On the discrete-time Schroedinger equation for the linear harmonic oscillator

In this paper is studied a mixed equation, i.e. a finite-difference one with respect to time and a differential one with respect to a continuous variable x, which reduces to the Schroedinger equation for the linear harmonic oscillator in the proper continuum limit. We construct its solution explicitly by a suitable algorithm, which is also shown to have applications in the theory of Hermite polynomials