Cardiff University | Prifysgol Caerdydd ORCA
Online Research @ Cardiff 
WelshClear Cookie - decide language by browser settings

Critical partitions and nodal deficiency of billiard eigenfunctions

Berkolaiko, Gregory, Kuchment, Peter and Smilansky, Uzy 2012. Critical partitions and nodal deficiency of billiard eigenfunctions. Geometric and Functional Analysis 22 (6) , pp. 1517-1540. 10.1007/s00039-012-0199-y

Full text not available from this repository.

Abstract

The paper addresses the the number of nodal domains for eigenfunctions of Schr\"{o}dinger 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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: Springer Verlag
ISSN: 1016-443X
Last Modified: 19 Mar 2016 22:57
URI: http://orca.cf.ac.uk/id/eprint/33115

Citation Data

Cited 11 times in Google Scholar. View in Google Scholar

Cited 8 times in Scopus. View in Scopus. Powered By Scopus® Data

Actions (repository staff only)

Edit Item Edit Item