Discrete Schrodinger operators and topology

Abstract

This work is a continuation and extension of the note published in the Russian Math Surveys 1997 n 6. For any pair of solutions of the spectral problem for the second order selfadjoint real Schrodinger Operator on the graph their Symplectic Wronskian is a 1-cycle in the graph. This is a vector-valued symplectic 2-form on the space of solutions. This construction was applied to the Scattering Theory on the graphs with finite number of tails. The asymptotic values of solutions is a Lagrangian Plane of half dimension. This property determines all unitary properties of Scattering Matrix, which is also symmetric. All higher order discrete operators and operators on higher dimensional simplicial complexes are included in this scheme. Nonlinear analog of that was invented by the present author in collaboration with A.S.Schwarz.