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31
Asymptotic completeness in quantum field theory. Massive PauliFierz Hamiltonians
, 1997
"... Spectral and scattering theory of massive PauliFierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of ob ..."
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Cited by 54 (7 self)
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Spectral and scattering theory of massive PauliFierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of observables. Its main result is what we call geometric asymptotic completeness. Finally, the last part is a proof of asymptotic completeness itself. 1 Introduction Our paper is devoted to a class of Hamiltonians used in physics to describe a quantum system ("matter" or "an atom") interacting with a bosonic field ("radiation"). K and K are respectively the Hilbert space and the Hamiltonian describing the matter. The bosonic field is described by a Fock space \Gamma(h) with the oneparticle space eg. h = L 2 (IR d ; dk), where IR d is the momentum space, and a free Hamiltonian of the form d\Gamma(!(k)) = Z !(k)a (k)a(k)dk: The function !(k) is called the dispersion relation. The inte...
Fractal upper bounds on the density of semiclassical resonances
 Duke Math. Journal
"... Let P = −h 2 ∆g +V (x) be a selfadjoint Schrödinger operator on a compact Riemannian nmanifold, (X, g), V ∈ C ∞ (X; R). The spectral asymptotics as h → 0 are given by the celebrated Weyl law – see [10] and [16] for recent advances and numerous references. If we assume that the zero energy surface ..."
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Cited by 35 (16 self)
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Let P = −h 2 ∆g +V (x) be a selfadjoint Schrödinger operator on a compact Riemannian nmanifold, (X, g), V ∈ C ∞ (X; R). The spectral asymptotics as h → 0 are given by the celebrated Weyl law – see [10] and [16] for recent advances and numerous references. If we assume that the zero energy surface is nondegenerate,
The functional calculus
 J. London Math. Soc
, 1995
"... One of the interesting developments in spectral theory recently has been the application of a new formula for a function j{H) of a selfadjoint operator H by Helffer and Sjostrand [8]. This formula, equation (5) below, depends upon the use of almost analytic extensions of functions initially defined ..."
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Cited by 32 (1 self)
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One of the interesting developments in spectral theory recently has been the application of a new formula for a function j{H) of a selfadjoint operator H by Helffer and Sjostrand [8]. This formula, equation (5) below, depends upon the use of almost analytic extensions of functions initially defined on the real line, an idea due
Elementary linear algebra for advanced spectral problems
"... The purpose of this article is to discuss a simple linear algebraic tool which has proved itself very useful in the mathematical study of spectral problems arising in elecromagnetism and quantum mechanics. Roughly speaking it amounts to replacing an operator of interest by a suitably chosen invertib ..."
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Cited by 29 (13 self)
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The purpose of this article is to discuss a simple linear algebraic tool which has proved itself very useful in the mathematical study of spectral problems arising in elecromagnetism and quantum mechanics. Roughly speaking it amounts to replacing an operator of interest by a suitably chosen invertible system of operators.
The Magnetic Weyl Calculus
, 2004
"... In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle ” at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a ..."
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Cited by 21 (18 self)
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In the presence of a variable magnetic field, the Weyl pseudodifferential calculus must be modified. The usual modification, based on “the minimal coupling principle ” at the level of the classical symbols, does not lead to gauge invariant formulae if the magnetic field is not constant. We present a gauge covariant quantization, relying on the magnetic canonical commutation relations. The underlying symbolic calculus is a deformation, defined in terms of the magnetic flux through triangles, of the classical Moyal product.
Operator Kernel Estimates For Functions Of Generalized Schrödinger Operators
 Proc. Amer. Math. Soc
, 2001
"... We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of Mathematical Physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave ..."
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Cited by 20 (8 self)
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We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of Mathematical Physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved CombesThomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.
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. Global c ..."
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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.
Asymptotic of the Density of States for the Schrödinger Operator with Periodic Electric Potential
, 1996
"... We analyze in this article the spectral properties of the Schrodinger operator with periodic potential on L 2 (IR n ). It is proven that the integrated density of states N(¯) has an asymptotic expansion of the form N(¯) = a n ¯ n=2 + a n\Gamma2 ¯ n\Gamma2 2 +O(¯ (n\Gamma3+ffl)=2 ); 8ffl ..."
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Cited by 18 (0 self)
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We analyze in this article the spectral properties of the Schrodinger operator with periodic potential on L 2 (IR n ). It is proven that the integrated density of states N(¯) has an asymptotic expansion of the form N(¯) = a n ¯ n=2 + a n\Gamma2 ¯ n\Gamma2 2 +O(¯ (n\Gamma3+ffl)=2 ); 8ffl ? 0 : This gives also a proof of the BetheSommerfeld conjecture for n 4. 1 Introduction We are interested in the study of the spectral properties of the Schrodinger operator on L 2 (IR n ); with n 2; P V = n X j=1 D 2 x j + V (x); (1.1) where D x j = \Gammai@ x j = 1 i @ @x j , for j = 1; \Delta \Delta \Delta ; n. The potential x 7! V (x) is real and V (\Delta) 2 C 1 (IR n ; IR): (1.2) Let \Gamma be a lattice on IR n : \Gamma = f n X j=1 k j e j ; k = (k 1 ; : : : ; k n ) 2 ZZ n g ; (1.3) where fe 1 ; : : : ; e n g is a basis of IR n . We assume that the potential is periodic: V (x + a) = V (x) ; 8a 2 \Gamma : (1.4) It is well known that P V is essentiall...
A TimeDependent Theory of Quantum Resonances
, 1999
"... In this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues. We apply this theory to manybody Schrodinger operators and to the problem of quasiclassical tunneling. 1. ..."
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Cited by 18 (0 self)
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In this paper we further develop a general theory of metastable states resulting from perturbation of unstable eigenvalues. We apply this theory to manybody Schrodinger operators and to the problem of quasiclassical tunneling. 1.
On the scattering theory of massless Nelson models
, 2001
"... We study the scattering theory for a class of nonrelativistic quantum field theory models describing a confined nonrelativistic atom interacting with a relativistic bosonic field. We construct invariant spaces H c which are defined in terms of propagation properties for large times and which consi ..."
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Cited by 13 (1 self)
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We study the scattering theory for a class of nonrelativistic quantum field theory models describing a confined nonrelativistic atom interacting with a relativistic bosonic field. We construct invariant spaces H c which are defined in terms of propagation properties for large times and which consist of states containing a finite number of bosons in the region fjxj ctg for t ! 1. We show the existence of asymptotic fields and we prove that the associated asymptotic CCR representations preserve the spaces H c and induce on these spaces representations of Fock type. For these induced representations, we prove the property of geometric asymptotic completeness, which gives a characterization of the vacuum states in terms of propagation properties. Finally we show that a positive commutator estimate imply the asymptotic completeness property, ie the fact that the vacuum states of the induced representations coincide with the bound states of the Hamiltonian.