Vortex Lattices and the Bogoliubov-De Gennes Equations.

We consider the Bogoliubov-de Gennes equations giving an equivalent formulation of the BCS theory of superconductivity. We are interested in static solutions with the magnetic field present. We carefully formulate the equations in the basis independent form, discuss their general features and isolate key physical classes of solutions (normal and vortex lattice states) which are the candidates for the ground state. We prove existence of the normal and vortex lattice states and stability of the normal states for large temperature or magnetic fields and their instability for small temperature and small magnetic fields.