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Ayman Kachmar

Publications and source records attributed to Ayman Kachmar.

At least 55 records · Page 3Linked to original sources
arXiv2016

Decay of superconductivity away from the magnetic zero set

We establish exponential bounds on the Ginzburg-Landau order parameter away from the curve where the applied magnetic field vanishes. In the units used in this paper, the estimates are valid when the parameter measuring the strength of the applied magnetic field is comparable with the Ginzburg-Landau parameter. This completes a previous work by the authors analyzing the case when this strength was much higher. Our results display the distribution of surface and bulk superconductivity and are valid under the assumption that the magnetic field is Hölder continuous.

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arXiv2016

Existence of Surface Smectic States of Liquid Crystals

The Landau-de Gennes model of liquid crystals is a functional acting on wave functions (order parameters) and vector fields (director fields). In a specific asymptotic limit of the physical parameters, we construct critical points such that the wave function (order parameter) is localized near the boundary of the domain, and we determine a sharp localization of the boundary region where the wave function concentrates. Furthermore, we compute the asymptotics of the energy of such critical points along with a boundary energy that may serve in localizing the director field. In physical terms, our results prove the existence of a surface smectic state.

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arXiv2016

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 domain's geometry. Our approach is close to the Born-Oppenheimer one and yields, as a byproduct, a Weyl formula of independent interest.

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arXiv2015

The influence of magnetic steps on bulk superconductivity

We study the distribution of bulk superconductivity in presence of an applied magnetic field, supposed to be a step function, modeled by the Ginzburg-Landau theory. Our results are valid for the minimizers of the two-dimensional Ginzburg-Landau functional with a large Ginzburg-Landau parameter and with an applied magnetic field of intensity comparable with the Ginzburg-Landau parameter.

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arXiv2015

The distribution of 3D superconductivity near the second critical field

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.

math.AP↗
arXiv2015

A new formula for the energy of bulk superconductivity

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 $L^4$-norm.

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arXiv2015

Eigenvalues for the Robin Laplacian in domains with variable curvature

We determine accurate asymptotics for the low-lying eigenvalues of the Robin Laplacian when the Robin parameter goes to $-\infty$. The two first terms in the expansion have been obtained by K. Pankrashkin in the $2D$-case and by K. Pankrashkin and N. Popoff in higher dimensions. The asymptotics display the influence of the scalar curvature and the splitting between every two consecutive eigenvalues. The analysis is based on the approach developed by Fournais-Helffer for the semi-classical magnetic Laplacian. We also propose a WKB construction as a candidate for the ground state energy.

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arXiv2015

From constant to non-degenerately vanishing magnetic fields in superconductivity

We explore the relationship between two reference functions arising in the analysis of the Ginzburg-Landau functional. The first function describes the distribution of superconductivity in a type II superconductor subjected to a constant magnetic field. The second function describes the distribution of superconductivity in a type II superconductor submitted to a variable magnetic field that vanishes non-degenerately along a smooth curve.

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arXiv2014

Semi-classical trace asymptotics for magnetic Schrodinger operators with Robin condition

We compute the sum and number of eigenvalues for a certain class of magnetic Schrodinger operators in a domain with boundary. Functions in the domain of the operator satisfy a (magnetic) Robin condition. The calculations are valid in the semi-classical asymptotic limit and the eigenvalues concerned correspond to eigenstates localized near the boundary of the domain. The formulas we derive display the influence of the boundary and the boundary condition and are valid under a weak regularity assumption of the boundary function. Our approach relies on three main points: reduction to the boundary; construction of boundary coherent states; handling the boundary term as a surface electric potential and controlling the errors by various Lieb-Thirring inequalities.

math.AP↗
arXiv2014

Energy of a rotating Bose-Einstein condensate in a harmonic trap

The state of a rotating Bose-Einstein condensate in a harmonic trap is modeled by a wave function that minimizes the Gross-Pitaevskii functional. The resulting minimization problem has two new features compared to other similar functionals arising in condensed matter physics, such as the Ginzburg-Landau functional. Namely, the wave function is defined in all the plane and is normalized relative to the $L^2$-norm. This paper deals with the situation when the coupling constant tends to $0$ (Thomas-Fermi regime) and the rotation speed is large compared with the first critical speed. It is given the leading order estimate of the ground state energy together with the location of the vortices of the minimizing wave function in the bulk of the condensate. When the rotation speed is inversely proportional to the coupling constant, the condensate is confined in an elliptical region whose conjugate diameter shrinks and whose transverse diameter expands as the rotation speed increases.

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arXiv2014

The Ginzburg-Landau functional with vanishing magnetic field

We study the infimum of the Ginzburg-Landau functional in the case of a vanishing external magnetic field in a two dimensional simply connected domain. We obtain an energy asymptotics which is valid when the Ginzburg-Landau parameter is large and the strength of the external field is comparable with 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.

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arXiv2014

Clusters of eigenvalues for the magnetic Laplacian with Robin condition

We study the Schrödinger 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.

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arXiv2010

The ground state energy of the three dimensional Ginzburg-Landau model in the mixed phase

We consider the Ginzburg-Landau functional defined over a bounded and smooth three dimensional domain. Supposing that the strength of the applied magnetic field varies between the first and second critical fields, in such a way that $H_{C_1}\ll H\ll H_{C_2}$, we estimate the ground state energy to leading order as the Ginzburg-Landau parameter tends to infinity.

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arXiv2009

A non-existence result for the Ginzburg-Landau equations

We consider the stationary Ginzburg-Landau equations in $\R^d$, $d=2,3$. We exhibit a class of applied magnetic fields (including constant fields) such that the Ginzburg-Landau equations do not admit finite energy solutions.

math.AP↗
arXiv2009

Remark on magnetic Schroedinger operators in exterior domains

We study the Schroedinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (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 appears to be independent from the boundary condition.

math.SP↗