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102
Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations
"... We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to ..."
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Cited by 50 (8 self)
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We present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations. The method does not rely on a priori estimates such as the maximum principle. It applies to systems of coupled equations, to boundary conditions at infinity creating a front, and to higher (possibly fractional) differential linear terms. We present in detail the analysis for nonlinear diffusiontype equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity. In an accompanying paper [5], the method is applied to systems of equations where some variables are "slaved", such as the complex GinzburgLandau equation. 1. Introduction The time evolution of many physical quantities is described by nonlinear, parabolic, partial differential equations. For most of these equations, to obtain a closed form solution seems to be a hopeless task. Therefore, one tries to determine certain qualitative prop...
Continuity properties of Schrödinger semigroups with magnetic fields
 MATHEMATICAL PHYSICS PREPRINT ARCHIVE
, 2000
"... Published in slightly different form in Rev. Math. Phys. 12, 181–225 (2000) The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on th ..."
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Cited by 39 (9 self)
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Published in slightly different form in Rev. Math. Phys. 12, 181–225 (2000) The objects of the present study are oneparameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Katolike conditions. The configuration space is supposed to be an arbitrary open subset of multidimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show localnormcontinuity of the semigroup in the potentials. Finally, we prove that the semigroup has a continuous integral kernel given by a Brownianbridge expectation. Altogether, the article is meant to extend some of the results in B. Simon’s landmark paper [Bull. Amer. Math. Soc. (N.S.) 7, 447–526
Moment Analysis for Localization in Random Schrödinger Operators
, 2005
"... We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the ..."
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Cited by 38 (12 self)
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We study localization effects of disorder on the spectral and dynamical properties of Schrödinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonancediffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the LifshitzKrein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weakL¹ estimate concerning the boundaryvalue distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.
Selfadjointness of the PauliFierz Hamiltonian for arbitrary coupling constants
, 2001
"... The PauliFierz Hamiltonian describes a system of N electrons interacting with a quantized radiation field. The electrons have spin and an ultraviolet cutoff is imposed on the quantized radiation field. For arbitrary coupling constants, selfadjointness and essential selfadjointness of the PauliFi ..."
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Cited by 35 (8 self)
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The PauliFierz Hamiltonian describes a system of N electrons interacting with a quantized radiation field. The electrons have spin and an ultraviolet cutoff is imposed on the quantized radiation field. For arbitrary coupling constants, selfadjointness and essential selfadjointness of the PauliFierz Hamiltonian are proven under a class of ultraviolet cutoffs. 1 Introduction The purpose of this paper is to establish the selfadjointness and the essential selfadjointness of the PauliFierz Hamiltonian [24] for arbitrary coupling constants. The PauliFierz Hamiltonian governs a system of N electrons interacting with a quantized radiation field. The N electrons are assumed to have spin and the quantized radiation field is smeared by an ultraviolet cutoff. The dynamics of the system is determined by the oneparameter unitary timeevolution generated by the PauliFierz Hamiltonian. So, as a first step, it is necessary to establish the selfadjointness of the PauliFierz Hamiltonian. Gener...
Trace Class Perturbations and the Absence of Absolutely Continuous Spectra
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1989
"... We show that various Hamiltonians and Jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices. ..."
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Cited by 25 (9 self)
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We show that various Hamiltonians and Jacobi matrices have no absolutely continuous spectrum by showing that under a trace class perturbation they become a direct sum of finite matrices.
Exponential Mixing of the 2D Stochastic NavierStokes Dynamics
 Comm. Math. Phys
, 2000
"... We consider the NavierStokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity , and grows like \Gamma3 when goes to zero. We prove that this Markov process has a ..."
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Cited by 23 (1 self)
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We consider the NavierStokes equation on a two dimensional torus with a random force which is white noise in time, and excites only a finite number of modes. The number of excited modes depends on the viscosity , and grows like \Gamma3 when goes to zero. We prove that this Markov process has a unique invariant measure and is exponentially mixing in time. 1 Introduction Homogenous isotropic turbulence is often mathematically modelled by Navier Stokes equation subjected to an external stochastic driving force which is stationary in space and time and "large scale", which in particular means smooth in space. The status of the existence and uniqueness of solutions to the stochastic PDE parallels that of the deterministic one. In particular, in two dimensions, it holds under very general conditions. However, for physical reasons, one is interested in the existence, uniqueness and properties of the stationary state of the resulting Markov process. While the existence of such a state fo...
The Classical Limit of Quantum Partition Functions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1980
"... We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approach S² classical spins as L> ∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight ..."
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Cited by 22 (0 self)
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We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approach S² classical spins as L> ∞] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has "diagonal form").
Existence and uniqueness of the integrated density of states for Schrödinger operators with magnetic fields and unbounded random potentials, Rev
 Math. Phys
"... The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic rando ..."
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Cited by 22 (6 self)
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The object of the present study is the integrated density of states of a quantum particle in multidimensional Euclidean space which is characterized by a Schrödinger operator with a constant magnetic field and a random potential which may be unbounded from above and from below. For an ergodic random potential satisfying a simple moment condition, we give a detailed proof that the infinitevolume limits of spatial eigenvalue concentrations of finitevolume operators with different boundary conditions exist almost surely. Since all these limits are shown to coincide with the expectation of the trace of the spatially localized spectral family of the infinitevolume operator, the integrated density of states is almost surely nonrandom and independent of the chosen boundary condition. Our proof of the
The Absolute Continuity of the Integrated Density of States for Magnetic Schrödinger Operators with Certain Unbounded Random Potentials
, 2001
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Ground state properties of the Nelson Hamiltonian  A Gibbs measurebased approach
 Rev. Math. Phys
, 2001
"... The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in positi ..."
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Cited by 19 (12 self)
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The Nelson model describes a quantum particle coupled to a scalar Bose field. We study properties of its ground state through functional integration techniques in case the particle is confined by an external potential. We obtain bounds on the average and the variance of the Bose field both in position and momentum space, on the distribution of the number of bosons, and on the position space distribution of the particle.