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99
An invitation to random Schrödinger operators
, 2007
"... This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are m ..."
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Cited by 50 (7 self)
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This review is an extended version of my mini course at the États de la recherche: Opérateurs de Schrödinger aléatoires at the Université Paris 13 in June 2002, a summer school organized by Frédéric Klopp. These lecture notes try to give some of the basics of random Schrödinger operators. They are meant for nonspecialists and require only minor previous knowledge about functional analysis and probability theory. Nevertheless this survey includes complete proofs of Lifshitz tails and Anderson localization. Copyright by the author. Copying for academic purposes is permitted.
Existence and instability of spike layer solutions to singular perturbation problems
, 2002
"... An abstract framework is given to establish the existence and compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov–Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold, and the related linearized ..."
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Cited by 26 (4 self)
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An abstract framework is given to establish the existence and compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov–Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold, and the related linearized problem is also analyzed using this reduction. As an application, we show the existence of a multipeak spike layer solution with peaks on the boundary of the domain, and we also obtain precise estimates of the small eigenvalues of the operator obtained by linearizing at a
Isoperimetric and universal inequalities for eigenvalues, in Spectral Theory and Geometry
 London Mathematical Society Lecture Note Series
, 1999
"... PaynePólyaWeinberger conjecture, Sperner’s inequality, biharmonic operator, biLaplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the PólyaSzegő conjecture, universal inequalities for eigenvalues, HileProtter inequality, H. C. Yang’s inequality. Short title: Isoperimetric ..."
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Cited by 24 (5 self)
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PaynePólyaWeinberger conjecture, Sperner’s inequality, biharmonic operator, biLaplacian, clamped plate problem, Rayleigh’s conjecture, buckling problem, the PólyaSzegő conjecture, universal inequalities for eigenvalues, HileProtter inequality, H. C. Yang’s inequality. Short title: Isoperimetric and Universal Inequalities This paper reviews many of the known inequalities for the eigenvalues of the Laplacian and biLaplacian on bounded domains in Euclidean space. In particular, we focus on isoperimetric inequalities for the low eigenvalues of the Dirichlet and Neumann Laplacians and of the vibrating clamped plate problem (i.e., the biharmonic operator with “Dirichlet ” boundary conditions). We also discuss the known universal inequalities for the eigenvalues of the Dirichlet Laplacian and the vibrating clamped plate and buckling problems and go on to
ITERATED BROWNIAN MOTION IN AN OPEN SET
, 2004
"... Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If τ is the first exit time of iterated Brownian motion from the solid, then P(τ ..."
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Cited by 20 (0 self)
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Suppose a solid has a crack filled with a gas. If the crack reaches the surrounding medium, how long does it take the gas to diffuse out of the crack? Iterated Brownian motion serves as a model for diffusion in a crack. If τ is the first exit time of iterated Brownian motion from the solid, then P(τ> t) can be viewed as a measurement of the amount of contaminant left in the crack at time t. We determine the large time asymptotics of P(τ> t) for both bounded and unbounded sets. We also discuss a strange connection between iterated Brownian motion and the parabolic operator 1 8 ∆2 − ∂ ∂t.
Sigal : Time dependent scattering theory for Nbody quantum systems
, 1997
"... We give a full and self contained account of the basic results in Nbody scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r−µ, µ> √ 3 − 1. Our approach is a synthesis of earlier work and of new ideas. Globa ..."
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Cited by 18 (2 self)
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We give a full and self contained account of the basic results in Nbody scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r−µ, µ> √ 3 − 1. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t2 as t →±∞for any orbit ψt which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre’s theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase–space analysis. As a topic of general interest we describe a method of commutator expansions. 0.
Bound States in Curved Quantum Layers
 Comm. Math. Phys
"... We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a noncompact surface embedded in R 3 . We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hardwall boundary. Under the assumption that the s ..."
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Cited by 17 (4 self)
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We consider a nonrelativistic quantum particle constrained to a curved layer of constant width built over a noncompact surface embedded in R 3 . We suppose that the latter is endowed with the geodesic polar coordinates and that the layer has the hardwall boundary. Under the assumption that the surface curvatures vanish at infinity we find sufficient conditions which guarantee the existence of geometrically induced bound states. KeyWords: waveguides, layers, constrained systems, Dirichlet Laplacian, bound states, surface geometry, curvature, integral curvatures, geodesic polar coordinates 1 Introduction Relations between the geometry of a region\Omega in R n , boundary conditions at @ and spectral properties of the corresponding Laplacian are one of the vintage problems of mathematical physics. Recent years brought new motivations and focused attention to aspects of the problem which attracted little attention earlier. A strong impetus comes from mesoscopic physics, where new e...
Topologically nontrivial quantum layers
 J. Math. Phys
"... Given a complete noncompact surface Σ embedded in R 3, we consider the Dirichlet Laplacian in the layer Ω that is defined as a tubular neighbourhood of constant width about Σ. Using an intrinsic approach to the geometry of Ω, we generalise the spectral results of the original paper [1] by Duclos et ..."
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Cited by 15 (2 self)
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Given a complete noncompact surface Σ embedded in R 3, we consider the Dirichlet Laplacian in the layer Ω that is defined as a tubular neighbourhood of constant width about Σ. Using an intrinsic approach to the geometry of Ω, we generalise the spectral results of the original paper [1] by Duclos et al. to the situation when Σ does not possess poles. This enables us to consider topologically more complicated layers and state new spectral results. In particular, we are interested in layers built over surfaces with handles or several cylindrically symmetric ends. We also discuss more general regions obtained by compact deformations of certain Ω.
Nonparametric estimation of scalar diffusions based on low frequency data is illposed
, 2002
"... We study the problem of estimating the coefficients of a diffusion (Xt,t≥0); the estimation is based on discrete data Xn∆,n = 0,1,...,N. The sampling frequency ∆ −1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both ..."
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Cited by 13 (0 self)
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We study the problem of estimating the coefficients of a diffusion (Xt,t≥0); the estimation is based on discrete data Xn∆,n = 0,1,...,N. The sampling frequency ∆ −1 is constant, and asymptotics are taken as the number N of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient (the volatility) and the drift in a nonparametric setting is illposed: the minimax rates of convergence for Sobolev constraints and squarederror loss coincide with that of a, respectively, first and secondorder linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. Our approach is based on the spectral analysis of the associated Markov semigroup. A rateoptimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue–eigenfunction pair of the transition operator of the discrete time Markov chain
A GEOMETRICAL VERSION OF HARDY’S INEQUALITY
, 2001
"... We proof a version of Hardy’s type inequality in a domain Å�which involves the distance to the boundary and the volume of Å. In partucular, we obtain a result which gives a positive answer to a question asked by H.Brezis and M.Marcus. ..."
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Cited by 13 (3 self)
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We proof a version of Hardy’s type inequality in a domain Å�which involves the distance to the boundary and the volume of Å. In partucular, we obtain a result which gives a positive answer to a question asked by H.Brezis and M.Marcus.