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424
Determinants of elliptic pseudo–differential operators
, 1994
"... Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained an explici ..."
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Cited by 48 (1 self)
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Determinants of invertible pseudodifferential operators (PDOs) close to positive selfadjoint ones are defined through the zetafunction regularization. We define a multiplicative anomaly as the ratio det(AB)/(det(A)det(B)) considered as a function on pairs of elliptic PDOs. We obtained an explicit formula for the multiplicative anomaly in terms of symbols of operators. For a certain natural class of PDOs on odddimensional manifolds generalizing the class of elliptic differential operators, the multiplicative anomaly is identically 1. For elliptic PDOs from this class a holomorphic determinant and a determinant for zero orders PDOs are introduced. Using various algebraic, analytic, and topological tools we study local and global properties of the multiplicative anomaly and of the determinant Lie group closely related with it. The Lie algebra for the determinant Lie group has a description in terms of symbols only. Our main discovery is that there is a quadratic nonlinearity hidden in the definition of determinants of PDOs through zetafunctions. The natural explanation of this nonlinearity follows from complexanalytic properties of a new trace functional TR on PDOs of noninteger orders. Using TR we easily reproduce known facts about noncommutative residues of PDOs and obtain several new results. In particular, we describe a structure of derivatives of zetafunctions at zero as of functions on logarithms of elliptic PDOs. We propose several definitions extending zetaregularized determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex orders we introduce a canonical determinant in its natural domain of definition.
Derivation of the GrossPitaevskii Equation for rotating Bose gases
, 2005
"... We prove that the GrossPitaevskii equation correctly describes the ground state energy and corresponding oneparticle density matrix of rotating, dilute, trapped Bose gases with repulsive twobody interactions. We also show that there is 100 % BoseEinstein condensation. While a proof that the GP e ..."
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Cited by 48 (4 self)
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We prove that the GrossPitaevskii equation correctly describes the ground state energy and corresponding oneparticle density matrix of rotating, dilute, trapped Bose gases with repulsive twobody interactions. We also show that there is 100 % BoseEinstein condensation. While a proof that the GP equation correctly describes nonrotating or slowly rotating gases was known for some time, the rapidly rotating case was unclear because the Bose (i.e., symmetric) ground state is not the lowest eigenstate of the Hamiltonian in this case. We have been able to overcome this difficulty with the aid of coherent states. Our proof also conceptually simplifies the previous proof for the slowly rotating case. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.
Central Limit Theorem for Non Linear Filtering and Interacting Particle Systems
 Ann. Appl. Probab
, 1999
"... Several random particle systems approaches were recently suggested to solve numerically non linear filtering problems. The present analysis is concerned with genetictype interacting particle systems. Our aim is to study the fluctuations on path space of such particle approximating models. Keywords ..."
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Cited by 43 (8 self)
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Several random particle systems approaches were recently suggested to solve numerically non linear filtering problems. The present analysis is concerned with genetictype interacting particle systems. Our aim is to study the fluctuations on path space of such particle approximating models. Keywords : Central Limit, Interacting random processes, Filtering, Stochastic approximation. code A.M.S : 60F05, 60G35, 93E11, 62L20. 1 Introduction 1.1 Background and motivations The Non Linear Filtering problem consists in recursively computing the conditional distributions of a non linear signal given its noisy observations. This problem has been extensively studied in the literature and, with the notable exception of the linearGaussian situation or wider classes of models (B`enes filters [2]) optimal filters have no finitely recursive solution (ChaleyatMaurel /Michel [7]). Although Kalman filtering ([26],[29]) is a popular tool in handling estimation problems its optimality heavily depends on...
Charge Deficiency, Charge Transport and Comparison of Dimensions
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1994
"... We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. ..."
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Cited by 41 (0 self)
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We study the relative index of two orthogonal infinite dimensional projections which, in the finite dimensional case, is the difference in their dimensions. We relate the relative index to the Fredholm index of appropriate operators, discuss its basic properties, and obtain various formulas for it. We apply the relative index to counting the change in the number of electrons below the Fermi energy of certain quantum systems and interpret it as the charge deficiency. We study the relation of the charge deficiency with the notion of adiabatic charge transport that arises from the consideration of the adiabatic curvature. It is shown that, under a certain covariance, (homogeneity), condition the two are related. The relative index is related to Bellissard's theory of the Integer Hall effect. For Landau Hamiltonians the relative index is computed explicitly for all Landau levels.
Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices
 Comm. Math. Phys
, 1983
"... Abstract. We consider the integrated density of states, k(E) 9 of a general operator on / 2(Z V) of the form h = hQ + v, where (h0u)(n) = Σ u ( n + 0 ancl lϋ = ι (vu)(n) = υ(n)u(n \ where v is a general bounded ergodic stationary process on Z v. We show that \k(E) k(E'} \ g C[ log( £ E&ap ..."
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Cited by 41 (1 self)
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Abstract. We consider the integrated density of states, k(E) 9 of a general operator on / 2(Z V) of the form h = hQ + v, where (h0u)(n) = Σ u ( n + 0 ancl lϋ = ι (vu)(n) = υ(n)u(n \ where v is a general bounded ergodic stationary process on Z v. We show that \k(E) k(E'} \ g C[ log( £ E'\Y l when \E ~ £'  ^i The key is a "Thouless formula for the strip." 1.
A new approach to the modelling of local defects in crystals: the reduced HartreeFock case
 Commun. Math. Phys
"... Abstract. This article is concerned with the derivation and the mathematical study of a new meanfield model for the description of interacting electrons in crystals with local defects. We work with a reduced HartreeFock model, obtained from the usual HartreeFock model by neglecting the exchange t ..."
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Cited by 38 (13 self)
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Abstract. This article is concerned with the derivation and the mathematical study of a new meanfield model for the description of interacting electrons in crystals with local defects. We work with a reduced HartreeFock model, obtained from the usual HartreeFock model by neglecting the exchange term. First, we recall the definition of the selfconsistent Fermi sea of the perfect crystal, which is obtained as a minimizer of some periodic problem, as was shown by Catto, Le Bris and Lions. We also prove some of its properties which were not mentioned before. Then, we define and study in detail a nonlinear model for the electrons of the crystal in the presence of a defect. We use formal analogies between the Fermi sea of a perturbed crystal and the Dirac sea in Quantum Electrodynamics in the presence of an external electrostatic field. The latter was recently studied by Hainzl, Lewin, Séré and Solovej, based on ideas from Chaix and Iracane. This enables us to define the ground state of the selfconsistent Fermi sea in the presence of a defect.
Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach
, 2004
"... complex approach: the case with boundary ..."