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Ayman Kachmar

Publications and source records attributed to Ayman Kachmar.

60 records · Page 4Linked to original sources
arXiv2008

Weyl asymptotics for magnetic Schrödinger operators and de Gennes' boundary condition

This paper is concerned with the discrete spectrum of the self-adjoint realization of the semi-classical Schrödinger 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)].

math.SP↗
arXiv2008

Limiting jump conditions for Josephson junctions in Ginzburg-Landau theory

We consider a superconducting/normal/superconducting Josephson junction modeled through the Ginzburg-Landau theory. When the normal material is sufficiently thin and the applied magnetic field is below the critical field of vortex nucleation, we prove to leading order that jump boundary conditions of the type predicted by de Gennes are satisfied across the junction.

math.AP↗
arXiv2007

Magnetic vortices for a Ginzburg-Landau type energy with discontinuous constraint

This paper is devoted to an analysis of vortex-nucleation for a Ginzburg-Landau functional with discontinuous constraint. This functional has been proposed as a model for vortex-pinning, and usually accounts for the energy resulting from the interface of two superconductors. The critical applied magnetic field for vortex nucleation is estimated in the London singular limit, and as a by-product, results concerning vortex-pinning and boundary conditions on the interface are obtained.

math.AP↗
arXiv2007

On the stability of normal states for a generalized Ginzburg-Landau model

We formulate a spectral problem related to the onset of superconductivity for a generalized Ginzburg-Landau model, where the order parameter and the magnetic potential are defined in the whole space. This model is devoted to the `proximity effect' for a superconducting sample surrounded by a normal material. In the regime when the Ginzburg-Landau parameter (of the superconducting material) is large, we estimate the critical applied magnetic field for which the normal state will lose its stability, a result that has some roots in the physical literature. In some asymptotic situations, we recover results related to the `standard' Ginzburg-Landau model, where we mention in particular the two-term expansion for the upper critical field obtained by Helffer-Pan.

math-ph↗
arXiv2007

On the perfect superconducting solution for a generalized Ginzburg-Landau equation

We study a generalized Ginzburg-Landau equation that models a sample formed of a superconducting/normal junction and which is not submitted to an applied magnetic field. We prove the existence of a unique positive (and bounded) solution of this equation. In the particular case when the domain is the entire plane, we determine the explicit expression of the solution (and we find that it satisfies a Robin (de Gennes) boundary condition on the boundary of the superconducting side). Using the result of the entire plane, we determine for the case of general domains, the asymptotic behavior of the solution for large values of the Ginzburg-Landau parameter. The main tools are Hopf's Lemma, the Strong Maximum Principle, elliptic estimates and Agmon type estimates.

math-ph↗