Workshop on Spectral Theory

• 11:15 Uhr, Raum 7.530 - Dr. Helmut Linde, Covestro Deutschland AG

      Vortragstitel: An improved bound for the ground state energy of a Schrödinger operator on a loop

       Abstract:  Consider a closed curve of length $2\pi$ with curvature $\kappa(s)$ and the Schrödinger operator $H$ with $\kappa^2$ as the potential term. Let $\lambda_\Gamma$ be the lowest eigenvalue of $H$. The question about the minimal possible value of $\lambda_\Gamma$ has attracted attention since the year 2004, when Benguria and Loss established a connection between this problem and the Lieb-Thirring conjecture in one dimension. The also conjectured that $\lambda_\Gamma \ge 1$, but a proof of this estimate has not been found yet. In the talk I will show how to establish a new lower bound of $0.81$ on $\lambda_\Gamma$, improving on the best previous estimate of approximately $0.60$.  

• Mittagessen, Kaffee und Diskussion

• 14:00 Uhr, Raum 7.527 - Prof. Dr. Rupert Frank, LMU München

    Vortragstitel: Riesz means for some explicitly solvable operators and applications to LT inequalities

     Abstract: Recently, Carvalho Corso, Weidl and Zeng have shown that the semiclassical approximation provides an upper bound for Riesz means of order $\gamma\geq 1$ for the Coulomb Hamiltonian. We show that their restriction $\gamma\geq 1$ is optimal in dimensions $d\geq 4$ and we present both positive and negative partial results in dimensions 2 and 3. Our strategy is applicable to other examples of operators with explicitly computable spectrum and provides, among other things, a partially alternative proof of the Helffer—Robert result.

The talk is based on joint work with Bernard Helffer and Francois Nicoleau.

Prof. Dr. Rupert Frank, LMU München

Prof. Dr. Timo Weidl, Universität Stuttgart

Die Veranstaltung wird unterstützt durch den Sonderforschungsbereich: 

CRC TRR 352: Mathematics of Many-Body Quantum Systems and Their Collective Phenomena

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