Quantum Mechanics
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Measure-valued weak solutions to some kinetic equations with singular kernels for quantum particles
(2018-12-19)In this thesis, we present a mathematical study of three problems arising in the kinetic theory of quantum gases. In the first part, we consider a Boltzmann equation that is used to describe the time evolution of the ... -
On a Boltzmann equation for Compton scattering, from non relativistic electrons at low density.
(2018-08-15)A Boltzmann equation, used to describe the Compton scattering in the non-relativistic limit is considered. A truncation of the very singular redistribution function is introduced and justified. The existence of weak solutions ... -
On a system of equations for the normal fluid - condensate interaction in a Bose gas
(2018-03-27)The existence of global solutions for a system of differential equations is proved, and some of their properties are described. The system involves a kinetic equation for quantum particles. It is a simplified version of ... -
Existence of “$d$-wave” Pairs and Density Waves in a Class of Microscopic Models for High Transition Temperature Superconductors
(2018-03-21)High-temperature superconductors have different properties than conventional superconductors, one of these important properties is non-isotropic symmetry of the order parameter. In this work we present a model that shows ... -
Universal bounds for large determinants from non-commutative Hölder inequalities in fermionic constructive quantum field theory
(Mathematical Models and Methods in Applied Sciences (M3AS), 2017-08-02)Efficiently bounding large determinants is an essential step in non-relativistic fermionic constructive quantum field theory to prove the absolute convergence of the perturbation expansion of correlation functions in terms ... -
Isotropic Bipolaron-Fermion-Exchange Theory and Unconventional Pairing in Cuprate Superconductors
(2017-05-03)The discovery of high-temperature superconductors in 1986 represented a major experimental breakthrough (Nobel Prize 1987), but their theoretical explanation is still a subject of much debate. These materials have many ... -
Geometric inequalities from phase space translations
(2016-07-22)We establish a quantum version of the classical isoperimetric inequality relating the Fisher information and the entropy power of a quantum state. The key tool is a Fisher information inequality for a state which results ... -
D-Wave pairing driven by bipolaric modes related to giant electron-phonon anomalies in high-Tc superconductors
(Journal of Statistical Mechanics: Theory and Experiment, 2015-12-31)Taking into account microscopic properties of most usual high-Tc superconductors, like cuprates, we define a class of microscopic model Hamiltonians for two fermions (electrons or holes) and one boson (bipolaron) on the ... -
Heat production of noninteracting fermions subjected to electric fields
(Communications on Pure and Applied Mathematics, 2014-07-21)Electric resistance in conducting media is related to heat (or entropy) production in the presence of electric fields. In this paper, by using Araki's relative entropy for states, we mathematically define and analyze the ... -
From the 2nd Law of Thermodynamics to AC–Conductivity Measures of Interacting Fermions in Disordered Media
(Mathematical Models and Methods in Applied Sciences, 2015-05-20)We study the dynamics of interacting lattice fermions with random hopping amplitudes and random static potentials, in presence of time-dependent electromagnetic fields. The interparticle interaction is short-range and ... -
AC-Conductivity Measure from Heat Production of Free Fermions in Disordered Media
(Archive for Rational Mechanics and Analysis, 2015-12-31)We extend (Bru et al. in J Math Phys 56:051901-1-51, 2015) in order to study the linear response of free fermions on the lattice within a (independently and identically distributed) random potential to a macroscopic electric ... -
Characterization of the Quasi-Stationary State of an Impurity Driven by Monochromatic Light II: Microscopic Foundations
(Annales Henri Poincare, 2015-12-31)From quantum mechanical first principles only, we rigorously study the time-evolution of a N-level atom (impurity) interacting with an external monochromatic light source within an infinite system of free electrons at ... -
Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles
(Letters in Mathematical Physics, 2016-01-01)We apply Lieb–Robinson bounds for multi-commutators we recently derived (Bru and de Siqueira Pedra, Lieb–Robinson bounds for multi-commutators and applications to response theory, 2015) to study the (possibly non-linear) ... -
Diagonalizing quadratic bosonic operators by non-autonomous flow equations volker bach
(Memoirs of the American Mathematical Society, 2016-01-01)We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics ... -
Microscopic Conductivity of Lattice Fermions at Equilibrium. Part II: Interacting Particles
(Letters in Mathematical Physics, 2015-12-31)We apply Lieb–Robinson bounds for multi-commutators we recently derived (Bru and de Siqueira Pedra, Lieb–Robinson bounds for multi-commutators and applications to response theory, 2015) to study the (possibly non-linear) ... -
Lieb–Robinson Bounds for Multi–Commutators and Applications to Response Theory
(2016-01-01)We generalize to multi–commutators the usual Lieb–Robinson bounds for commutators. In the spirit of constructive QFT, this is done so as to allow the use of combinatorics of minimally connected graphs (tree expan- sions) ... -
Microscopic conductivity of lattice fermions at equilibrium. I. Non-interacting particles
(Journal of Mathematical Physics, 2015-12-31)We consider free lattice fermions subjected to a static bounded potential and a timeand space-dependent electric field. For any bounded convex region R ⊂ ℠(d (d ≥ 1) of space, electric fields ε within R drive currents. ... -
Universal Bounds for Large Determinants from Non–Commutative Ho ̈lder Inequalities in Fermionic Constructive Quantum Field Theory
(2016-01-01)Efficiently bounding large determinants is an essential step in non–relati- vistic fermionic constructive quantum field theory, because, together with the summability of the interaction and the covariance, it implies the ... -
Kinetic energy estimates for the accuracy of the time-dependent Hartree-Fock approximation with Coulomb interaction
(Journal des Mathematiques Pures et Appliquees, 2016-01-01)We study the time evolution of a system of $N$ spinless fermions in $\mathbb{R}^3$ which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation ... -
Macroscopic conductivity of free fermions in disordered media
(Reviews in Mathematical Physics, 2014-12-31)We conclude our analysis of the linear response of charge transport in lattice systems of free fermions subjected to a random potential by deriving general mathematical properties of its conductivity at the macroscopic ...
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