Pleijel’s theorem for Schrödinger operators with radial potentials

Published in Pavillon André-Aisenstadt, room AA-5183, 2018

The study of nodal domains for eigenfunctions of the Laplacian has garnered some attention over the years. First, Courant proved in 1923 that the $n$-th eigenfunction cannot have more than $n$ nodal domains. Then, Pleijel proved in 1956 that the $n$-th eigenfunction has at most $C(d) n + o(n)$ nodal domains with $C(d) < 1$ depending only on the dimension. This holds for the Laplacian with Dirichlet boundary condition. This was extended to smooth manifolds with or without bondary par Berard and Meyer in 1982, and to the Laplacian with Neumann boundary condition by Polterovich in 2009 and Léna in 2016. In this talk we will generalize these results for a large class of Schrödinger operators with radial potentials in $R^d$. This is based on joint work with Bernard Helffer and Thomas Hoffmann-Ostenhof.