Superconductivity Near the Normal State Under the Action of Electric Currents and Induced Magnetic Fields in {\mathbb{R}^2}

We consider the linearization of the time-dependent Ginzburg-Landau system near the normal state. We assume that an electric current is applied through the sample, which captures the whole plane, inducing thereby, a magnetic field. We show that independently of the current, the normal state is always stable. Using Fourier analysis the detailed behaviour of solutions is obtained as well. Relying on semi-group theory we then obtain the spectral properties of the steady-state elliptic operator.     

Nodal Sets for Groundstates of Schrödinger Operators with Zero Magnetic Field in Non Simply Connected Domains

We investigate nodal sets of magnetic Schr?dinger operators with zero magnetic field, acting on a non simply connected domain in ℝ². For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a characterisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole and a fixed electric potential, we show that the first eigenvalue takes its highest value for circulation 1/2. Received: 23 July 1998 / Accepted: 17 November 1998     

The density of superconductivity in domains with corners

We compute the \(L^2\)-norm of any minimizer of the Ginzburg–Landau functional in a planar domain with a finite number of corners. Our computations are valid for a uniform applied magnetic field, large Ginzburg–Landau parameter and in the regime where superconductivity is confined near the corners of the domain.     

On the semiclassical analysis of the ground state energy of the Dirichlet Pauli operator III: magnetic fields that change sign

We consider the semiclassical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm–Kova?ík–Portmann and Helffer–Sundqvist for the asymptotics of the ground state energy in the semiclassical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semiclassical parameter tends to zero and give lower bounds and upper bounds for this decay rate. Concrete examples of magnetic fields changing sign on the unit disk are discussed. Various natural conjectures are disproved, and this leaves the research of an optimal result in the general case still open.

     

Courant-Sharp Eigenvalues for the Equilateral Torus,and for the Equilateral Triangle

We address the question of determining the eigenvalues \({\lambda_{n}}\) (listed in nondecreasing order, with multiplicities) for which Courant’s nodal domain theorem is sharp i.e., for which there exists an associated eigenfunction with \({n}\) nodal domains (Courant-sharp eigenvalues). Following ideas going back to Pleijel (1956), we prove that the only Courant-sharp eigenvalues of the flat equilateral torus are the first and second, and that the only Courant-sharp Dirichlet eigenvalues of the equilateral triangle are the first, second, and fourth eigenvalues. In the last section we sketch similar results for the right-angled isosceles triangle and for the hemiequilateral triangle.     

Calcul fonctionnel par la transformation de Mellin et opérateurs admissibles

Mellin's transform is used to establish a functional calculus of a class of pseudodifferential-operators depending on a small parameter h > 0. We apply for exeample this result to prove the semi-classical behaviour of the discrete spectrum of Schrödinger operators ?h² · Δ + V, and of Dirac operators h ∑_{j = 1}³α_jD_j + α₄ ? V.     

On Spectral Minimal Partitions: A Survey

Given a bounded open set Ω in \mathbbRⁿ{\mathbb{R}^n} (or a Riemannian manifold) and a partition of Ω by k open sets D _j , we can consider the quantity max _j λ(D _j ) where λ(D _j ) is the groundstate energy of the Dirichlet realization of the Laplacian in D _j . If we denote by \mathfrakL_k(W){\mathfrak{L}_k(\Omega)} the infimum over all the k-partitions of max _j λ(D _j ), a minimal (spectral) k-partition is then a partition which realizes the infimum. Although the analysis is rather standard when k = 2 (we find the nodal domains of a second eigenfunction), the analysis of higher k’s becomes non trivial and quite interesting.     

On laplace integrals and transfer operators in large dimension: Examples in the nonconvex case

This Letter gives detailed proofs concerning the analysis of the pair correlations for a nonconvex model. Using the transfer matrix approach, the problem is reduced to the analysis of the spectral properties of this transfer operator. Although the problem is similar to the semiclassical study of the Kac operator presented in a paper with M. Brunaud, which was devoted to the study of exp-(v/2) exp h ² exp-(v/2) for h small, new features appear for the model exp-(v/2h) exp h exp-(v/2h). Our principal results concern the splitting of this operator between the two largest eigenvalues. We give an upper and a lower bound for this splitting in the semi-classical regime. As a corollary, we get good control of the decay of the pair correlation. Some of the results were announced in a previous paper. Related WKB constructions will be developed in a later paper.Inspired by papers by M. Kac [15, 16].     

On the Third Critical Field in Ginzburg-Landau Theory

Using recent results by the authors on the spectral asymptotics of the Neumann Laplacian with magnetic field, we give precise estimates on the critical field, , describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near only a small part, near the boundary points where the curvature is maximal, of the sample carries superconductivity. We give precise estimates on the size of this zone and decay estimates in both the normal (to the boundary) and parallel variables.The two authors are supported by the European Research Network ‘Postdoctoral Training Program in Mathematical Analysis of Large Quantum Systems’ with contract number HPRN-CT-2002-00277, and the ESF Scientific Programme in Spectral Theory and Partial Differential Equations (SPECT). Part of this work was carried out while S.F. visited CIMAT, Mexico.