  We consider the Laplacian on a periodic metric graph and obtain its
decomposition into a direct fiber integral in terms of the corresponding
discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric
Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues
of the corresponding fiber 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-defined and is analytic in the upper half-plane without any modification
for any dimension, c) the determinant and the corresponding S-matrix satisfy
the Birman-Krein identity.

Spectral Theory