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23
NonEquilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
, 1999
"... . We study the statistical mechanics of a finitedimensional nonlinear 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 differ ..."
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Cited by 53 (14 self)
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. We study the statistical mechanics of a finitedimensional nonlinear 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 noncompact 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...
Isotropic hypoellipticity and trend to the equilibrium for the FokkerPlanck equation with high degree potential
, 2002
"... ..."
Magnetic Bottles in Connection With Superconductivity
, 2001
"... 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 semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here ..."
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Cited by 24 (14 self)
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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 semiclassical regime of the lowest eigenvalue of the Schrodinger operator with magnetic field have appeared recently. Here we would like to mention the works by BernoffSternberg, LuPan and Del PinoFelmerSternberg. 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 BernoffSternberg concerning the localization of the ground state inside the boundary in the case with constant magnetic fields.
Semiclassical Analysis for the Ground State Energy of a Schrödinger Operator with Magnetic Wells
, 1995
"... Motivated by a recent paper by Montgomery [?], we give the asymptotic behavior, in the semiclassical 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 stud ..."
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Cited by 17 (9 self)
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Motivated by a recent paper by Montgomery [?], we give the asymptotic behavior, in the semiclassical 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.
Heisenberg calculus and spectral theory of hypoelliptic operators on Heisenberg manifolds
, 2005
"... This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, including CR and contact manifolds. In this context the main differential operators at stake include the Hörmander’s sum of squares, the Kohn Laplacian, the horizontal sublaplacian, the CR conformal operators of GoverGraham ..."
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Cited by 11 (6 self)
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This memoir deals with the hypoelliptic calculus on Heisenberg manifolds, including CR and contact manifolds. In this context the main differential operators at stake include the Hörmander’s sum of squares, the Kohn Laplacian, the horizontal sublaplacian, the CR conformal operators of GoverGraham and the contact Laplacian. These operators cannot be elliptic and the relevant pseudodifferential calculus to study them is provided by the Heisenberg calculus of BealsGreiner and Taylor. The Heisenberg manifolds generalize CR and contact manifolds and their name stems from the fact that the relevant notion of tangent space in this setting is rather that of a bundle of graded twostep nilpotent Lie groups. Therefore, the idea behind the Heisenberg calculus, which goes back to Stein, is to construct a pseudodifferential calculus modelled on homogeneous leftinvariant
NONCOMMUTATIVE RESIDUE FOR HEISENBERG MANIFOLDS AND APPLICATIONS IN CR AND CONTACT GEOMETRY
, 2007
"... Abstract. This paper has four main parts. In the first part we construct a noncommutative residue for hypoelliptic calculus on Heisenberg manifolds, that is, for the class of ΨHDO operators introdcued by BealsGreiner and Taylor. This noncommutative residue appears as the residual trace induced on Ψ ..."
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Cited by 9 (7 self)
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Abstract. This paper has four main parts. In the first part we construct a noncommutative residue for hypoelliptic calculus on Heisenberg manifolds, that is, for the class of ΨHDO operators introdcued by BealsGreiner and Taylor. This noncommutative residue appears as the residual trace induced on ΨHDOs of integer orders by the analytic extension of the usual trace to ΨHDOs of noninteger orders and it agrees with the integral of the density defined by the logarithmic singularity of the Schwartz kernel of the corresponding operator. In addition, we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order ΨHDOs. In the second part, we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators and logarithmic metric estimates for Green kernels of hypoelliptic operators, and we show that this noncommutative residue allows us to extend the Dixmier trace to the whole algebra of integer order ΨHDOs. In the third part, we present examples of computations of noncommutative residues for suitable powers of the horizontal sublaplacian and of Rumin’s contact Laplacian. In the fourth part, we present several applications in CR geometry. First, we give some examples of geometric computations of noncommutative residues of suitable powers of the horizontal sublaplacian and of the Kohn Laplacian. We then show that the logarithmic singularities of the Green kernels of the GoverGraham are local CR invariants in the sense of Fefferman. Finally, we make use of framework of noncommutative geometry and of our noncommutative residue to define lower dimensional volumes in CR, e.g., we can give sense to the area of any 3dimensional CR manifold. On the way we obtain a spectral interpretation of the EinsteinHilbert action in CR geometry. 1.
Models for Free Nilpotent Lie Algebras
 J. Algebra
, 1988
"... this paper we are interested in explicit computations of the Lie algebras g M,r . It is well known [19] that that there is a representation of g M,r on upper triangular N by N matrices. The problem is that many computations are di#cult using this representation. In this paper we present an algorithm ..."
