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Search: MSC category 35P ( Spectral theory and eigenvalue problems [See also 47Axx, 47Bxx, 47F05] )

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1. CJM 2014 (vol 66 pp. 1110)

Li, Dong; Xu, Guixiang; Zhang, Xiaoyi
 On the Dispersive Estimate for the Dirichlet SchrÃ¶dinger Propagator and Applications to Energy Critical NLS We consider the obstacle problem for the SchrÃ¶dinger evolution in the exterior of the unit ball with Dirichlet boundary condition. Under the radial symmetry we compute explicitly the fundamental solution for the linear Dirichlet SchrÃ¶dinger propagator $e^{it\Delta_D}$ and give a robust algorithm to prove sharp $L^1 \rightarrow L^{\infty}$ dispersive estimates. We showcase the analysis in dimensions $n=5,7$. As an application, we obtain global well-posedness and scattering for defocusing energy-critical NLS on $\Omega=\mathbb{R}^n\backslash \overline{B(0,1)}$ with Dirichlet boundary condition and radial data in these dimensions. Keywords:Dirichlet SchrÃ¶dinger propagator, dispersive estimate, Dirichlet boundary condition, scattering theory, energy criticalCategories:35P25, 35Q55, 47J35

2. CJM 2013 (vol 65 pp. 1095)

Sambou, Diomba
 RÃ©sonances prÃ¨s de seuils d'opÃ©rateurs magnÃ©tiques de Pauli et de Dirac Nous considÃ©rons 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 magnÃ©tique non constant, $V$ Ã©tant un potentiel Ã©lectrique qui dÃ©croÃ®t super-exponentiellement dans la direction du champ magnÃ©tique. Nous montrons que dans des espaces de Banach appropriÃ©s, les rÃ©solvantes de $H$ et $D$ dÃ©finies sur le demi-plan supÃ©rieur admettent des prolongements mÃ©romorphes. Nous dÃ©finissons les rÃ©sonances de $H$ et $D$ comme Ã©tant les pÃ´les de ces extensions mÃ©romorphes. D'une part, nous Ã©tudions la rÃ©partition des rÃ©sonances de $H$ prÃ¨s de l'origine $0$ et d'autre part, celle des rÃ©sonances de $D$ prÃ¨s de $\pm m$ oÃ¹ $m$ est la masse d'une particule. Dans les deux cas, nous obtenons d'abord des majorations du nombre de rÃ©sonances dans de petits domaines au voisinage de $0$ et $\pm m$. Sous des hypothÃ¨ses supplÃ©mentaires, nous obtenons des dÃ©veloppements asymptotiques du nombre de rÃ©sonances qui entraÃ®nent leur accumulation prÃ¨s des seuils $0$ et $\pm m$. En particulier, pour une perturbation $V$ de signe dÃ©fini, nous obtenons des informations sur la rÃ©partition des valeurs propres de $H$ et $D$ prÃ¨s de $0$ et $\pm m$ respectivement. Keywords:opÃ©rateurs magnÃ©tiques de Pauli et de Dirac, rÃ©sonancesCategories:35B34, 35P25

3. CJM 2011 (vol 63 pp. 961)

Bouclet, Jean-Marc
 Low Frequency Estimates for Long Range Perturbations in Divergence Form 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. Keywords:resolvent estimates, thresholds, scattering theory, Riesz transformCategory:35P25

4. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
 Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps 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. Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identitiesCategories:28A80, , , , 35P20, 35J05, 43A05, 47A75

5. CJM 2010 (vol 62 pp. 808)

Legendre, Eveline
 Extrema of Low Eigenvalues of the Dirichlet-Neumann Laplacian on a Disk 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. Keywords: Laplacian, eigenvalues, Dirichlet-Neumann mixed boundary condition, Zaremba's problemCategories:35J25, 35P15

6. CJM 2009 (vol 61 pp. 548)

Girouard, Alexandre
 Fundamental Tone, Concentration of Density, and Conformal Degeneration on Surfaces 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. Categories:35P, 58J

7. CJM 2008 (vol 60 pp. 572)

Hitrik, Michael; Sj{östrand, Johannes
 Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point 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. Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularityCategories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40

8. CJM 2007 (vol 59 pp. 393)

Servat, E.
 Le splitting pour l'opÃ©rateur de Klein--Gordon: une approche heuristique et numÃ©rique 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. Categories:35P05, 34L16, 34E05, 47A10, 47A70

9. CJM 2004 (vol 56 pp. 794)

Michel, Laurent
 Semi-Classical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy 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. Categories:35P25, 35B34, 35B40

10. CJM 2002 (vol 54 pp. 998)

Dimassi, Mouez
 Resonances for Slowly Varying Perturbations of a Periodic SchrÃ¶dinger Operator 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 $h^{\frac12}$. Categories:35P99, 47A60, 47A40

11. CJM 2000 (vol 52 pp. 119)

Edward, Julian
 Corrigendum to `Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends'' 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. Categories:35P25, 58G25, 47F05

12. CJM 1997 (vol 49 pp. 232)

Edward, Julian
 Spectral theory for the Neumann Laplacian on planar domains with horn-like ends 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. Categories:35P25, 58G25
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