Hermitian symplectic geometry and the Schr"{o}dinger operator on the graph

M. Harmer


The theory of self-adjoint extensions is closely related to the theory of
hermitian symplectic geometry cite{Pav,Kost:Sch,Nov3}. Here we develop
this idea, showing that it may also be used to consider symmetric
extensions of a symmetric operator. Furthermore we find an explicit
parameterisation of the Lagrange Grassmannian in terms of the unitary
matrices $U (n)$. This allows us to explicitly describe all self-adjoint
boundary conditions for the Schr"{o}dinger operator on
the graph in terms of a unitary matrix. We show that the asymptotics
of the scattering matrix can be simply expressed in terms of this
unitary matrix. \
Using the construction of the asymptotic hermitian
symplectic space cite{Nov1,Nov3} we derive a formula for the scattering
matrix of a graph in terms of the scattering matrices of its subgraphs.
This also provides a characterisation of the discrete eigenvalues
embedded in the continuous spectrum.

Schr"{o}dinger operator, graphs, extension theory, hermitian symplectic geometry, graph factorisation the graph

Math Review Classification

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39 pages

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