Discrete Schrodinger operators and topology

TL;DR

This paper explores the spectral theory of discrete Schrödinger operators on graphs, revealing symplectic structures and their implications for scattering theory, with extensions to higher-order and nonlinear operators.

Contribution

It introduces a symplectic Wronskian framework for solutions of Schrödinger operators on graphs and extends this to higher-order and nonlinear operators.

Findings

Symplectic Wronskian is a 1-cycle in the graph.

Asymptotic solutions form a Lagrangian plane.

Scattering matrix properties are derived from symplectic structure.

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.