This paper proposes a fairly general new point of view on the question of asymptotic stability of (topological) solitons. Our approach is based on the use of the distorted Fourier transform at the nonlinear level; it does not rely only on Strichartz or virial estimates and is therefore able to treat low-power nonlinearities (hence also nonlocalised solitons) and capture the global (in space and time) behaviour of solutions.

More specifically, we consider quadratic nonlinear Klein-Gordon equations with a regular and decaying potential in one space dimension. Additional assumptions are made so that the distorted Fourier transform of the solution vanishes at zero frequency. Assuming also that the associated Schrödinger operator has no negative eigenvalues, we obtain global-in-time bounds, including sharp pointwise decay and modified asymptotics, for small solutions.

These results have some direct applications to the asymptotic stability of (topological) solitons, as well as several other potential applications to a variety of related problems. For instance, we obtain full asymptotic stability of kinks with respect to odd perturbations for the double sine-Gordon problem (in an appropriate range of the deformation parameter). For the \phi ^4$\phi ^4$ problem, we obtain asymptotic stability for small odd solutions, provided the nonlinearity is projected on the continuous spectrum. Our results also go beyond these examples since our framework allows for the presence of a fully coherent phenomenon (a space-time resonance) at the level of quadratic interactions, which creates a degeneracy in distorted Fourier space. We devise a suitable framework that incorporates this and use multilinear harmonic analysis in the distorted setting to control all nonlinear interactions.

We show that there is nonuniqueness for the Calderón problem with partial data for Riemannian metrics with Hölder continuous coefficients in dimension greater than or equal to three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only Hölder continuous of order \unicode[STIX]{x1D70C}<1$\unicode[STIX]{x1D70C}<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring (M,g)$(M,g)$ and we show that there exist in the conformal class of g$g$ an infinite number of Riemannian metrics \tilde{g}=c^{4}g$\tilde{g}=c^{4}g$ such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric g$g$ and do not satisfy the unique continuation principle.

We obtain generalizations of the classical Menchov–Rademacher theorem to the case of continuous orthogonal systems. These results are applied to show the existence of Moller wave operators in Schrödinger evolution.

This paper is concerned with the invisibility cloaking in acoustic wave scattering from a new perspective. We are especially interested in achieving the invisibility cloaking by completely regular and isotropic mediums. It is shown that an interior transmission eigenvalue problem arises in our study, which is the one considered theoretically in Cakoni et al. (Transmission eigenvalues for inhomogeneous media containing obstacles, Inverse Problems and Imaging, 6 (2012), 373–398). Based on such an observation, we propose a cloaking scheme that takes a three-layer structure including a cloaked region, a lossy layer and a cloaking shell. The target medium in the cloaked region can be arbitrary but regular, whereas the mediums in the lossy layer and the cloaking shell are both regular and isotropic. We establish that if a certain non-transparency condition is satisfied, then there exists an infinite set of incident waves such that the cloaking device is nearly invisible under the corresponding wave interrogation. The set of waves is generated from the Herglotz approximation of the associated interior transmission eigenfunctions. We provide both theoretical and numerical justifications.

We consider the obstacle problem for the Schrödinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet Schrödinger propagator {{e}^{it{{\Delta }_{D}}}}${{e}^{it{{\Delta }_{D}}}}$ and give a robust algorithm to prove sharp {{L}^{1}}\,\to \,{{L}^{\infty }}${{L}^{1}}\,\to \,{{L}^{\infty }}$ dispersive estimates. We showcase the analysis in dimensions n\,=\,5,\,7$n\,=\,5,\,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical \text{NLS}$\text{NLS}$ on \Omega \,=\,{{\mathbb{R}}^{n}}\backslash \overline{B\left( 0,\,1 \right)}$\Omega \,=\,{{\mathbb{R}}^{n}}\backslash \overline{B\left( 0,\,1 \right)}$ with Dirichlet boundary condition and radial data in these dimensions.

We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

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

We show improved local energy decay for the wave equation on asymptotically Euclidean manifolds in odd dimensions in the short range case. The precise decay rate depends on the decay of the metric towards the Euclidean metric. We also give estimates of powers of the resolvent of the wave propagator between weighted spaces.

