The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

Certain solutions to Harper's equation are discrete analogues of (and appro ximations to) the Hermite-Gaussian functions. They are the energy eigenfunc tions of a-discrete algebraic analogue of the harmonic oscillator, and they lead to a definition of a discrete fractional Fourier transform (FT). The discrete fractional FT is essentially the time-evolution operator of the di screte harmonic oscillator.