Pleijel's theorem for Schrödinger operators with radial potentials (after P. Charron, B. Helffer and T. Hoffmann-Ostenhof)
In 1956, Å. Pleijel gave his celebrated theorem showing that the inequality in Courant's theorem on the number of nodal domains is strict for large eigenvalues of the Laplacian. This was a consequence of a stronger result giving an asymptotic upper bound for the number of nodal domains of the eigenfunction as the eigenvalue tends to infinity. A similar question occurs naturally for the case of the Schrödinger operator. The first significant result has been obtained recently by P. Charron for the case of the harmonic oscilllator. The purpose of this talk is to consider more general potentials which are radial. We will analyze either the case when the potential tends to infinity or the case when the potential tends to zero, the considered eigenfunctions being associated with the eigenvalues below the essential spectrum.