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

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

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