Probabilistic representation of the Cauchy problem solution for the discrete nonstationary Schrödinger equation

We study the one-dimensional free Schrödinger equation on $\mathbb{Z}$, which describes the quantum evolution of a discrete wave function $u(n,t)$ with continuous time. The initial state $\varphi(n)$ is prescribed, and the wave function admits the standard interpretation: namely, $|u(n,t)|^2$ represents the probability of observing a free particle at site $n$ at time $t$. A new approach to solving such an evolution equation is developed, based on the use of discrete analytic functions and symmetric random walks.