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Статья

Scattering on periodic metric graphs

Evgeny Korotyaev, Saburova N.

We consider the Laplacian on a periodic metric graph and obtain its decomposition into a
direct ber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and
eigenvalues of the ber metric Laplacian are expressed explicitly in terms of eigenfunctions
and eigenvalues of the corresponding ber discrete Laplacian and eigenfunctions
of the Dirichlet problem on the unit interval. We show that all these eigenfunctions are
uniformly bounded. We apply these results to the periodic metric Laplacian perturbed
by real integrable potentials. We prove the following: (a) the wave operators exist and
are complete, (b) the standard Fredholm determinant is well-de ned and is analytic in
the upper half-plane without any modi cation for any dimension, (c) the determinant
and the corresponding S-matrix satisfy the Birman{Krein identity.