Fractal Weyl laws for quantum resonances
We define the wave front set of a distribution vector of a unitary representation in terms of pseudo-differential-like operators [M2] for any real Lie group G. This refines the notion of wave front set of a representation introduced by R. Howe [Hw]. We give as an application a necessary condition so that a distribution vector remains a distribution vector for the restriction of the representation to a closed subgroup H, and we give a propagation of singularities theorem for distribution vectors.
We study the Cauchy problem in the hyperbolic space for the semilinear heat equation with forcing term, which is either of KPP type or of Allen-Cahn type. Propagation and extinction of solutions, asymptotical speed of propagation and asymptotical symmetry of solutions are addressed. With respect to the corresponding problem in the Euclidean space new phenomena arise, which depend on the properties of the diffusion process in . We also investigate a family of travelling wave solutions, named...
En adaptant les méthodes algébriques et géométriques qu’utilisent M. Sato, T. Kawai et M. Kashiwara pour obtenir le faisceau des microfonctions, nous construisons de manière fonctorielle, donc intrinsèque, un faisceau sur la sphère cotangente à un espace vectoriel réel de dimension finie . Les sections de ce faisceau jouent vis-à-vis des fonctions analytiques sur un rôle analogue à celui des microfonctions vis-à-vis des hyperfonctions. Nous en déduisons une notions de front d’onde à l’infini...
We consider a continuous curve of linear elliptic formally self-adjoint differential operators of first order with smooth coefficients over a compact Riemannian manifold with boundary together with a continuous curve of global elliptic boundary value problems. We express the spectral flow of the resulting continuous family of (unbounded) self-adjoint Fredholm operators in terms of the Maslov index of two related curves of Lagrangian spaces. One curve is given by the varying domains, the other by...
2000 Mathematics Subject Classification: 35P25, 35R30, 58J50.Using the free-space translation representation (modified Radon transform) of Lax and Phillips in odd dimensions, it is shown that the generalized backscattering transform (so outgoing angle w = Sq in terms of the incoming angle with S orthogonal and Id-S invertible) may be further restricted to give an entire, globally Fredholm, operator on appropriate Sobolev spaces of potentials with compact support. As a corollary we show that the...