Lower part of the spectrum for the two-dimensional Schrödinger operator periodic in one variable and application to quantum dimers

We study the semiclassical asymptotic approximation of the spectrum of the two-dimensional Schrödinger operator with a potential periodic in $x$ and increasing at infinity in $y$. We show that the lower part of the spectrum has a band structure (where bands can overlap) and calculate their widths and dispersion relations between energy and quasimomenta. The key role in the obtained asymptotic approximation is played by librations, i.e., unstable periodic trajectories of the Hamiltonian system with an inverted potential. We also present an effective numerical algorithm for computing the widths of bands and discuss applications to quantum dimers.