Schrödinger Degeneracies of Lattice Potentials

We study the spectral theory of Schrödinger operators $H = -\Delta + V(x)$ where the potential $V$ is periodic with respect to a lattice $\Lambda$ and has additional symmetries. We first focus on the impact of these symmetries on the multiplicity of Floquet-Bloch eigenvalues using results from lattice and perturbation theory. Our main result then states that the multiplicities of the eigenvalues of $H_z = -\Delta+ zV$ are constant in $z$ on an open neighborhood of $\mathbb{R}$ except for a discrete set.