We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4- V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x> 0). We then extend our results to higher dimensional systems periodic, except for a localised impurity, in all but one space dimension. A new method, "the twisting trick", is presented for proving the absence of singular continuous spectrum, and some independent applications of this trick are given in an appendix.

We investigate the dynamics of an N-level system linearly coupled to a field of mass-less bosons at positive temperature. Using complex deformation techniques, we develop time-dependent perturbation theory and study spectral properties of the total

Abstract. We survey some aspects of the theory of the integrated density of states (IDS) of random Schrödinger operators. The first part motivates the problem and introduces the relevant models as well as quantities of interest. The proof of the existence of this interesting quantity, the IDS, is discussed in the second section. One central topic of this survey is the asymptotic behavior of the integrated density of states at the boundary of the spectrum. In particular, we are interested in Lifshitz tails and the occurrence of a classical and a quantum regime. In the last section we discuss regularity properties of the IDS. Our emphasis is on the discussion of fundamental problems and central ideas to handle them. Finally, we discuss further developments and problems of current

We present an introduction (also for non-experts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²-spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, so-called generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 --semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p --conditions, singular and fractional diffusion operators. Subsequently, we analyz...

Abstract. We justify the WKB analysis for the semiclassical nonlinear Schrödinger equation with a subquadratic potential. This concerns subcritical, critical, and supercritical cases as far as the geometrical optics method is concerned. In the supercritical case, this extends a previous result by E. Grenier; we also have to restrict to nonlinearities which are defocusing and cubic at the origin, but besides subquadratic potentials, we consider initial phases which may be unbounded. For this, we construct solutions for some compressible Euler equations with unbounded source term and unbounded initial velocity. (1.1) (1.2)

Introduction Let\Omega be a bounded region in a complete Riemannian manifold with smooth boundary @ Let C 1 (\Omega ); C 1 0 (\Omega\Gamma and C 1 c (\Omega\Gamma denote respectively the space of smooth functions on\Omega , the subspace consisting of such functions which vanish on @ and the subspace of such functions which vanish in a neighbourhood of @ \Omega\Gamma For those whose main interest is in spectral theory in Euclidean space we mention that our main results are also new in that context. We investigate the existence and explicit determination of constants c and weights X and Y on\Omega such that the Rellich inequality Z \Omega Xjuj p c Z \Omega Y

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We present an interacting particle system methodology for the numerical solving of the Lyapunov exponent of Feynman-Kac semigroups and for estimating the principal eigenvalue of Schrodinger generators. The continuous or discrete time models studied in this work consists of N interacting particles evolving in an environment with soft obstacles related to a potential function V . These models are related to genetic algorithms and Moran type particle schemes. Their choice is not unique. We will examine a class of models extending the hard obstacle model of K. Burdzy, R. Holyst and P. March and including the Moran type scheme presented by the authors in a previous work. We provide precise uniform estimates with respect to the time parameter and we analyse the fluctuations of continuous time particle models. Keywords: Feynman-Kac formula, Schrodinger operator, spectral radius, Lyapunov exponent, spectral decomposition, semigroups on a Banach space, interacting particle systems, genetic algorithms, asymptotic stability, central limit theorems. 1

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We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner–Wienholtz–Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.