Density Functional Theory (DFT)
attempts to describe all the relevant information
about the ground state of a many-body quantum system
in terms of its one-body reduced density. It is
widely and successfully used in practice for
computations in quantum chemistry. In these
lectures, we shall explain the mathematical
structure of DFT, and review known bounds on the
universal density functional (including the
Lieb-Thirring Inequality for the kinetic energy, and
the Lieb-Oxford Inequality for the electrostatic
energy). We shall also outline recent results
concerning the validity of the local density
approximation, and the equivalence of various
possible formulations of the uniform electron gas.
Much (but not all) of the material of the lectures
is contained in the review paper
https://arxiv.org/pdf/1912.10424
The structure of complex atoms and
understanding the periodic table of the elements
Lecture 1: We will
discuss the phenomenology of the structure of
atoms, i.e., their size, ionization energy, and
maximal negative ionization. We will also discuss
the periodic table and the way it structures,
i.e., the Aufbau principle or what is also
referred to as the Madelung Rule. A recurrent
question is whether we can give a mathematical
explanation of the pattern seen in the structure
of atoms and the periodic table. We end Lecture 1
with describing the full many-body
(non-relativistic) quantum description of an atom
and the mathematical definition of the total
energy size, ionization energy, and maximal
negative ionization of an atom. We formulate the
ionization conjecture, i.e., uniform bounds on the
size and maximal ionization for heavy atoms.
Lecture 2: We discuss
different simplified models, e.g., the
Hartree-Fock model, the reduced Hartree-Fock
model, and the Thomas-Fermi model. We quantify
the approximation of the total energy in the
limit of heavy atoms.
Lecture 3: We analyze
the Thomas-Fermi model in greater details and
give the exact total energy, size, and maximal
ionization in this model. We show how that the
accuracy of the Thomas-Fermi model as an
approximation relies on a semiclassical
analysis.
Lecture 4: We give more
details of the semiclassical analysis and show
that the phenomenological Aufbau principle fails
for large atoms. It should be replaced by
another principle the “Fermi Aufbau formula”
Lecture 5: We describe a
model, the Thomas-Fermi Mean-field model, in
which there is an exact asymptotic periodicity
for heavy atoms. We describe the “infinite
periodic table”. If time permits, we sketch a
proof of the ionization conjecture for the
Hartree-Fock model
Slides
Stefan Teufel
Mathematical aspects of quantum Hall
physics in microscopic models of interacting
fermions
The quantum Hall effect
(QHE) refers to the fact that the Hall
conductance of a two-dimensional electron gas
takes on only quantised values, i.e. integer
(or fractional) multiples of e^2/h. First
discovered in 1980, it has sparked
significant experimental, theoretical and
mathematical research, with Nobel Prizes being
awarded to both experimentalists and
theorists. The effect has also inspired a
mathematical problem, the solution to which
earned its author a Fields Medal. In this
lecture series, I will discuss recent
developments that demonstrate how certain
experimentally relevant features of the QHE
can be rigorously understood within
microscopic models of interacting
fermions. Most importantly, I will
present results that establish a rigorous
Ohm’s law for macroscopic Hall currents in
interacting, spectrally gapped systems at zero
temperature.
Slides: Lecture
1,
Lecture
2,
Lecture
3,
Lectures
4 and 5