I don't have a complete answer, but the math physics literature has many papers on "discrete Schrödinger operators," which are partial difference operators on a discrete space. These are analyzed in their own right, not as an approximation of some continuum model. Look for "discrete Schrödinger operators." Also look for "tight binding approximation" or "tight binding Schrödinger operators." Also try spelling Schrödinger as Schroedinger.
I don't know of a "canonical reference" along the lines of Evans for PDE's, say. On the other hand, the basic theory of discrete difference operators (existence, uniqueness, etc.) is usually easier since there are no local regularity issues.
