Geodesic
Nodal domain theorems à la Courant
Documenta mathematica, Tome 9 (2004), pp. 283-299.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: Let $H(\Om_0)=-\Delta+V$ be a Schrödinger operator on a bounded domain $\Om_0\subset \mathbb R^d (d\geq 2)$ with Dirichlet boundary condition. Suppose that $\Om_\ell (\ell \in \{1,\dots,k\})$ are some pairwise disjoint subsets of $\Om_0$ and that $H(\Om_\ell)$ are the corresponding Schrödinger operators again with Dirichlet boundary condition. 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 where 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 $H(\Omega_0)$.
Classification : 35B05 
@article{DOCMA_2004__9__a14,
     author = {Ancona, A. and Helffer, B. and Hoffmann-Ostenhof, T.},
     title = {Nodal domain theorems \`a la {Courant}},
     journal = {Documenta mathematica},
     pages = {283--299},
     publisher = {mathdoc},
     volume = {9},
     year = {2004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2004__9__a14/}
}
TY  - JOUR
AU  - Ancona, A.
AU  - Helffer, B.
AU  - Hoffmann-Ostenhof, T.
TI  - Nodal domain theorems à la Courant
JO  - Documenta mathematica
PY  - 2004
SP  - 283
EP  - 299
VL  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DOCMA_2004__9__a14/
LA  - en
ID  - DOCMA_2004__9__a14
ER  - 
%0 Journal Article
%A Ancona, A.
%A Helffer, B.
%A Hoffmann-Ostenhof, T.
%T Nodal domain theorems à la Courant
%J Documenta mathematica
%D 2004
%P 283-299
%V 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DOCMA_2004__9__a14/
%G en
%F DOCMA_2004__9__a14
Ancona, A.; Helffer, B.; Hoffmann-Ostenhof, T. Nodal domain theorems à la Courant. Documenta mathematica, Tome 9 (2004), pp. 283-299. http://geodesic.mathdoc.fr/item/DOCMA_2004__9__a14/