The Schr\"odinger operator on graphs and topology
S.P. Novikov
TL;DR
This paper develops a symplectic scalar product on solutions of the discrete Schrödinger equation on graphs, linking it to graph topology and enabling a fundamental understanding of scattering theory through symplectic geometry.
Contribution
It introduces a novel symplectic scalar product on solution spaces of the discrete Schrödinger equation on graphs, connecting spectral properties with topological invariants.
Findings
Scalar product takes values in the first homology group of the graph
Unitarity properties of scattering are derived from symplectic geometry
The approach links spectral theory with graph topology
Abstract
Symplectic vector-valued scalar product is constructed on the spaces of solutions of the real discrete Shrodinger equation with fixed value of the spectral parameter on graphs. It takes values in the first homology group of the graph. This scalar product plays fundamental role in the Scattering theory for the graphs with tails. In particular, all unitarity properties of scattering follow from elementary symplectic geometry
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Taxonomy
Topicsadvanced mathematical theories