Talk by Hanne Van Den Bosch
On December 7th, 2022, Prof. Dr. Hanne Van Den Bosch (Santiago de Chile) gave a talk about "A Keller-Lieb-Thirring Inequality for Dirac operators" as part of the research seminar Analysis of the FernUniversität in Hagen.
The classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrödinger equation.
This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equation.
This is joint work with Jean Dolbeault, David Gontier and Fabio Pizzichillo.