Pleijel's theorem for Schr\"odinger operators

We are concerned in this paper with the real eigenfunctions of Schr"odinger operators. We prove an asymptotic upper bound for the number of their nodal domains, which implies in particular that the inequality stated in Courant’s theorem is strict, except for finitely many eigenvalues. Results of this type originated in 1956 with Pleijel’s Theorem on the Dirichlet Laplacian and were obtained for some classes of Schr"odinger operators by the first author, alone and in collaboration with B. Helffer and T. Hoffmann-Ostenhof. Using methods in part inspired by work of the second author on Neumann and Robin Laplacians, we greatly extend the scope of these previous results.