  Bardeen-Cooper-Schrieffer (BCS) theory describes a superconducting transition
as a single critical point where the gap function or, equivalently, the order
parameter vanishes uniformly in the entire system. We demonstrate that in
superconductors described by standard BCS models, the superconducting gap
survives near the sample boundaries at higher temperatures than
superconductivity in the bulk. Therefore, conventional superconductors have
multiple critical points associated with separate phase transitions at the
boundary and in the bulk. We show this by revising the Caroli-De
Gennes-Matricon theory of a superconductor-vacuum boundary and finding
inhomogeneous solutions of the BCS gap equation near the boundary, which
asymptotically decay in the bulk. This is demonstrated for a BCS model of
almost free fermions and for lattice fermions in a tight-binding approximation.
The analytical results are confirmed by numerical solutions of the microscopic
model. The existence of these boundary states can manifest itself as
discrepancies between the critical temperatures observed in calorimetry and
transport probes.
