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Resolvent and Scattering Matrix at the Maximum of the Potential

Alexandrova, Ivana, Bony, Jean-François, Ramond, Thierry (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 35P25, 81U20, 35S30, 47A10, 35B38.We study the microlocal structure of the resolvent of the semiclassical Schrödinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semiclassical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing...

Résonances près d’une énergie critique

Jean-François Bony (2001/2002)

Séminaire Équations aux dérivées partielles

Dans cet exposé, on décrit un travail effectué sous la direction de J. Sjöstrand. On prouve des majorations et des minorations du nombre de résonances d’un opérateur de Schrödinger semi-classique P = - h 2 Δ + V ( x ) dans des petits disques centrés en E 0 > 0 , une valeur critique de p ( x , ξ ) = ξ 2 + V ( x ) .

Rieffel’s pseudodifferential calculus and spectral analysis of quantum Hamiltonians

Marius Măntoiu (2012)

Annales de l’institut Fourier

We use the functorial properties of Rieffel’s pseudodifferential calculus to study families of operators associated to topological dynamical systems acted by a symplectic space. Information about the spectra and the essential spectra are extracted from the quasi-orbit structure of the dynamical system. The semi-classical behavior of the families of spectra is also studied.

SAK principle for a class of Grushin-type operators.

Lidia Maniccia, Marco Mughetti (2006)

Revista Matemática Iberoamericana

We prove Fefferman's SAK Principle for a class of hypoelliptic operators on R2 whose nonnegative symbol vanishes anisotropically on the characteristic manifold.

Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations

Tanya Christiansen, M. S. Joshi (2003)

Annales de l’institut Fourier

The scattering matrix is defined on a perturbed stratified medium. For a class of perturbations, its main part at fixed energy is a Fourier integral operator on the sphere at infinity. Proving this is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix determines the asymptotics of a large class of perturbations.

Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Grzegorz Karch (2000)

Studia Mathematica

Large time behavior of solutions to the generalized damped wave equation u t t + A u t + ν B u + F ( x , t , u , u t , ∇ u ) = 0 for ( x , t ) ∈ ℝ n × [ 0 , ∞ ) is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where A u t = u t , B u = - Δ u , and the nonlinear term is either | u t | q - 1 u t or | u | α - 1 u . In this case, the asymptotic profile of solutions is given...

Some decay properties for the damped wave equation on the torus

Nalini Anantharaman, Matthieu Léautaud (2012)

Journées Équations aux dérivées partielles

This article is a proceedings version of the ongoing work [1], and has been the object of a talk of the second author during the Journées “Équations aux Dérivées Partielles” (Biarritz, 2012).We address the decay rates of the energy of the damped wave equation when the damping coefficient b does not satisfy the Geometric Control Condition (GCC). First, we give a link with the controllability of the associated Schrödinger equation. We prove that the observability of the Schrödinger group implies that...

Some remarks on almost-positivity of ψ ⁢ do 's

Cesare Parenti, Alberto Parmeggiani (1998)

Bollettino dell'Unione Matematica Italiana

Per una classe di operatori pseudodifferenziali a caratteristiche multiple vengono date condizioni necessarie e sufficienti per la validità di stime dal basso «ottimali»

Currently displaying 421 – 440 of 624