Since the discovery of type II superconductivity by Abrikosov, the distribution of superconductivity near the second critical field is a challenging mathematical problem. In this talk, I will discuss the asymptotic behavior of the Abrikosov energy in a large domain, and the connection between the Abrikosov and Ginzburg-Landau energies. In this discussion, a key role is played by the spectrum of the Schrödinger operator with a constant magnetic field in a lattice. As a matter of application, I will write a new formula for the distribution of superconductivity near the second critical field. I will conclude the talk with a hierarchy of energies: the Abrikosov energy, the bulk energy of superconductivity, and the reference energy for a superconducting surface. Part of these results are obtained jointly with S. Fournais and with B. Helffer.
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