Superconductivity in domains with corners

We study the two-dimensional Ginzburg-Landau functional in a domain with corners for exterior magnetic field strengths near the critical field where the transition from the superconducting to the normal state occurs. We discuss and clarify the definition of this field and obtain a complete asymptotic expansion for it in the large $\kappa$ regime. Furthermore, we discuss nucleation of superconductivity at the boundary

On spectral minimal partitions II, the case of the rectangle

In continuation of \cite{HHOT}, we discuss the question of spectral minimal 3-partitions for the rectangle $]-\frac a2,\frac a2[\times ] -\frac b2,\frac b2[$, with $0< a\leq b$. It has been observed in \cite{HHOT} that when $0<\frac ab < \sqrt{\frac 38}$ the minimal 3-partition is obtained by the three nodal domains of the third eigenfunction corresponding to the three rectangles $]-\frac a2,\frac a2[\times ] -\frac b2,-\frac b6[$, $]-\frac a2,\frac a2[\times ] -\frac b6,\frac b6[$ and $]-\frac a2,\frac a2[\times ] \frac b6, \frac b2[$. We will describe a possible mechanism of transition for increasing $\frac ab$ between these nodal minimal 3-partitions and non nodal minimal 3-partitions at the value $\sqrt{\frac 38}$ and discuss the existence of symmetric candidates for giving minimal 3-partitions when $\sqrt{\frac 38}<\frac ab \leq 1$. Numerical analysis leads very naturally to nice questions of isospectrality which are solved by introducing Aharonov-Bohm Hamiltonians or by going on the double covering of the punctured rectangle

Efficient numerical calculation of drift and diffusion coefficients in the diffusion approximation of kinetic equations

In this paper we study the diffusion approximation of a swarming model given by a system of interacting Langevin equations with nonlinear friction. The diffusion approximation requires the calculation of the drift and diffusion coefficients that are given as averages of solutions to appropriate Poisson equations. We present a new numerical method for computing these coefficients that is based on the calculation of the eigenvalues and eigenfunctions of a Schr\"odinger operator. These theoretical results are supported by numerical simulations showcasing the efficiency of the method

Far from equilibrium steady states of 1D-Schrödinger-Poisson systems with quantum wells I

We describe the asymptotic of the steady states of the out-of equilibrium Schrödinger-Poisson system, in the regime of quantum wells in a semiclassical island. After establishing uniform estimates on the nonlinearity, we show that the nonlinear steady states lie asymptotically in a finite-dimensional subspace of functions and that the involved spectral quantities are reduced to a finite number of so-called asymptotic resonant energies. The asymptotic finite dimensional nonlinear system is written in a general setting with only a partial information on its coefficients. After this first part, a complete derivation of the asymptotic nonlinear system will be done for some specific cases in a forthcoming article. UNE VERSION MODIFIEE DE CE TEXTE EST PARUE DANS LES ANNALES DE L'INSTITUT H. POINCARE, ANALYSE NON LINEAIRE

The Schr\"odinger operator on an infinite wedge with a tangent magnetic field

We study a model Schr\"odinger operator with constant magnetic field on an infinite wedge with Neumann boundary condition. The magnetic field is assumed to be tangent to a face. We compare the bottom of the spectrum to the model spectral quantities coming from the regular case. We are particularly motivated by the influence of the magnetic field and the opening angle of the wedge on the spectrum of the model operator and we exhibit cases where the bottom of the spectrum is smaller than in the regular case. Numerical computations enlighten the theoretical approach

Weyl asymptotics for magnetic Schr\"odinger operators and de Gennes' boundary condition

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schr\"odinger operator with constant magnetic field and associated with the de Gennes (Fourier/Robin) boundary condition. We derive an asymptotic expansion of the number of eigenvalues below the essential spectrum (Weyl-type asymptotics). The methods of proof relies on results concerning the asymptotic behavior of the first eigenvalue obtained in a previous work [A. Kachmar, J. Math. Phys. Vol. 47 (7) 072106 (2006)].Comment: 28 pages (revised version). to appear in Rev Math Phy

Simulation of resonant tunneling heterostructures: numerical comparison of a complete Schr{ö}dinger-Poisson system and a reduced nonlinear model

Two different models are compared for the simulation of the transverse electronic transport through an heterostructure: a $1D$ self-consistent Schr{ö}dinger-Poisson model with a numerically heavy treatment of resonant states and a reduced model derived from an accurate asymptotic nonlinear analysis. After checking the agreement at the qualitative and quantitative level on quite well understood bifurcation diagrams, the reduced model is used to tune double well configurations for which nonlinearly interacting resonant states actually occur in the complete self-consistent model

Hybridization and interference effects for localized superconducting states in strong magnetic field

Within the Ginzburg-Landau model we study the critical field and temperature enhancement for crossing superconducting channels formed either along the sample edges or domain walls in thin-film magnetically coupled superconducting - ferromagnetic bilayers. The corresponding Cooper pair wave function can be viewed as a hybridization of two order parameter (OP) modes propagating along the boundaries and/or domain walls. Different momenta of hybridized OP modes result in the formation of vortex chains outgoing from the crossing point of these channels. Near this crossing point the wave functions of the modes merge giving rise to the increase in the critical temperature for a localized superconducting state. The origin of this critical temperature enhancement caused by the wave function squeezing is illustrated for a limiting case of approaching parallel boundaries and/or domain walls. Using both the variational method and numerical simulations we have studied the critical temperature dependence and OP structure vs the applied magnetic field and the angle between the crossing channels.Comment: 12 pages, 13 figure

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, $H_{C_3}$, describing the appearance of superconductivity in superconductors of type II. Furthermore, we prove that the local and global definitions of this field coincide. Near $H_{C_3}$ 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