F. A. Grünbaum, The eigenvectors of the discrete Fourier transform: a version of the Hermite functions, J. Math. Anal. Appl., 88 (1982) 355-363.
• G. B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, New Jersey, 1989.
• T. H. Seligman, Group Theory and its Applications in Physics, AIP, New York, 1980.
• N. M. Atakishiyev and P. Winternitz, Bases for representations of quantum algebras, J. Phys. A: Math. Gen., 33 (2000) 5303-5313.
• L. Auslander and R. Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics, Bull. Amer. Math. Soc.,1 (1979) 847-897.
• I. Kra and S. R. Simanca, On Circulant Matrices, Notes of the AMS, 59 (2012) 368-377.
• H. Georgi, Lie Algebras in Particle Physics, Westview Press, USA, 1999.
• T. S. Santhanam and A. R. Tekumalla, Quantum Mechanics in Finite Dimensions, Foundations of Physics, 6 (1975) 583-587.
• M. C. Pereyra and L. A. Word, Harmonic analysis: from Fourier to wavelets, AMS, Providence, Rhode Island, 2012.
• J. Alvarez, A Mathematical Presentation of Laurent Schwartz Distributions, Surveys in Mathematics and its Applications, 15 (2020) 1-37.
• J. Alvarez and M. Guzman-Partida, Properties of the Dirichlet kernel, Electronic J. Math. Anal. Appl., 11 (2023) 96-110.
• N. L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space, Birkhäuser, Basel, 2008.
• A. Perälä, J.Taskinen and J.Virtanen, Toeplitz operators with distributional symbols on Fock spaces, Functiones et Approximatio Commentarii Mathematici, 44 (2011) 203-213.
• A. Ch. Ganchev and T. D. Palev, A Lie superalgebraic interpretation of the para-Bose statistics, J. Math. Phys., 21 (1979) 797-799.
• H. S. Green, A Generalized Method of Field Quantization, Phys. Rev., 90 (1953) 270-273.
• E. I. Jafarov, S. Lievens and J. Van der Jeugt, The Wigner distribution function for the one-dimensional parabose oscillator, J. Phys A: Math.Theor., 41 (2008) 235301.
• E. I. Jafarov, N. I. Stoilova and J. Van der Jeugt Finite oscillator models: the Hahn oscil- lator, J. Phys. A: Math. Theor., 44 (2011) 265203.
• M. Arik, N. M. Atakishiyev and K. B. Wolf, Quantum algebraic structures compatible with the harmonic oscillator Newton equation, J. Phys. A: Math. Gen., 32 (1999) L371- L376.
• E. P. Wigner, Do the equations of motion determine the quantum mechanical commutation relations?, Phys. Rev., 77 (1950) 711-712.
• M. K. Atakishiyeva, N. M. Atakishiyev and A. Zhedanov, An algebraic interpretation of the intertwining operators associated with the discrete Fourier transform, J. Math. Phys., 62 (2021) 101704.
• G. L. Naber, Topology, Geometry and Gauge Fields: Interactions, Springer-Verlag, New York, 2000.
• J. M. Bogoya, A. Bättcher, S. M. Grudsky and E. A. Maximenko, Eigenvalues of Hermitian Toeplitz matrices with smooth simple-loop symbols, J. Math. Anal. Appl., 42 (2015) 1308-1334.
• V. E. Tarasov, Exact Discrete Analogs of Canonical Commutation and Uncertainty Rela- tions, Mathematics, 4 (2016) 44.