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.