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

We derive a perfectly matched layer (PML) for the Schrödinger equation using a modal ansatz. We derive approximate error formulas for the modeling error from the outer boundary of the PML and the error from the discretization in the layer and show how to choose layer parameters so that these errors are matched and optimal performance of the PML is obtained. Numerical computations in 1D and 2D demonstrate that the optimized PML works efficiently at a prescribed accuracy for the zero potential case, with a layer of width less than a third of the de Broglie wavelength corresponding to the dominating frequency.

We obtain solvability conditions for some elliptic equations involving non-Fredholm operators with the methods of spectral theory and scattering theory for Schrödinger-type operators. One of the main results of the paper concerns solvability conditions for the equation –Δu + V(x)u–au = f where a ≥ 0. The conditions are formulated in terms of orthogonality of the function f to the solutions of the homogeneous adjoint equation.

That vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the A–B (Aharonov–Bohm) effect. We study scattering by two solenoidal magnetic fields (point-like magnetic fields) in two dimensions and analyze the asymptotic behavior of the scattering amplitude in the semiclassical limit. The corresponding classical mechanical system has a trajectory oscillating between the centers of the two fields. We derive the asymptotic formula with the first three terms and make clear how such a trapping trajectory is reflected in the asymptotic formula through the A–B effect. We also make a brief comment on an extension to the scattering by many solenoidal fields. The result depends on the location of the centers of the fields. In particular, the A–B effect appears strongly when the centers are on an even line.

Nous étudions les résonances de Rayleigh créées par une boule en dimension deux et trois. Nous savons qu’elles convergent exponentiellement vite vers l’axe réel. Nous calculons exactement les fonctions résonantes associées puis nous les estimons asymptotiquement en fonction de la partie réelle des résonances. L’application de la formule de Green nous donne alors le taux de décroissance exponentielle de la partie imaginaire des résonances.

We apply the method of complex scaling to give a natural proof of a formula relating the multiplicity of a resonance to the multiplicity of a pole of the scattering matrix.

We study the semi-classical behavior as h\,\to \,0$h\,\to \,0$ of the scattering amplitude f(\theta ,\,\omega ,\,\lambda ,\,h)$f(\theta ,\,\omega ,\,\lambda ,\,h)$ associated to a Schrödinger operator P(h)\,=\,-\,\frac{1}{2}{{h}^{2}}\Delta \,+\,V\,(x)$P(h)\,=\,-\,\frac{1}{2}{{h}^{2}}\Delta \,+\,V\,(x)$ 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)$V(x)$ in a domain lying behind the barrier \left\{ x\,:\,V(x)\,>\,\lambda \right\}$\left\{ x\,:\,V(x)\,>\,\lambda \right\}$, the scattering amplitude f(\theta ,\,\omega ,\,\lambda ,\,h)$f(\theta ,\,\omega ,\,\lambda ,\,h)$ changes by a term of order \mathcal{O}({{h}^{\infty }})$\mathcal{O}({{h}^{\infty }})$. Under an escape assumption on the classical trajectories incoming with fixed direction \omega $\omega$, we obtain an asymptotic development of f(\theta ,\,\omega ,\,\lambda ,\,h)$f(\theta ,\,\omega ,\,\lambda ,\,h)$ similar to the one established in the non-trapping case.

The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrödinger operators with short and long range potentials are computed. A kernel expansion for the Schrödinger semigroup is derived, and a connection with non-commutative Taylor formulas is established.

The growth properties at infinity for eigenfunctions corresponding to embedded eigenvalues of the Neumann Laplacian on horn-like domains are studied. For domains that pinch at polynomial rate, it is shown that the eigenfunctions vanish at infinity faster than the reciprocal of any polynomial. For a class of domains that pinch at an exponential rate, weaker, {{L}^{2}}${{L}^{2}}$ bounds are proven. A corollary is that eigenvalues can accumulate only at zero or infinity.

Errors to a previous paper (Canad. J. Math. (2) 49(1997), 232–262) are corrected. A non-standard regularisation of the auxiliary operator A$A$ 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 ∞. The proof uses Mourre theory.