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Cited by 8 (4 self)
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this paper we are interested in explicit computations of the Lie algebras g M,r . It is well known [19] that that there is a representation of g M,r on upper triangular N by N matrices. The problem is that many computations are di#cult using this representation. In this paper we present an algorithm that yields vector fields E 1 , ... , EM defined in R with the property that they generate a Lie algebra isomorphic to g M,r . See [6] for another approach to this problem
Semiclassical analysis for Schrödinger operator with magnetic wells, in Quasiclassical methods
 The IMA Volumes in Mathematics and its applications vol. 95, Springer–Verlag New–York
, 1997
"... Abstract. We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schrödinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the individual eigenvalues for operators on closed manifolds ..."
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Cited by 6 (3 self)
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Abstract. We give a survey of some results, mainly obtained by the authors and their collaborators, on spectral properties of the magnetic Schrödinger operators in the semiclassical limit. We focus our discussion on asymptotic behavior of the individual eigenvalues for operators on closed manifolds and existence of gaps in intervals close to the bottom of the spectrum of periodic operators. 1. Preliminaries 1.1. The magnetic Schrödinger operators. Let (M, g) be an oriented Riemannian manifold of dimension n ≥ 2. Let B be a realvalued closed C ∞ 2form on M. Assume that B is exact and choose a realvalued C ∞ 1form A on M such that dA = B. Thus, one has a natural mapping u ↦ → i du + Au from C ∞ c (M) to the space Ω1c (M) of smooth, compactly supported oneforms on M. The Riemannian metric allows to define scalar products in these spaces and consider the adjoint operator (i d + A) ∗ : Ω 1 c (M) → C ∞ c (M). A Schrödinger operator with magnetic potential A is defined by the formula HA = (i d + A) ∗ (i d + A). From the geometric point of view, we may regard A as a connection one form of a Hermitian connection on the trivial line bundle L over M, defining the covariant derivative ∇A = d − iA. The curvature of this connection is −iB. Then the operator HA coincides with the covariant (or Bochner) Laplacian: HA = ∇ ∗ A∇A. Choose local coordinates X = (X1,..., Xn) on M. Write the 1form A in the local coordinates as n∑ A = Aj(X)dXj, j=1 the matrix of the Riemannian metric g as g(X) = (gjℓ(X))1≤j,ℓ≤n
Functional calculus and spectral asymptotics for hypoelliptic operators on Heisenberg manifolds
"... Abstract. This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of BealsGreiner and Taylor to analyze the spectral theory of hypoell ..."
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Cited by 5 (3 self)
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Abstract. This paper is part of a series papers devoted to geometric and spectral theoretic applications of the hypoelliptic calculus on Heisenberg manifolds. More specifically, in this paper we make use of the Heisenberg calculus of BealsGreiner and Taylor to analyze the spectral theory of hypoelliptic operators on Heisenberg manifolds. The main results of this paper include: (i) Obtaining complex powers of hypoelliptic operators as holomorphic families of ΨHDO’s, which can be used to define a scale of weighted Sobolev spaces interpolating the weighted Sobolev spaces of FollandStein and providing us with sharp regularity estimates for hypoelliptic operators on Heisenberg manifolds; (ii) Criterions on the principal symbol of P to invert the heat operator P + ∂t and to derive the small time heat kernel asymptotics for P; (iii) Weyl asymptotics for hypoelliptic operators which can be reformulated geometrically for the main geometric operators on CR and contact manifolds, that is, the Kohn Laplacian, the horizontal sublaplacian and its conformal powers, as well as the contact Laplacian. For dealing we cannot make use of the standard approach of Seeley, so we rely on a new approach based on the pseudodifferential approach representation of the heat kernel. This is especially suitable for dealing with positive hypoelliptic operators. We will deal with more general operator in a forthcoming paper using another new approach. The results of this paper will be used in another forthcoming paper dealing with an analogue for the Heisenberg calculus of the noncommutative geometry which, in particular, will allow us to make use in the Heisenberg setting of Connes ’ noncommutative geometry, including the operator theoretic framework for the local index formula of ConnesMoscovici. 1.
Spectral gaps for periodic Schrödinger operators with strong magnetic fields
 Commun. Math. Phys
"... periodic magnetic field B = dA on covering spaces of compact manifolds. Under some assumptions on B, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field h → 0. ..."
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Cited by 5 (2 self)
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periodic magnetic field B = dA on covering spaces of compact manifolds. Under some assumptions on B, we prove that there are arbitrarily large number of gaps in the spectrum of these operators in the semiclassical limit of strong magnetic field h → 0.