Spectral properties of two classes of periodic difference operators

A study is made of the iso-energetic spectral problem for two classes of multidimensional periodic difference operators. The first class of operators is defined on a regular simplicial lattice. The second class is defined on a standard rectangular lattice and is the difference analogue of a multidimensional Schrödinger operator. The varieties arising in the direct spectral problem are described, along with the divisor of an eigenfunction, defined on the spectral variety, of the corresponding operator. Multidimensional analogues are given for the Veselov–Novikov correspondences connecting the divisors of the eigenfunction with the canonical divisor of the spectral variety. Also, a method is proposed for solving the inverse spectral problem in terms of $\theta$-functions of curves lying “at infinity” on the spectral variety.