Geometric methods for nonlinear many-body quantum systems

Abstract

Geometric techniques have played an important role in the seventies, for the study of the spectrum of many-body Schrödinger operators. In this paper we provide a formalism which also allows to study nonlinear systems. We start by defining a weak topology on many-body states, which appropriately describes the physical behavior of the system in the case of lack of compactness, that is when some particles are lost at infinity. We provide several important properties of this topology and use them to provide a simple proof of the famous HVZ theorem in the repulsive case. In a second step we recall the method of geometric localization in Fock space as proposed by Dereziński and Gérard, and we relate this tool to our weak topology. We then provide several applications. We start by studying the so-called finite-rank approximation which consists in imposing that the many-body wavefunction can be expanded using finitely many one-body functions. We thereby emphasize geometric properties of Hartree-Fock states and prove nonlinear versions of the HVZ theorem, in the spirit of works of Friesecke. In the last section we study translation-invariant many-body systems comprising a nonlinear term, which effectively describes the interactions with a second system. As an example, we prove the existence of the multi-polaron in the Pekar-Tomasevich approximation, for certain values of the coupling constant.

Final version to appear in Journal of Functional Analysis

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Cited by in corpus (61)

• The Local Density Approximation in Density Functional Theory
• Kinetic Energy Estimates for the Accuracy of the Time-Dependent Hartree-Fock Approximation with Coulomb Interaction
• Statistical Mechanics of the Uniform Electron Gas
• Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystals
• Mean-field limit of Bose systems: rigorous results
• On the dynamics of polarons in the strong-coupling limit
• Higher order Schrodinger and Hartree-Fock equations
• Universal Functionals in Density Functional Theory
• Local incompressibility estimates for the Laughlin phase
• Classical field theory limit of many-body quantum Gibbs states in 2D and 3D
• Building Kohn-Sham potentials for ground and excited states
• On the Magnetic Pekar Functional and the Existence of Bipolarons
• Hohenberg-Kohn theorems for interactions, spin and temperature
• The nonlinear Schrödinger equation for orthonormal functions: II. Application to Lieb-Thirring inequalities
• Ground state energy of mixture of Bose gases
• The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states
• Coupled-Cluster Theory Revisited. Part II: Analysis of the single-reference Coupled-Cluster equations
• Existence of Hartree-Fock excited states for atoms and molecules
• The ground state energy of the multi-polaron in the strong coupling limit
• Some contributions to many-body quantum mathematics
• Dynamical Ionization Bounds for Atoms
• Eigenvalue asymptotics for the one-particle kinetic energy density operator
• A Numerical Perspective on Hartree-Fock-Bogoliubov Theory
• Symmetry of bipolaron bound states for small Coulomb repulsion
• Some properties of the potential-to-ground state map in quantum mechanics
• Bogoliubov spectrum of interacting Bose gases
• Binding of Polarons and Atoms at Threshold
• Derivation of Hartree's theory for generic mean-field Bose systems
• Ground state energy is not always convex in the number of electrons
• Théorèmes de de Finetti, limites de champ moyen et condensation de Bose-Einstein
• Derivation of renormalized Gibbs measures from equilibrium many-body quantum Bose gases
• Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Scaling limits of bosonic ground states, from many-body to nonlinear Schr{ö}dinger
• Orthonormal bases of extreme quantumness
• The semi-classical limit of large fermionic systems
• A new definition of the Dirac-Fock ground state
• Classical Density Functional Theory: The Local Density Approximation
• Differentiability of the van der Waals interaction between two atoms
• The Strong-Coupling Polaron in Electromagnetic Fields
• The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
• Validity of Bogoliubov's approximation for translation-invariant Bose gases
• The Validity of the Local Density Approximation for Smooth Short Range Interaction Potentials
• Classical Density Functional Theory: Representability and Universal Bounds
• Grand-Canonical Optimal Transport
• The Lieb--Thomas strategy for strongly coupled fermionic multipolarons with general external fields
• Regularity of the (N-1)-particle electronic reduced density matrix for molecules with fixed nuclei and N electrons
• Analysis of fluctuations around non linear effective dynamics
• Charge renormalisation in a mean-field approximation of QED
• Remarks on the quantum de Finetti theorem for bosonic systems
• On the Strong Coupling Limit of Many-Polaron Systems in Electromagnetic Fields
• A many-body RAGE theorem
• Differential Equations of Quantum Mechanics
• Coupled-Cluster Theory Revisited. Part I: Discretization
• Asymptotics for Two-dimensional Atoms
• Absence of a Ground State for Bosonic Coulomb Systems with Critical Charge
• Convergence of states for polaron models in the classical limit
• Blow-up profile of neutron stars in the Hartree-Fock-Bogoliubov theory
• Weyl laws for interacting particles
• Absence of binding in a mean-field approximation of quantum electrodynamics
• Asymptotics of the Kantorovich Potential for the Optimal Transport with Coulomb Cost
• Mixed state representability of entropy-density pairs