Nodal domains a la Courant

2004/03/02 by A. Ancona, Alano Ancona, Bernard Helffer +6
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #FOS: Physical sciences #Graph theory and applications #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.math/0403038

arxiv created 2004/03/02 · openalex publication_date 2004/03/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H(\Om₀)=-Δ+V be a Schrödinger operator on a bounded domain \Om₀⊂ \mathbb R^d with Dirichlet boundary conditions. Suppose that the \Om_ℓ (ℓ ∈ \1,...,k\) are some pairwise disjoint subsets of \Om₀ and that H(\Om_ℓ) are the corresponding Schrödinger operators again with Dirichlet boundary conditions. We investigate the relations between the spectrum of H(\Om₀) and the spectra of the H(\Om_ℓ). 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_ℓ to the nodal domains of some eigenfunction of H(Ω₀).

Related

• Nodal domains and spectral minimal partitions
• Nodal count for Dirichlet-to-Neumann operators with potential
• Pleijel's theorem for Schrödinger operators
• Pleijel's theorem for Schrödinger operators with radial potentials
• On nodal domains in Euclidean balls