Superconductivity in a high magnetic field. Excitation spectrum and de Haas-van-Alphen oscillation

The excitation spectrum of a superconductor placed in the magnetic field high enough so that the inter-Landau level pairing can be ignored is shown to be gapless. The gap turns to zero at the set of points in the magnetic Brillouin zone which are in correspondence to the chosen vortex lattice in the real space. This leads to the algebraic temperature dependences of the thermodynamic potential, heat capacity and the amplitude of the dHvA oscillations at low temperatures.