This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.

. In this article, we prove and exploit a trace identity for the spectra of Schrodinger operators and similar operators. This identity leads to universal bounds on the spectra, which apply to low-lying eigenvalues, eigenvalue asymptotics, and to partition functions (traces of heat operators). In many cases they are sharp in the sense that there are specific examples for which the inequalities are saturated. Special cases corresponding to known inequalities include those of Hile and Protter and of Yang. Introduction In this article, we prove and exploit an identity for the spectra of self-adjoint operators H modeled on the Dirichlet Laplacian or, more generally, on Schrodinger operators of the form (1) (p \Gamma A(x)) 2 + V (x); where p = 1 i r is the usual momentum operator in convenient units and A(x) is the magnetic vector potential. We recover and extend several known inequalities involving sums, differences, and ratios of eigenvalues. Let j , j = 1; : : : ; denote the ordere...

Let ¯ gc L; denote the grand canonical Gibbs measure of a lattice gas in a cube of size L with the chemical potential and a fixed boundary condition. Let ¯ c L;n be the corresponding canonical measure defined by conditioning ¯ gc L; on P x2 j x = n. Consider the lattice gas dynamics for which each particle performs random walk with rates depending on near-by particles. The rates are chosen such that, for every n and L fixed, ¯ c L ;n is a reversible measure. Suppose that the Dobrushin-Shlosman mixing conditions holds for ¯ L; for all chemical potentials 2 R I . We prove that R f log fd¯ c L ;n const.L 2 D( p f ) for any probability density f with respect to ¯ c L;n ; here the constant is independent of n or L and D denotes the Dirichlet form of the dynamics. The dependence on L is optimal. Keywords: Dobrushin-Shlosman mixing conditions, Interacting random walks, Lattice gas dynamics, Logarithmic Sobolev inequality Research partially supported by U...

Abstract. Suppose that Ω ⊂ R n is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Ω with various boundary conditions are quantum ergodic if the classical billiard map β on the ball bundle B ∗ (∂Ω) is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfunction φλ, ∆φλ = λ 2 φλ is an eigenfunction of an operator Fh on the boundary of Ω with h = λ −1. In the case of the Neumann boundary condition, Fh is the boundary integral operator induced by the double layer potential. We show that Fh is a semiclassical Fourier integral operator quantizing the billiard map plus a ‘small ’ remainder; the quantum dyanmics defined by Fh can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of Zelditch-Zworski (1996). Novelties include the facts that Fh is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under β and which depend on the boundary conditions. Ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by S. Ozawa (1993), and was almost simultaneously proved by Gerard-Leichtnam (1993) in the case of convex C 1,1 domains (with continuous tangent planes) and with Dirichlet boundary conditions. Our methods seem to be quite different. Motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form. Contents

. We prove the compactness of the imbedding of the Sobolev space W 1;2 0 (\Omega\Gamma into L 2(\Omega\Gamma for any relatively compact open subset\Omega of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous Dirichlet heat kernel. 1. Introduction Consider a family M of n-dimensional closed Riemannian manifolds with a uniform lower bound of sectional curvature and a uniform upper bound of diameter for a fixed n 2 N . In order to investigate various properties of manifolds in M, it is very useful to study its closure M with respect to the Gromov-Hausdorff distance dGH , which is compact by the Gromov compactness theorem [15]. Since the closure M consists of Alexandrov spaces introduced in [2], the study of Alexandrov spaces is nowadays an important topic i...

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

Sharp smoothing estimates are proven for magnetic Schrödinger semigroups in two dimensions under the assumption that the magnetic field is bounded below by some positive constant B0. As a consequence the L ∞ norm of the associated integral kernel is bounded by the L ∞ norm of the Mehler kernel of the Schrödinger semigroup with the constant magnetic field B0.

2. Eigenvalues and eigenfunctions 516 2.1 Spectral asymptotics 516 2.2 The isoperimetric problem 518

We consider Schrodinger semigroups e.- IH, H =-A+V on Iw ” with V---cIxl- ’ as 1x1--rco, O<c<[(l/2)(n-2)] * with H>O. We determine the exact power law divergence of I~e-‘Hi~p,p and of some IIe-‘Hlly,p as maps from Lp to Lq. The results are expressed most naturally in terms of the power a for which there exists a positive resonance 9 such that Hq = 0, q(x)- 1.x-‘.:Ta 1991 Academic Press, Inc. 1.

We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.