A. Ancona, B. Helffer, T. Hoffmann-Ostenhof
Nodal Domain Theorems a la Courant
Preprint series: ESI preprints

MSC:

35B05 General behavior of solutions of PDE (comparison theorems; oscillation, zeros and growth of solutions; mean value theorems)

Abstract: 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 $\Om_\ell$ ($\ell \in \{1,
\dots,k\}$) 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 $H(\Omega_0)$.