  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.
