Thin Spectrum States in Superconductors

We show that finite size superconductors have a spectrum of states at extremely low energy, i.e. inside the superconducting gap. The presence of this {\it thin spectrum} is a generic feature and related to the fact that in a superconductor the global phase rotation symmetry is spontaneously broken. For a strong coupling superconductor we find the spectrum by exactly solving the Lieb-Mattis type Hamiltonian onto which the problem maps. One of the physical consequences of the presence of thin states is that they cause quantum decoherence in superconducting qubits of finite extent.