Ayman Kachmar

On the ground state energy for a magnetic Schrödinger operator and the effect of the DeGennes boundary condition

Motivated by the Ginzburg-Landau theory of superconductivity, we estimate in the semiclassical limit the ground state energy of a magnetic Schrodinger operator with De Gennes boundary condition and we study the localization of the ground states. We exhibit cases when the De Gennes boundary condition has strong effects on this localization.

The Ground State Energy of the Three Dimensional Ginzburg-Landau Functional Part I: Bulk Regime

We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the magnetic field is comparable with the second critical field and that the Ginzburg-Landau parameter is large, we determine a precise asymptotic formula for the minimizing energy. In particular, this shows how bulk superconductivity decreases in average as the applied magnetic field approaches the second critical field from below. Other estimates are also obtained which allow us to obtain, in a subsequent paper [19], a fine characterization of the second critical field. The approach relies on a careful analysis of several limiting energies, which is of independent interest.

Tunneling for the Robin Laplacian in smooth planar domains

We study the low-lying eigenvalues of the semiclassical Robin Laplacian in a smooth planar domain symmetric with respect to an axis. In the case when the curvature of the boundary of the domain attains its maximum at exactly two points away from the axis of symmetry, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues. This is a rigorous derivation of the semiclassical tunneling effect induced by the domains geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

The Ginzburg–Landau Functional with Vanishing Magnetic Field

We study the infimum of the Ginzburg–Landau functional in a two dimensional simply connected domain and with an external magnetic field allowed to vanish along a smooth curve. We obtain energy asymptotics which are valid when the Ginzburg–Landau parameter is large and the strength of the external field is below the third critical field. Compared with the known results when the external magnetic field does not vanish, we show in this regime a concentration of the energy near the zero set of the external magnetic field. Our results complete former results obtained by K. Attar and X.B. Pan–K.H. Kwek.

The Ginzburg--Landau Order Parameter near the Second Critical Field

In the Ginzburg--Landau theory of superconductivity, the density and location of the superconducting electrons are measured by a complex-valued wave function, the order parameter. In this paper, when the intensity of the applied magnetic field is close to the second critical field, and when the order parameter minimizes the Ginzburg--Landau functional defined over a two-dimensional domain, the leading order approximation of its

On the energy of bound states for magnetic Schrödinger operators

L^2

Clusters of eigenvalues for the magnetic Laplacian with Robin condition

-norm in “small” squares is given as the Ginzburg--Landau parameter tends to infinity.

A New Formula for the Energy of Bulk Superconductivity

We provide a leading order semiclassical asymptotics of the energy of bound states for magnetic Neumann Schr¨odinger operators in two-dimensional (exterior) domains with smooth boundaries. The asymptotics is valid all the way up to the bottom of the essential spectrum. When the spectral parameter is varied near the value where bound states become allowed in the interior of the domain, we show that the energy has a boundary and a bulk component. The estimates rely on coherent states, in particular on the construction of �boundary coherent states�, and magnetic Lieb�Thirring estimates

Spectral asymptotics for magnetic Schrödinger operators in domains with corners

We study the Schrodinger operator with a constant magnetic field in the exterior of a compact domain in Euclidean space. Functions in the domain of the operator are subject to a boundary condition of the third type (a magnetic Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which is independent of the boundary condition.

STRENGTH OF SUPERCONDUCTIVITY CLOSE TO CRITICAL MAGNETIC FIELD

The energy of a type II superconductor submitted to an external magnetic field of intensity close to the second critical field is given by the celebrated Abrikosov energy. If the external magnetic field is comparable to and below the second critical field, the energy is given by a reference function obtained as a special (thermodynamic) limit of a non-linear energy. In this note, we give a new formula for this reference energy. In particular, we obtain it as a special limit of a {\it linear} energy defined over configurations normalized in the