Critical partitions and nodal deficiency of billiard eigenfunctions

Abstract

The paper addresses the the number of nodal domains for eigenfunctions of Schrödinger operators with Dirichlet boundary conditions in bounded domains. In dimension one, the $n$th eigenfunction has $n$ nodal domains. The Courant Theorem claims that in any dimension, the number of nodal domains of the $n$th eigenfunction cannot exceed $n$. However, in dimensions higher than 1 the equality can hold for only finitely many eigenfunctions. Thus, a "nodal deficiency" arises. Examples are known of eigenfunctions with arbitrarily large index $n$ that have just two nodal domains. It was suggested in the recent years to look at the partitions of the domain, rather than eigenfunctions. It was shown in a recent paper by Helffer, Hoffmann-Ostenhof and Terracini that (under some natural conditions) bipartite partitions minimizing the maximum of the ground-state energies in sub-domains of the partition, correspond to the "Courant sharp" eigenfunctions, i.e. to those with zero nodal deficiency. In this paper, the authors show, under some genericity conditions, among the bipartite equipartitions, the nodal ones correspond exactly to the critical points of an analogous functional, with the nodal deficiency being equal to the Morse index at this point. This explains, in particular, why all the minimal partitions must be Courant sharp.

In the 2nd version minor modifications were implemented

References in corpus (7)

• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal count of graph eigenfunctions via magnetic perturbation
• Magnetic interpretation of the nodal defect on graphs
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• Counting nodal domains on surfaces of revolution
• Inverse Nodal Problems
• $C^{\infty }$ spectral rigidity of the ellipse

Cited by in corpus (16)

• Geometrical structure of Laplacian eigenfunctions
• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• Nodal count of graph eigenfunctions via magnetic perturbation
• Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Stability of nodal structures in graph eigenfunctions and its relation to the nodal domain count
• Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
• Courant-sharp eigenvalues of Neumann 2-rep-tiles
• Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
• Spectral shift via "lateral" perturbation
• Quantum graphs -- Generic eigenfunctions and their nodal count and Neumann count statistics
• Stability of spectral partitions and the Dirichlet-to-Neumann map
• Isospectral discrete and quantum graphs with the same flip counts and nodal counts
• Computing nodal deficiency with a refined Dirichlet-to-Neumann map
• Topics in elliptic problems: from semilinear equations to shape optimization
• First-order asymptotic perturbation theory for extensions of symmetric operators