Counting function of the embedded eigenvalues for some manifold with cusps, and magnetic Laplacian

We consider a non compact, complete manifold {\bf{M}} of finite area with cuspidal ends. The generic cusp is isomorphic to ${\bf{X}}\times ]1,+\infty [$ with metric $ds^2=(h+dy^2)/y^{2\delta}.$ {\bf{X}} is a compact manifold with nonzero first Betti number equipped with the metric $h.$ For a one-form $A$ on {\bf{M}} such that in each cusp $A$ is a non exact one-form on the boundary at infinity, we prove that the magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ satisfies the Weyl asymptotic formula with sharp remainder. We deduce an upper bound for the counting function of the embedded eigenvalues of the Laplace-Beltrami operator $-\Delta =-\Delta_0.

Semiclassical analysis for a Schr\"odinger operator with a U(2) artificial gauge: the periodic case

We consider a Schr\"odinger operator with a Hermitian 2x2 matrix-valued potential which is lattice periodic and can be diagonalized smoothly on the whole $R^n.$ In the case of potential taking its minimum only on the lattice, we prove that the well-known semiclassical asymptotic of first band spectrum for a scalar potential remains valid for our model

Accuracy on eigenvalues for a Schrodinger operator with a degenerate potential in the semi-classical limit

We consider a semi-classical Schrodinger operator with a degenerate potential V(x,y) =f(x) g(y) . g is assumed to be a homogeneous positive function of m variables and f is a strictly positive function of n variables, with a strict minimum. We give sharp asymptotic behaviour of low eigenvalues bounded by some power of the parameter h, by improving Born-Oppenheimer approximation

Semiclassical analysis for a Schrödinger operator with a U(2) artificial gauge: the periodic case

International audienceWe consider a Schrödinger operator with a Hermitian 2x2 matrix-valued potential which is lattice periodic and can be diagonalized smoothly on the whole $R^n.$ In the case of potential taking its minimum only on the lattice, we prove that the well-known semiclassical asymptotic of first band spectrum for a scalar potential remains valid for our model

Magnetic bottles on geometrically finite hyperbolic surfaces

We consider a magnetic Laplacian on a geometrically finite hyperbolic surface, when the corresponding magnetic field is infinite at the boundary at infinity. We prove that the counting function of the eigenvalues has a particular asymptotic behaviour when the surface has an infinite area

Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface \mM with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0.

Magnetic Bottles in Connection With Superconductivity

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan and Del Pino-Felmer-Sternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by Bernoff-Sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields