. We study the statistical mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...

We give the first Lieb-Thirring type estimate on the sum of the negative eigenvalues of the Pauli operator that behaves as the corresponding semiclassical expression even in the case of strong non-homogeneous magnetic fields. This enables us, in the companion paper [ES-II], to obtain the leading order semiclassical eigenvalue asymptotic, which, in turn, leads to the proof of the validity of the magnetic Thomas-Fermi theory of [LSY-II]. Our work generalizes the results of [LSY-II] to non-homogeneous magnetic fields. Contents 1 Introduction 2 2 The geometry of the magnetic field 12 2.1 Approximating the magnetic field : : : : : : : : : : : : : : : : : : : : : : : : : 12 2.2 Choice of a good gauge : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 16 3 The Neumann problem with a constant field on a cylinder 18 3.1 Supersymmetry : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18 3.2 Estimate of the Neumann ground state density : : : : : : : : : : : : ...

The standard eigenfunctions φλ = e i〈λ,x 〉 on flat tori R n /L have L ∞-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L 2-normalized eigenfunctions have uniformly bounded L ∞-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum completely integrable Laplacians. This paper is concerned with the relation between the dynamics of the geodesic flow G t on the unit sphere bundle S ∗ M of a compact Riemannian manifold (M, g) and the growth rate of the L ∞-norms of its L 2-normalized �-eigenfunctions (or “modes”) {φλ}. Let Vλ: = {φ: �φλ = λφλ} denote the λ-eigenspace for λ ∈ Sp(�), and define

Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, a lot of papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan and Del Pino-Felmer-Sternberg. This recovers partially questions analyzed in a different context by the authors around the question of the so called magnetic bottles. Our aim is to analyze the former results, to treat them in a more systematic way and to improve them by giving sharper estimates of the remainder. In particular, we improve significatively the lower bounds and as a byproduct we solve a conjecture proposed by Bernoff-Sternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.

We study the quantum motion of a charged particle in a half plane, subject to a perpendicular constant magnetic field B and to an arbitrary weak impurity potential WB (i.e. jjW B jj 1 ! ffiB, for some ffi small enough). We show that there exist states propagating with a speed of size B 1=2 along the edge, no matter how fast WB fluctuates. As a consequence, the spectrum of the Hamiltonian is purely absolutely continuous in a spectral interval of size flB ( 0 ! fl ! 1) between the Landau levels of the system without edge or potential, so that the corresponding eigenstates are extended. This then provides a rigorous proof of a phenomenon pointed out by Halperin in his work on the quantum Hall effect. Edge states for a magnetic Hamiltonian 2 1 Introduction It is well known that a classical charged particle, constrained to a plane and subjected to a perpendicular magnetic field will move along physical boundaries when those are present. In the case of a particle moving in a half plane...

We investigate nodal sets of magnetic Schrödinger operators with zero magnetic field, acting on a non simply connected domain in R². For the case of circulation 1/2 of the magnetic vector potential around each hole in the region, we obtain a characterisation of the nodal set, and use this to obtain bounds on the multiplicity of the groundstate. For the case of one hole and a fixed electric potential, we show that the first eigenvalue takes its highest value for circulation 1/2.

Motivated by a recent paper by Montgomery [?], we give the asymptotic behavior, in the semi-classical sense, of the ground state energy for the Schrödinger operator with a magnetic field. We consider the case when the locus of the minima of the intensity of the magnetic field is compact and our study is sharper when this locus is an hypersurface or a finite union of points.

Diamagnetism of the magnetic Schrodinger operator and paramagnetism of the Pauli operator are rigorously proven for nonhomogeneous magnetic fields in the large field, in the large temperature and in the semiclassical asymptotic regimes. New counterexamples are presented which show that neither dia- nor paramagnetism are true in a robust sense (without asymptotics). In particular, we demonstrate that the recent diamagnetic comparison result by Loss and Thaller [1] is essentially the best one can hope. Running title: Dia- and paramagnetism Appeared: Journal of Math. Phys. 38(3) 1289-1317 (1997) Contents 1 Introduction 2 2 Diamagnetism for the Schrodinger operator 6 2.1 Constant field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Nonhomogeneous field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Short time and semiclassics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Large time and/or large field asymptotics . . ....

Abstract. Suppose that M is a compact Riemannian manifold with boundary and u is an L2-normalized Dirichlet eigenfunction with eigenvalue λ. Let ψ be its normal derivative at the boundary. Scaling considerations lead one to expect that the L2 norm of ψ will grow as λ1/2 as λ → ∞. We prove an upper bound of the form ‖ψ‖2 2 ≤ Cλ for any Riemannian manifold, and a lower bound cλ ≤ ‖ψ‖2 2 provided that M has no trapped geodesics (see the main Theorem for a precise statement). Here c and C are positive constants that depend on M, but not on λ. The proof of the upper bound is via a Rellich-type estimate and is rather simple, while the lower bound is proved via a positive commutator estimate. 1.

In this paper we introduce a class of semicalssical Fourier integral operators with global complex phases approximating the fundamental solutions (propagators) for timedependent Schroedinger equations. Our construction is elementary, it is inspired by the joint work of the first author with Yu. Safarov and D. Vasiliev. We consider several simple but basic examples. Keywords: Fourier integral operator, global phase, Schroedinger equation, fundamental solution, parametrix, semiclassical asymptotics, Maslov index, magnetic field. 1. Introduction The notion of Fourier integral operator (FIO) plays an important role in partial differential equations. If pseudo-differential operators can be said to arise from quantization of classical observables, FIO's are due to quantization of classical canonical transformations (Egorov's theorem). More precisely, a transformation of a pseudo-differential operator, P , induced by a change of its symbol under a canonical transformation can be realized (to...