Oscar Mickelin: On Spectral Inequalities in Quantum Mechanics and Conformal Field Theory

Following Exner et al. (Commun. Math. Phys. 26 (2014), no. 2, 531–541), this thesis proves new Lieb-Thirring inequalities for a general class of self-adjoint, second order differential operators with matrix-valued potentials, acting in one space-dimension. This class contains, but is not restricted to, the magnetic and non-magnetic Schrödinger operators. We consider the three cases of functions defined on all reals, all positive reals, and an interval, respectively, and acquire three different kinds of bounds.
The thesis also investigates the spectral properties of a family of operators from conformal field theory, by proving an asymptotic phase-space bound on the eigenvalue counting function and establishing a number of spectral inequalities. These bound the Riesz-means of eigenvalues for these operators, together with each individual eigenvalue, and are applied to a few physically interesting examples.

The seminar will give an introduction to the stability of matter problem, highlighting the use of the Lieb-Thirring inequalities. We then summarize the generalized Lieb-Thirring inequalities acquired, with full details available in the thesis.