Bloch theory and spectral gaps for linearized water waves

The system of equations for water waves, when linearized about an equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet--Neumann operator of the unperturbed free surface. This spectral problem is fundamental in questions of stability, as well as to the perturbation theory of the evolution of the free surface in such settings. In addition, the Dirichlet--Neumann operator is self-adjoint when given an appropriate definition and domain, and it is a novel but very natural spectral problem for a nonlocal operator. In the case in which the bottom boundary varies periodically, $\{y = -h + b(x)\}$ where $b(x+\gamma) = b(x)$, $\gamma \in \Gamma$ is a lattice, this spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. In this article we describe this analytic construction in the case of a spatially periodic bottom variation from constant depth in two space dimensional water w...