Nous considérons les perturbations $H := H_{0} + V$ et $D := D_{0} + V$ des Hamiltoniens libres $H_{0}$ de Pauli et $D_{0}$ de Dirac en dimension 3 avec champ magnétique non constant, $V$ étant un potentiel électrique qui décroît super-exponentiellement dans la direction du champ magnétique. Nous montrons que dans des espaces de Banach appropriés, les résolvantes de $H$ et $D$ définies sur le demi-plan supérieur admettent des prolongements méromorphes. Nous définissons les résonances de $H$ et $D$ comme étant les pôles de ces extensions méromorphes. D'une part, nous étudions la répartition des résonances de $H$ près de l'origine $0$ et d'autre part, celle des résonances de $D$ près de $\pm m$ où $m$ est la masse d'une particule. Dans les deux cas, nous obtenons d'abord des majorations du nombre de résonances dans de petits domaines au voisinage de $0$ et $\pm m$. Sous des hypothèses supplémentaires, nous obtenons des développements asymptotiques du nombre de résonances qui entraînent leur accumulation près des seuils $0$ et $\pm m$. En particulier, pour une perturbation $V$ de signe défini, nous obtenons des informations sur la répartition des valeurs propres de $H$ et $D$ près de $0$ et $\pm m$ respectivement.

We prove a uniform control as $ z \rightarrow 0 $ for the resolvent $ (P-z)^{-1} $ of long range perturbations $ P $ of the Euclidean Laplacian in divergence form by combining positive commutator estimates and properties of Riesz transforms. These estimates hold in dimension $d \geq 3 $ when $ P $ is defined on $ \mathbb{R}^d $ and in dimension $ d \geq 2 $ when $ P $ is defined outside a compact obstacle with Dirichlet boundary conditions.

We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet--Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact $1$-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition.

We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a periodic classical flow. Assuming that the strength $\epsilon$ of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$ (and may sometimes reach even smaller values), we get an asymptotic description of the eigenvalues in rectangles $[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point value of the flow average of the leading perturbation.

Dans cet article on \'etudie la diff\'erence entre les deux premi\`eres valeurs propres, le splitting, d'un op\'erateur de Klein--Gordon semi-classique unidimensionnel, dans le cas d'un potentiel sym\'etrique pr\'esentant un double puits. Dans le cas d'une petite barri\`ere de potentiel, B. Helffer et B. Parisse ont obtenu des r\'esultats analogues \`a ceux existant pour l'op\'erateur de Schr\"odinger. Dans le cas d'une grande barri\`ere de potentiel, on obtient ici des estimations des tranform\'ees de Fourier des fonctions propres qui conduisent \`a une conjecture du splitting. Des calculs num\'eriques viennent appuyer cette conjecture.

We study the semi-classical behavior as $h\rightarrow 0$ of the scattering amplitude $f(\theta,\omega,\lambda,h)$ associated to a Schr\"odinger operator $P(h)=-\frac 1 2 h^2\Delta +V(x)$ with short-range trapping perturbations. First we realize a spatial localization in the general case and we deduce a bound of the scattering amplitude on the real line. Under an additional assumption on the resonances, we show that if we modify the potential $V(x)$ in a domain lying behind the barrier $\{x:V(x)>\lambda\}$, the scattering amplitude $f(\theta,\omega,\lambda,h)$ changes by a term of order $\O(h^{\infty})$. Under an escape assumption on the classical trajectories incoming with fixed direction $\omega$, we obtain an asymptotic development of $f(\theta,\omega,\lambda,h)$ similar to the one established in thenon-trapping case.

We study the resonances of the operator $P(h) = -\Delta_x + V(x) + \varphi(hx)$. Here $V$ is a periodic potential, $\varphi$ a decreasing perturbation and $h$ a small positive constant. We prove the existence of shape resonances near the edges of the spectral bands of $P_0 = -\Delta_x + V(x)$, and we give its asymptotic expansions in powers of $h^{\frac12}$.

Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997), 232--262) are corrected. A non-standard regularisation of the auxiliary operator $A$ appearing in Mourre theory is used.

The spectral theory for the Neumann Laplacian on planar domains with symmetric, horn-like ends is studied. For a large class of such domains, it is proven that the Neumann Laplacian has no singular continuous spectrum, and that the pure point spectrum consists of eigenvalues of finite multiplicity which can accumulate only at $0$ or $\infty$. The proof uses Mourre theory.