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230
Ground States in Nonrelativistic Quantum Electrodynamics
, 2000
"... The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state — one that minimizes the energy and satisfies the Schrödinger equation. We prove quite generally that this state exis ..."
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Cited by 132 (15 self)
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The excited states of a charged particle interacting with the quantized electromagnetic field and an external potential all decay, but such a particle should have a true ground state — one that minimizes the energy and satisfies the Schrödinger equation. We prove quite generally that this state exists for all values of the finestructure constant and ultraviolet cutoff. We also show the same thing for a manyparticle system under physically natural conditions.
Pointwise semigroup methods and stability of viscous shock waves
 Indiana Univ. Math. J
, 1998
"... Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator abou ..."
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Cited by 119 (53 self)
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Abstract. Considered as rest points of ODE on L p, stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective except in the scalar or totally compressive case ([Sat], [K.2], resp.), each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We overcome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard [H.1], Kapitula [K.12], and Zeng [Ze,LZe]. These techniques allow us to do \hard &quot; analysis in PDE within the dynamical systems/semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications
Resonances, Radiation Damping and Instability in Hamiltonian Nonlinear Wave Equations
, 1998
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Nonequilibrium statistical mechanics of strongly anharmonic chains of oscillators
 Comm. Math. Phys
"... We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with ..."
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Cited by 75 (17 self)
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We study the model of a strongly nonlinear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, ReyBellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hörmander’s theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.
Uniformly elliptic operators on Riemannian manifolds
 J. Diff. Geom
, 1992
"... Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiiso ..."
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Cited by 68 (3 self)
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Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g. Typical examples of such operators are the Laplace operators of Riemannian structures which are quasiisometric to g. We first prove some Poincare and Sobolev inequalities on geodesic balls. Then we use Moser's iteration to obtain Harnack inequalities. Gaussian estimates, uniqueness theorems, and other applications are also discussed. These results involve local or global lower bound hypotheses on the Ricci curvature of g. Some of them are new even when applied to the Laplace operator of (M, g). 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 60 (17 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.
Stable manifolds for all monic supercritical focusing nonlinear Schrödinger equations in one dimension
 J. Amer. Math. Soc
"... We consider the nonlinear Schrödinger equation (1.1) i∂tψ + ∂ 2 xψ = −ψ  2σ ψ on the line R with σ> 2. This is exactly the L 2supercritical case and these equations are locally wellposed in H 1 (R) = W 1,2 (R). Let φ = φ(·, α) be the ..."
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Cited by 50 (10 self)
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We consider the nonlinear Schrödinger equation (1.1) i∂tψ + ∂ 2 xψ = −ψ  2σ ψ on the line R with σ> 2. This is exactly the L 2supercritical case and these equations are locally wellposed in H 1 (R) = W 1,2 (R). Let φ = φ(·, α) be the
Localization of Classical Waves I: Acoustic Waves.
 Commun. Math. Phys
, 1996
"... We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existe ..."
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Cited by 44 (1 self)
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We consider classical acoustic waves in a medium described by a position dependent mass density %(x). We assume that %(x) is a random perturbation of a periodic function % 0 (x) and that the periodic acoustic operator A 0 = \Gammar \Delta 1 %0 (x) r has a gap in the spectrum. We prove the existence of localized waves, i.e., finite energy solutions of the acoustic equations with the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, with probability one. Localization of acoustic waves is a consequence of Anderson localization for the selfadjoint operators A = \Gammar \Delta 1 %(x) r on L 2 (R d ). We prove that, in the random medium described by %(x), the random operator A exhibits Anderson localization inside the gap in the spectrum of A 0 . This is shown even in situations when the gap is totally filled by the spectrum of the random operator; we can prescribe random environments that ensure localization in almost the wh...