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.