The Green's function of a five-point discretization of a two-dimensional finite-gap Schr\"{o}dinger operator: the case of four singular points of the spectral curve

Abstract

Consider a five-point discretization of a two-dimensional finite-gap for a fixed energy Schr\"{o}dinger operator. We construct the Green's function of the operator. In appears as the explicit formula in terms of the integral by the specific contour on the spectral curve of the differential which is constructed by the wave function and its double. The formula is parametrized by the point on the spectral curve and for almost every parameter value it gives the Green's function with known growth at infinity.