Molecular predissociation resonances below an energy level crossing
We study the resonances of $2\times 2$ systems of one dimensional Schrodinger operators which are related to the mathematical theory of molecular predissociation. We determine the precise positions of the resonances with real parts below the energy where bonding and anti-bonding potentials intersect transversally. In particular, we find that imaginary parts (widths) of the resonances are exponentially small and that the indices are determined by Agmon distances for the minimum of two potentials.
Matrice de scattering et résonances associées à une orbite hétérocline
Nous etudions la matrice de scattering et les resonances de l'operateur de Schrodinger sur R en regime semiclassique dans le cas ou le potentiel presente deux maximums de meme niveau V 0 . On calcule le developpement asymptotique quand h tend vers 0 des coefficients de la matrice de scattering. On etablit alors l'existence de resonances proches de V 0 et l'on decrit tres precisement leur position. En particulier, et par rapport au cas d'un unique maximum, la presence d'une orbite heterocline dans la courbe d'energie critique E = V 0 provoque une concentration des resonances: leur nombre dans un voisinage de V 0 de taille O(h) est d'ordre |log h| alors qu'il reste borne pour un maximum unique. Nous montrons enfin que ces resonances engendrent des oscillations dans le coefficient de transmission pour E = V 0 , qui n'existent pas dans le cas d'un unique maximum.
Metric measure spaces with Riemannian Ricci curvature bounded from below
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out Finsler geometries. It can be given in terms of an enforcement of the Lott, Sturm and Villani geodesic convexity condition for the entropy coupled with the linearity of the heat flow. Besides stability, it enjoys the same tensorization, global-to-local and local-to-global properties. In these spaces, that we call RCD(K,\infty) spaces, we prove that the heat flow (which can be equivalently characterized either as the flow associated to the Dirichlet form, or as the Wasserstein gradient flow of the entropy) satisfies Wasserstein contraction estimates and several regularity properties, in particular Bakry-Emery estimates and the L^\infty-Lip Feller regularization. We also prove that the distance induced by the Dirichlet form coincides with d, that the local energy measure has density given by the square of Cheeger's relaxed slope and, as a consequence, that the underlying Brownian motion has continuous paths. All these results are obtained independently of Poincar\'e and doubling assumptions on the metric measure structure and therefore apply also to spaces which are not locally compact, as the infinite-dimensional ones