Research Interests: The Mathematics of Quantum Systems

Topics
Description
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Topics

 

 

Nonlinear equations in quantum chemistry

Phase space bounds and eigenvalue asymptotics for Schrödinger operators

Spectral and scattering properties near thresholds for scalar-valued and matrix-valued Schrödinger operators
Resonances in Quantum Chemistry

Spectral and scattering properties of quantum wires

Spectral and scattering properties for Schrödinger and Dirac operators with a constant magnetic field

 

 

General description

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In my scientific research I study problems in mathematical analysis and functional analysis, which arise in mathematical physics. I focus on spectral theory and scattering theory of differential operators and integral operators with applications to quantum mechanics. A complete understanding of Schrödinger operators is necessary in order to obtain a thorough and satisfactory description of non-relativistic quantum mechanics. Therefore the rigorous study of Schrödinger operators has become a highly developed mathematical discipline during the past 50 years. The mathematical foundation of quantum mechanics is extremely complex and has branches into many areas of mathematics. Basically it deals with the spectral and scattering theory of unbounded operators in Hilbert spaces. These operators represent physical observables, such as position, momentum or energy. For instance, Schrödinger operators represent energy observables for various physical systems. The study of Schrödinger operators includes:

  • Scattering theory.

    The purpose is to describe collision experiments in physics in a rigorous mathematical way and clarify how the physical system evolves for large times. Since the collision experiment is probably the single most important experiment in physics, it is clear that the theoretical study is one of the main topics of mathematical physics.

  • Spectral theory

    Concerns the study of the spectra of the observables, e.g. the spectrum of the Schrödinger operator consisting of the allowed energies of the quantum system. In particular, the negative eigenvalues of the Schrödinger operator are important, since these eigenvalues corresponds to bound states, e.g. bound states of an atom or a molecule. The distribution of these eigenvalues has significant impact on the physical properties of the system. Consequently, their study is a central topic in mathematical physics.

I am interested in both kinds of studies; in practice, these studies intertwine in subtle ways.

In the past 20 years Schrödinger operators with external magnetic fields have attracted substantial attention and each of my works has been motivated by a problem within this area, if not dealt explicitly with such a problem. In one of my latest works, where I study Dirac operators with magnetic fields, relativistic effects are taken into account.