Nodal domain theorems à la Courant

Abstract

Let $H({\Omega }_0)=-\Delta +V$ be a Schrödinger operator on a bounded domain ${\Omega }_0\subset {\mathbb{R}}^d(d\ge 2)$ with Dirichlet boundary condition. Suppose that ${\Omega }_{\ell }(\ell \in \{1,\dots ,k\})$ are some pairwise disjoint subsets of ${\Omega }_0$ and that $H({\Omega }_{\ell })$ are the corresponding Schrödinger operators again with Dirichlet boundary condition. We investigate the relations between the spectrum of $H({\Omega }_0)$ and the spectra of the $H({\Omega }_{\ell })$. In particular, we derive some inequalities for the associated spectral counting functions which can be interpreted as generalizations of Courant's nodal theorem. For the case where equality is achieved we prove converse results. In particular, we use potential theoretic methods to relate the ${\Omega }_{\ell }$ to the nodal domains of some eigenfunction of $H({\Omega }_0)$.