Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing ear-lier work of Howard [H.1], Kapitula [K.1-2], and Zeng [Ze,LZe]. These techniques allow us to do \hard " analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications

The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state --- one that minimizes the energy and satisfies the Schrodinger equation. We prove quite generally that this state exists for all values of the fine-structure constant and ultraviolet cutoff. We also show the same thing for a many-particle system under physically natural conditions. 1 INTRODUCTION An established picture of an atom or molecule is that even in the presence of a quantized radiation field there is a ground state. The excited states that exist in the absence of coupling to the field are expected to melt into resonances, which means that they eventually decay with time into the ground state plus free photons. This picture has been established by Bach, Frohlich and Sigal in [8] for sufficiently small values of the various parameters that define the theory. Here we show that a ground state exists...

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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the self-adjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...

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We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hormander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains. Introduction In this paper, we study the statistical mechanics of a highly non-linear chain of oscillators coupled to two heat reservoirs which are at (arbitrary) different temperatures. We show that such systems have, under suitable conditions, a unique stationary state, in which heat flows from the hotter reservoir to the cooler one. These results are an extension of the same statements obtained by Eckmann, Pillet and ReyBellet in [EPR99a, EPR99b] where it was ass...

The purpose of this note is to give a survey of some recent work on dispersive estimates for the Schrödinger flow (1) e itH Pc, H = − △ + V on R d, d ≥ 1 where Pc is the projection onto the continuous spectrum of H. V is a real-valued potential that

this paper is a proxy for what deserves a book or at least a very long review article. In attempting to overview such a vast area in a few pages, I have had to focus on the high points. No proofs are given and I have settled for usually quoting the initial or especially significant papers. I have no doubt that I have left out some important papers, and if so, I ask the forgiveness of the reader (and their authors!).

We prove L p estimates for charge transfer Hamiltonians, including matrix and inhomogeneous generalizations; such equations appear naturally in the study of multi-soliton systems. 1

this paper various conditions --- abstract and explicit --- are given for (1.2), and exploiting these in the general strictly convex case, (1.2) is used to deduce (1.1) and to extend it to a version with f 2 H