The course will be in English, and will be self-contained, that is: for mathematicians it is not required previous knowledge of physics and for physicists and chemists is required only a basic knowledge of functional analysis.
it will be structured as follows:
The starting point is the mathematical model of electrons and nuclei.
The first step is the mathematical justification of why ordinary matter of electrons and nuclei can exist at all. This will be shown through the Kato theorem for the lower bound of the Hamiltonian operator (which will be previously defined) and will introduce the concept of "closure" of an operator.
Next we will discuss the mathematical necessary condition for the ordinary matter to exist in the form a chemist or a physicist would see, that is forming separated structures as we see around.This will be done by proving the theorem of Lieb and Thirring about stability of matter as a function of the number of electrons and nuclei (with its corresponding notions in functional analysis, e.g. Sobolev spaces and Sobolev inequalities).
The previous theorem sets a necessary condition , however this is not sufficient to prove that ordinary matter can exist in the form we see it around us. Thus the necessary condition of the previous section will be used to show the so called Lieb-Lebowitz theorem of existence of a finite thermodynamic limit, that is an exact description of why ordinary matter can build arbitrarily large systems as observed by chemists and physicists. The central core of this proof is a theorem of topology known as packing theorem for spheres, and concerns how to fully pack space by spheres of different sizes.
These first three subjects justify the use of the mathematical model of matter with electron and nuclei, however does not tell us how to solve the operator/Hamiltonian problem in Hilbert space. The problem of an explicit solution for electrons and nuclei in Hilbert space is monumental, however in the spirit of applied functional analysis we will discuss the approach of Density Functional Theory, that is the most successful approach for real calculations.
We will discuss the Hohenberg-Kohn theorem and the corresponding mathematical generalization of the Levy-Lieb constrained-search formulation and have an overview on how computational chemists have then transformed a complex functional minimization into a relatively simple self-consistent procedure, i.e. the Kohn-Sham method.
Finally, for the practical section we will have two open problems of current interest in functional analysis for molecular modeling. Working as single or in group, each student will try to contribute with mathematical as well as physical ideas to a possible solution of the open problem and write a brief report about her/his results.
(1) T.Kato, Transactions of the American Mathematical Society Vol. 70, No. 2 (Mar., 1951)
(2) Elliott H. Lieb, Rev. Mod. Phys. 48, 553, Published 1 October 1976
(3) Elliott H. Lieb and Joel L. Lebowitz, Advances in Mathematics 9, 316-398 (1972)
(4) Parr R. and Yang W. Density Functional Theory of Atoms and Molecules, Oxford Press, 1985
Interested students should send me a message before starting of the semester:
Luigi Delle Site