Acoustic Diffraction Problems on Periodic Graphs

We consider acoustic diffraction by graphs $\Gamma$ embedded in $\mathbb{R}^{2}$ and periodic with respect to an action of the group $\mathbb{Z}^{n}$, $n=1,2$. The diffraction problem is described by the Helmholtz equation with variable nonperiodic bounded coefficients and nonperiodic transmission conditions on the graph $\Gamma$. We introduce single and double layer potentials on $\Gamma$ generated by the Schwartz kernel of the operator inverse to the Helmholtz operator on $\mathbb{R}^{2}$ and reduce the diffraction problem to a boundary pseudodifferential equation on the graph. Necessary and sufficient conditions for the boundary operators to be Fredholm are obtained.