. 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...

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.

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We consider an open manifold which is the interior of a compact manifold with boundary. Assuming gauge invariance, we classify magnetic fields with compact support into being trapping or non-trapping. We study spectral properties of the associated magnetic Laplacian for a class of Riemannian metrics which includes complete hyperbolic metrics of finite volume. When B is non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using Mourre theory, we show the absence of singular continuous spectrum and the local finiteness of the point spectrum. When B is trapping, the spectrum is discrete and obeys the Weyl law. The existence of trapping magnetic fields with compact support depends on cohomological conditions, indicating a new and very strong long-range effect. In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On hyperbolic surfaces with at least two cusps, we show that the magnetic Laplacian associated to every magnetic field with compact support has purely discrete spectrum for some choices of the vector potential, while other choices lead to a situation of limiting absorption principle. We also study perturbations of the metric. We show that in the Mourre theory it is not necessary to require a decay of the derivatives of the perturbation. This very singular perturbation is then brought closer to the perturbation of a potential.

Abstract. In this paper we characterize compactness of the canonical solution operator to ∂ on weigthed L 2 spaces on C. For this purpose we consider certain Schrödinger operators with magnetic fields and use a condition which is equivalent to the property that these operators have compact resolvents. We also point out what are the obstructions in the case of several complex variables. 1. Introduction. Let ϕ: C − → R be a C2-weight function and consider the Hilbert spaces

Abstract. We consider a periodic magnetic Schrödinger operator on a noncompact Riemannian manifold M such that H 1 (M, R) = 0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We review a general scheme of a proof of existence of an arbitrary large number of gaps in the spectrum of such an operator in the semiclassical limit, which was suggested in our previous paper, and some applications of this scheme. Then we apply these methods to establish similar results in the case when the wells have regular hypersurface pieces.

This paper is devoted to the study of a class of C*-algebras graded by semilattices (see 3.6) which allow one to formalize in a convenient way the notion of a-connected component of an operator (this approach was motivated by the papers [KPR, Pol, PSS]). It is also shown there that (dispersive) N-body hamiltonians are affiliated to such algebras and that the decomposition of the resolvent according to the homogeneous components of the grading gives exactly the well known Weinberg-Van Winter equation. As a consequence, a purely algebraic proof of the HVZ theorem is obtained (see Theorem 3.25). The algebraic formalism was then extended in [BG2] such as to cover the proof of the Mourre estimate for N-body hamiltonians (not necessarily non-relativistic, the proof being given for a class of hamiltonians abstractly defined). The Mourre estimate for such systems is a highly technical and nontrivial inequality (see the first papers [PSS] and [FrH] devoted to this question) and it seems to us quite remarkable that a purely algebraic statement involving quotients of C*-algebras is relevant in this context. In fact, it was shown that one can realize a complete decoupling of channels by taking such a quotient (thus eliminating the Simon partition of unity, which gave only an approximative channel decoupling). This paper also contains new examples of graded C*-algebras associated to symplectic spaces, which allow one to treat N-body systems in constant magnetic fields. These methods were shown to be efficient in the treatment of very singular N-body systems (with hard-core interactions) in [BGS], where the Mourre estimate was proved for such systems, and in [If1], where the scattering theory was treated. A complete and unified pres...

In this paper we discuss compactness of the canonical solution operator to ∂ on weigthed L² spaces on C n. For this purpose we apply ideas which were used for the Witten Laplacian in the real case and various methods of spectral theory of these operators. We also point out connections to the theory of Dirac and Pauli operators.

Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential

We study generalised magnetic Schrödinger operators of the form Hh(A,V) = h(Π A)+ V, where h is an elliptic symbol, Π A = −i∇−A, with A a vector potential defining a variable magnetic field B, and V is a scalar potential. We are mainly interested in anisotropic functions B and V. The first step is to show that these operators are affiliated to suitable C ∗-algebras of (magnetic) pseudodifferential operators. A study of the quotient of these C ∗-algebras by the ideal of compact operators leads to formulae for the essential spectrum of Hh(A,V), expressed as a union of spectra of some asymptotic operators, supported by the quasi-orbits of a suitable dynamical system. The quotient of the same C ∗-algebras by other ideals give localization results on the functional calculus of the operators Hh(A,V), which can be interpreted as non-propagation properties of their unitary groups. 1