Spectral properties of hill-schrödinger operators with special distribution potentials

Let L be the Hill-Schrodinger operator considered with a singular complex-valued potential v of the form v = Q' where Q E L2oc(R) is ∏-periodic. Then for large enough n, there is a disc of radius n/4 around n2 which contains two eigenvalues A± of L considered on [∏,n] with periodic (for even n) or antiperiodic (for odd n) boundary conditions. In this thesis we consider Hill-Schrodinger operators with specific n-periodic potentials v of the form v = Q' where Q is n-periodic with Q(x) = ax + b on [∏, n). We provide asymptotics for the spectral gaps of L considered with these specific potentials.