  Let $H(\Om_0)=-\Delta+V$ be a Schr\"odinger operator on a bounded domain
$\Om_0\subset \mathbb R^d$ with Dirichlet boundary conditions. Suppose that the
$\Om_\ell$ ($\ell \in \{1,...,k\}$) are some pairwise disjoint subsets of
$\Om_0$ and that $H(\Om_\ell)$ are the corresponding Schr\"odinger operators
again with Dirichlet boundary conditions. We investigate the relations between
the spectrum of $H(\Om_0)$ and the spectra of the $H(\Om_\ell)$. 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 that equality is achieved we prove converse results. In particular, we use
potential theoretic methods to relate the $\Om_\ell$ to the nodal domains of
some eigenfunction of $H(\Omega_0)$.
