The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal reference function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface.

Abstract. We study the minimizers of the Ginzburg-Landau energy functional with a constant magnetic field in a three dimensional bounded domain. The functional depends on two positive parameters, the Ginzburg-Landau parameter and the intensity of the applied magnetic field, and acts on complex valued functions and vector fields. We establish a formula for the distribution of the L 2 -norm of the minimizing complex valued function (order parameter). The formula is valid in the regime where the Ginzburg-Landau parameter is large and the applied magnetic field is close to the second critical field-the threshold value corresponding to the transition from the superconducting to the normal phase in the bulk of the sample. Earlier results are valid in 2D domains and for the L 4 -norm in 3D domains.

Abstract. We establish a semi-classical formula for the sum of eigenvalues of a magnetic Schrödinger operator in a three-dimensional domain with compact smooth boundary and Neumann boundary conditions. The eigenvalues we consider have eigenfunctions localized near the boundary of the domain, hence they correspond to surface states. Using relevant coordinates that straighten out the boundary, the leading order term of the energy is described in terms of the eigenvalues of model operators in the half-axis and the half-plane.