Two-dimensional discrete operators and rational functions on algebraic curves

In this paper we study a connection between finite-gap on one energy level two-dimensional Schrödinger operators and two-dimensional discrete operators. We find spectral data for a new class of two-dimensional integrable discrete operators. These operators have eigenfunctions on zero level energy parameterized by points of algebraic spectral curves. In the case of genus one spectral curves we show that the finite-gap Schrödinger operators can be obtained as a limit of the discrete operators.