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69
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
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Pulé: Propagating edge states for a magnetic Hamiltonian
, 1999
"... We study the quantum mechanical motion of a charged particle moving in a half plane (x> 0) subject to a uniform constant magnetic field B directed along the zaxis and to an arbitrary impurity potential WB, assumed to be weak in the sense that WB ∞ < δB, for some δ small enough. We show rigorous ..."
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Cited by 29 (0 self)
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We study the quantum mechanical motion of a charged particle moving in a half plane (x> 0) subject to a uniform constant magnetic field B directed along the zaxis and to an arbitrary impurity potential WB, assumed to be weak in the sense that WB ∞ < δB, for some δ small enough. We show rigorously a phenomenon pointed out by Halperin in his work on the quantum Hall effect, namely the existence of current carrying and extended edge states in such a situation. More precisely, we show that there exist states propagating with a speed of size B 1/2 in the ydirection, no matter how fast WB fluctuates. As a result of this, we obtain that the spectrum of the Hamiltonian is purely absolutely continuous in a spectral interval of size γB (for some γ < 1) between the Landau levels of the unperturbed system (i.e. the system without edge or potential), so that the corresponding eigenstates are extended. Edge states for a magnetic Hamiltonian 2 1
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 eigenvalue estimates for the Pauli operator with strong nonhomogeneous magnetic fields  I. Nonasymptotic LiebThirring type estimate
, 1996
"... We give the first LiebThirring 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 nonhomogeneous magnetic fields. This enables us, in the companion paper [ESII], to obtain the leading ord ..."
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Cited by 19 (3 self)
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We give the first LiebThirring 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 nonhomogeneous magnetic fields. This enables us, in the companion paper [ESII], to obtain the leading order semiclassical eigenvalue asymptotic, which, in turn, leads to the proof of the validity of the magnetic ThomasFermi theory of [LSYII]. Our work generalizes the results of [LSYII] to nonhomogeneous 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 : : : : : : : : : : : : ...
Nodal sets for the groundstate of the Schrödinger operator with zero magnetic field in a non simply connected domain.
"... 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 ob ..."
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Cited by 18 (11 self)
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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.
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.
Riemannian manifolds with uniformly bounded eigenfunctions
 Duke Math. J
, 2000
"... 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 2normalized eigenfunctions have uniformly bounded L ∞norms. Similar bases exist on other flat manifolds. Does this property ..."
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Cited by 16 (7 self)
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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 2normalized 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 2normalized �eigenfunctions (or “modes”) {φλ}. Let Vλ: = {φ: �φλ = λφλ} denote the λeigenspace for λ ∈ Sp(�), and define
Between classical and quantum
, 2005
"... The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, inclu ..."
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Cited by 12 (3 self)
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The relationship between classical and quantum theory is of central importance to the philosophy of physics, and any interpretation of quantum mechanics has to clarify it. Our discussion of this relationship is partly historical and conceptual, but mostly technical and mathematically rigorous, including over 500 references. For example, we sketch how certain intuitive ideas of the founders of quantum theory have fared in the light of current mathematical knowledge. One such idea that has certainly stood the test of time is Heisenberg’s ‘quantumtheoretical Umdeutung (reinterpretation) of classical observables’, which lies at the basis of quantization theory. Similarly, Bohr’s correspondence principle (in somewhat revised form) and Schrödinger’s wave packets (or coherent states) continue to be of great importance in understanding classical behaviour from quantum mechanics. On the other hand, no consensus has been reached on the Copenhagen Interpretation, but in view of the parodies of it one typically finds in the literature we describe it in detail. On the assumption that quantum mechanics is universal and complete, we discuss three ways in which classical physics has so far been believed to emerge from quantum physics, namely
Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions
"... Abstract. Suppose that M is a compact Riemannian manifold with boundary and u is an L2normalized 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 b ..."
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Cited by 11 (3 self)
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Abstract. Suppose that M is a compact Riemannian manifold with boundary and u is an L2normalized 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 Rellichtype estimate and is rather simple, while the lower bound is proved via a positive commutator estimate. 1.