arXiv subjects
Publications and source records attributed to Ayman Kachmar.
The existence of bound states for the magnetic Laplacian in unbounded domains can be quite challenging in the case of a homogeneous magnetic field. We provide an affirmative answer for almost flat corners and slightly curved half-planes when the total curvature of the boundary is positive.
The semiclassical Laplacian with discontinuous magnetic field is considered in two dimensions. The magnetic field is sign changing with exactly two distinct values and is discontinuous along a smooth closed curve, thereby producing an attractive magnetic edge. Various accurate spectral asymptotics are established by means of a dimensional reduction involving a microlocal phase space localization allowing to deal with the discontinuity of the field.
We consider the magnetic Robin Laplacian with a negative boundary parameter. Among a certain class of domains, we prove that the disk maximizes the ground state energy under the fixed perimeter constraint provided that the magnetic field is of moderate strength. This class of domains includes, in particular, all domains that are contained upon translations in the disk of the same perimeter and all convex centrally symmetric domains.
We study the 3D Neuman magnetic Laplacian in the presence of a semi-classical parameter and a non-uniform magnetic field with constant intensity. We determine a sharp two term asymptotics for the lowest eigenvalue, where the second term involves a quantity related to the magnetic field and the geometry of the domain. In the special case of the unit ball and a helical magnetic field, the concentration takes place on two symmetric points of the unit sphere.
We establish accurate eigenvalue asymptotics and, as a by-product, sharp estimates of the splitting between two consecutive eigenvalues, for the Dirichlet magnetic Laplacian with a non-uniform magnetic field having a jump discontinuity along a smooth curve. The asymptotics hold in the semiclassical limit which also corresponds to a large magnetic field limit, and is valid under a geometric assumption on the curvature of the discontinuity curve.
Experimentally observed magnetic fields with nanoscale variations are theoretically modeled by a piece-wise constant function with jump discontinuity along a smooth curve, the magnetic edge. Assuming the edge is a closed curve with an axis of symmetry and the field is sign changing and with exactly two distinct values, we prove that semi-classical tunneling occurs and calculate the magnitude of this tunneling effect.
We estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the non-linear Ginzburg-Landau energy, and as a byproduct, yields a non-Gaussian trial state for the Dirichlet magnetic Laplacian.
We study the Schrödinger operator in the plane with a step magnetic field function. The bottom of its spectrum is described by the infimum of the lowest eigenvalue band function, for which we establish the existence and uniqueness of the non-degenerate minimum. We discuss the curvature effects on the localization properties of magnetic ground states, among other applications.
We consider the Aharonov-Bohm magnetic potential and study the transition from normal to superconducting solutions within the Ginzburg-Landau model of superconductivity. We obtain oscillations consistent with the Little-Parks effect. We study the same problem but for a regularization of the Aharonov-Bohm potential, which leads to an interesting Aharonov-Bohm like magnetic field, and we prove that the transition between superconducting and normal solutions is not monotone too. Our results show a mechanism to derive the Aharonov-Bohm magnetic potential starting from a step magnetic field thereby presenting a new aspect of magnetic steps, besides their favoring of the celebrated edge states.
We determine accurate asymptotics of the lowest eigenvalue for the Laplace operator with a smooth magnetic field and Robin boundary conditions in a smooth 3D domain, when the Robin parameter tends to $+\infty$. Our results identify a critical regime where the contribution of the magnetic field and the Robin condition are of the same order. In this critical regime, we derive an effective operator defined on the boundary of the domain.
We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty$.
We study the magnetic Laplacian and the Ginzburg-Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some non-linear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little-Parks experiment, for a thin cylindrical sample.
In the asymptotic limit of a large Ginzburg-Landau parameter, we give a new asymptotic formula for the $L^2$-norm of the Ginzburg-Landau order parameter. The formula is valid in the bulk regime where the intensity of the applied magnetic field is of the same order as the Ginzburg-Landau parameter and strictly below the second critical field. Our formula complements the celebrated one of Sandier-Serfaty for the $L^4$-norm.
On a compact Riemannian manifold $M$ with boundary, we give an estimate for the eigenvalues $(λ\_k(τ,α))\_k$ of the magnetic Laplacian with the Robin boundary conditions. Here, $τ$ is a positive number that defines the Robin condition and $α$ is a real differential 1-form on $M$ that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter $τ$ and a lower bound of the Ricci curvature of $M$ (see Theorem \ref{estimate1} and Corollary \ref{corestimate}). The main technique is to use the Bochner formula established in \cite{ELMP} for the magnetic Laplacian and to integrate it over $M$ (see Theorem \ref{bochnermagnetic1}). In the last part, we compare the eigenvalues $λ\_k(τ,α)$ with the first eigenvalue $λ\_1(τ)=λ\_1(τ,0)$ (i.e. without magnetic field) and the Neumann eigenvalues $λ\_k(0,α)$ (see Theorem \ref{thm:comp}) using the min-max principle.
We consider the Ginzburg-Landau functional, defined on a two-dimensional simply connected domain with smooth boundary, in the situation when the applied magnetic field is piecewise constant with a jump discontinuity along a smooth curve. In the regime of large Ginzburg-Landau parameter and strong magnetic field, we study the concentration of minimizing configurations along this discontinuity.
We compute the $L^2$-norm of the 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.
The energy of a type II superconductor placed in a strong non-uniform, smooth and signed magnetic field is displayed via a universal characteristic function defined by means of a simplified two dimensional Ginzburg-Landau functional. We study the asymptotic behavior of this functional in a specific asymptotic regime, thereby linking it to a one dimensional functional, using methods developed by Almog-Helffer and Fournais-Helffer devoted to the analysis of surface superconductivity in the presence of a uniform magnetic field. As a result, we obtain an asymptotic formula reminiscent of the one for the surface superconductivity regime, where the zero set of the magnetic field plays the role of the superconductor's surface.
In 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 $L^2$-norm in `small' squares is given as the Ginzburg-Landau parameter tends to infinity.