PhD Defense Sabiha Sibel Tokus

Title: Variational Methods for Quantum Hamiltonians

This thesis is concerned with variational methods in two different settings within the field of mathematical quantum theory.

The first part is based on a project about self-adjoint extensions of a certain type of operators that satisfy a gap condition. A famous example of such a gapped operator is the Dirac operator in three dimensions with a Coulomb potential term. Operators with such a gap condition can be seen as a generalisation of lower semibounded operators. For the latter there is known to exist a distinguished self-adjoint extension, the Friedrichs extension.

We present a distinguished self-adjoint extension for operators with a gap as a generalisation of the Friedrichs extension and formulate a corresponding variational principle, a min-max principle, which determines the eigenvalues of the operator in the gap.

For Dirac operators on a bounded cylinder we can relate this extension to the famous Atiyah—Patodi--Singer boundary conditions.

The second part is concerned with Bogoliubov’s well-known approximation theory for systems of interacting bosons. More specifically, we investigate a variational reformulation of Bogoliubov’s theory which recently was proposed in the literature. We test this approximation scheme by applying it to the Lieb—Liniger model of a one-dimensional gas of bosons interacting via a delta-potential, for which we compute the ground state energy as an expansion for weak and strong coupling.

Link to the thesis

Advisor: Jan Philip Solovej, Math, University of Copenhagen.

Assessment committee:
Prof. chair: Bergfinnur Durhuus, Math, University of Copenhagen
Directrice de recherche, Maria J. Esteba, CNRS
Prof. Horia D. Cornean, Aalborg Universitet