Upper bounds on eigenvalue multiplicities for surfaces of genus $0$ revisited (Work in progress)

Pierre Bérard; Bernard Helffer

Pré-Publication, Document De Travail Année : 2022

Upper bounds on eigenvalue multiplicities for surfaces of genus $0$ revisited (Work in progress)

(1) , (2)

1
2

Pierre Bérard

Fonction : Auteur
PersonId : 6509
IdHAL : pierrehberard

Institut Fourier

Bernard Helffer

Fonction : Auteur
PersonId : 1021621

Laboratoire de Mathématiques Jean Leray

Résumé

We revisit two 1999 papers: [1]~ \emph{M.~Hoffmann-Ostenhof, T.~Hoffmann-Ostenhof, and N.~Nadirashvili. On the multiplicity of eigenvalues of the Laplacian on surfaces. Ann. Global Anal. Geom. 17 (1999) 43--48} ~~and~~ [2]~ \emph{T.~Hoffmann-Ostenhof, P.~Michor, and N.~Nadirashvili. Bounds on the multiplicity of eigenvalues for fixed membranes. Geom. Funct. Anal. 9 (1999) 1169--1188}.\\ The main result of these papers is that the multiplicity $\mathrm{mult}(\lambda_k(M))$ of the $k$th eigenvalue of the surface $M$ is bounded from above by $(2k-3)$ provided that $k \geq 3$. Here, $M$ is [1] a closed surface with genus $0$, or [2] a planar domain with Dirichlet boundary condition (in both cases, the starting label of eigenvalues is $1$). The proofs given in [1,2] are not very detailed, and often rely on figures or special configurations of nodal sets. In the case of closed surfaces with genus $0$, we provide full details of the proof that $\mathrm{mult}(\lambda_k) \leq (2k-3)$ by introducing and carefully studying the \emph{combinatorial type} (defined in Subsection~5.2) of the nodal sets involved. In the case of planar domains, we consider the three boundary conditions, Dirichlet, Neumann, Robin, and we also carefully study the \emph{combinatorial types} of the nodal sets involved. We provide full details of the proof of the inequality $\mathrm{mult}(\lambda_k) \leq (2k-2)$ ($k \geq 3$). Unfortunately, we are so far unable to complete the proof of the inequality $\mathrm{mult}(\lambda_k) \leq (2k-3)$, even when the domain is simply connected. In the last section, we explain the difficulties we met. Our work on the subject is still ongoing.

Mots clés

Spectral theory Eigenvalue multiplicity Laplacian Partitions

Domaines

Equations aux dérivées partielles [math.AP] Théorie spectrale [math.SP] Géométrie différentielle [math.DG] Physique mathématique [math-ph]

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Soumis le : mardi 15 février 2022-15:08:45

Dernière modification le : mercredi 18 décembre 2024-10:14:56

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hal-03575434 , version 1 (15-02-2022)

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Pierre Bérard, Bernard Helffer. Upper bounds on eigenvalue multiplicities for surfaces of genus $0$ revisited (Work in progress). 2022. ⟨hal-03575434⟩

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Upper bounds on eigenvalue multiplicities for surfaces of genus $0$ revisited (Work in progress)

Résumé

Mots clés

Domaines

Dates et versions

Licence

Identifiants

Citer

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Collections

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