A sufficient condition for nonsingularity of two-dimensional finite-gap   Schr\"{o}dinger operator discretization of a quad graph
  We construct a two-dimensional Schr\"{o}dinger operator on a quad-graph using finite-gap approach. A guad-graph is a planar graph with only 4-sided faces. We define the non-singularity of the operator as the positivity of all of his coefficients. At last, we prove some sufficient conditions of such positivity on the spectral curve and the graph. 
Vasilevskiy Boris
math-ph math.MP