ar X iv : m at h - ph / 9 90 30 25 v 1 1 1 M ar 1 99 9 Dedicated to the 70 th birthday of Mikio

Introduction. During the last 3 years the present author made a series of works [1, 2, 3, 4, 5, 8, 6, 7, 9] dedicated to the study of the unusual spectral properties of low-dimensional continuous and discrete (difference) Schrodinger Operators. Some of these works were done in collaboration with A.Veselov, I.Taimanov and I.Dynnikov. First, let me briefly describe the list of problems discussed in these works. 1.Euler-Darboux-Backlund (EDB)-Transformations as nonstandard spectral symmetries for the 1-dimensional Schrodinger Operators and its discrete analogs on the lattice Z. Problem of cyclic chains, its solutions for the special cases. Exactly solvable spectral problems for some operators. EDB Transformations for the nonstationary Schrodinger Equation and the problem of cyclic chains (see [11, 12, 13, 14, 15, 2, 8, 16]). 2.Laplace Transformations for the 2D stationary Schrodinger operators in the double-periodic magnetic field and potential, acting on the space of eigenfunctions of one energy level. Problems of cyclic, semicyclic and quasicyclic chains. The possibility to have two exactly solvable highly degenerate energy levels as a maximal possible solvability for the spectral theory in the Hilbert space L2(R ) (except the Landau case in constant magnetic field and trivial potential)? Discretization of the Laplace transformations: square lattice is compatible with hyperbolic equations; equilateral triangle lattice is compatible with elliptic selfadjoint operators. Exactly solvable operators. See [1, 2, 3, 4, 6]. 3.The second order operators on simplicial complexes. Factorizations and Laplace Transformations. The cases of 2-manifolds with 2-colored triangulation and multidimensional equilateral lattices. Zero modes problem. First order equations in the simplicial complexes and nonstandard discretization of connections, combinatorial curvature. See [6, 7].