214 research outputs found
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
The Dirac-Frenkel Principle for Reduced Density Matrices, and the Bogoliubov-de-Gennes Equations
The derivation of effective evolution equations is central to the study of
non-stationary quantum many-body sytems, and widely used in contexts such as
superconductivity, nuclear physics, Bose-Einstein condensation and quantum
chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of
reduced density matrices, and apply it to fermionic and bosonic many-body
systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov
equations, respectively. While we do not prove quantitative error estimates,
our formulation does show that the approximation is optimal within the class of
quasifree states. Furthermore, we prove well-posedness of the
Bogoliubov-de-Gennes equations in energy space and discuss conserved
quantities.Comment: 46 pages, 1 figure; v2: simplified proof of conservation of particle
number, additional references; v3: minor clarification
On the uniqueness of solutions to the periodic 3D Gross-Pitaevskii hierarchy
In this paper, we present a uniqueness result for solutions to the
Gross-Pitaevskii hierarchy on the three-dimensional torus, under the assumption
of an a priori spacetime bound. We show that this a priori bound is satisfied
for factorized solutions to the hierarchy which come from solutions of the
nonlinear Schr\"{o}dinger equation. In this way, we obtain a periodic analogue
of the uniqueness result on previously proved by Klainerman and
Machedon, except that, in the periodic setting, we need to assume additional
regularity. In particular, we need to work in the Sobolev class
for . By constructing a specific counterexample, we show that, on
, the existing techniques don't apply in the endpoint case
. This is in contrast to the known results in the non-periodic
setting, where the these techniques are known to hold for all .
In our analysis, we give a detailed study of the crucial spacetime estimate
associated to the free evolution operator. In this step of the proof, our
methods rely on lattice point counting techniques based on the concept of the
determinant of a lattice. This method allows us to obtain improved bounds on
the number of lattice points which lie in the intersection of a plane and a set
of radius , depending on the number-theoretic properties of the normal
vector to the plane. We are hence able to obtain a sharp range of admissible
Sobolev exponents for which the spacetime estimate holds.Comment: 42 page
Analyticity and hp discontinuous Galerkin approximation of nonlinear Schr\"odinger eigenproblems
We study a class of nonlinear eigenvalue problems of Scr\"odinger type, where
the potential is singular on a set of points. Such problems are widely present
in physics and chemistry, and their analysis is of both theoretical and
practical interest. In particular, we study the regularity of the
eigenfunctions of the operators considered, and we propose and analyze the
approximation of the solution via an isotropically refined hp discontinuous
Galerkin (dG) method.
We show that, for weighted analytic potentials and for up-to-quartic
nonlinearities, the eigenfunctions belong to analytic-type non homogeneous
weighted Sobolev spaces. We also prove quasi optimal a priori estimates on the
error of the dG finite element method; when using an isotropically refined hp
space the numerical solution is shown to converge with exponential rate towards
the exact eigenfunction. In addition, we investigate the role of pointwise
convergence in the doubling of the convergence rate for the eigenvalues with
respect to the convergence rate of eigenfunctions. We conclude with a series of
numerical tests to validate the theoretical results
A Dirac field interacting with point nuclear dynamics
The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in H\u3c3 with \u3c3 08[1,32[. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities
Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential
49 pagesInternational audienceWe consider the quantum dynamics of many bosons systems in the mean field limit with a singular pair-interaction potential, including the attractive or repulsive Coulombic case in three dimensions. By using a measure transportation technique, we show that Wigner measures propagate along the nonlinear Hartree flow. Such property was previously proved only for bounded potentials in our previous works with a slightly different strategy
Classical and Quantum Mechanical Models of Many-Particle Systems
The topic of this meeting were non-linear partial differential and integro-differential equations (in particular kinetic equations and their macroscopic/fluid-dynamical limits) modeling the dynamics of many-particle systems with applications in physics, engineering, and mathematical biology. Typical questions of interest were the derivation of macro-models from micro-models, the mathematical analysis (well-posedness, stability, asymptotic behavior of solutions), and “to a lesser extent” numerical aspects of such equations. A highlight of this meeting was a mini-course on the recent mathematical theory of Landau damping
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