- The present work aims to give a detailed discussion of a recently introduced difference analogue of quantum number operator in terms of the raising and lowering difference operators, that governs eigenvectors of the N-dimensional discrete (finite) Fourier transform (DFT). In particular, we argue that the aforementioned discrete number operator has distinct eigenvalues only if it is associated with the DFT’s based on grids, with N odd. This means that in the cases of the DFT’s on grids with N even the discrete reflection symmetry in the space of eigenvectors of the discrete number operator is spontaneously broken. This essential distinction between even and odd dimensions is intimately related with the algebraic properties of the above DFT raising and lowering difference operators and consistent with the well-known formula for the multiplicities of the eigenvalues, associated with the N-dimensional discrete Fourier transform. © 2020, Springer Nature Switzerland AG.