Leydold, Josef (1993) On the number of nodal domains of spherical harmonics. Preprint Series / Department of Applied Statistics and Data Processing, 6. Institut für Statistik und Mathematik, Abt. f. Angewandte Statistik u. Datenverarbeitung, WU Vienna University of Economics and Business, Vienna.

PDF
document.pdf Download (520kB) 
Abstract
It is well known that the nth eigenfunction to onedimensional SturmLiouville eigenvalue problems has exactly n1 nodes, i.e. nondegenerate zeros. For higher dimensions, it is much more complicated to obtain general statements on the zeros of eigenfunctions. The author states a new conjecture on the number of nodal domains of spherical harmonics, i.e. of connected components of S^2 \ N(u) with the nodal set N(u) = (x in S^2 : u(x) = 0) of the eigenfunction u, and proves it for the first six eigenvalues. It is a sharp upper bound, thus improving known bounds as the Courant nodal domain theorem, see S. Y. Cheng, Comment. Math. Helv. 51, 4355 (1976; Zbl 334.35022). The proof uses facts on real projective plane algebraic curves (see D. A. Gudkov, Usp. Mat. Nauk 29(4), 379, Russian Math. Surveys 29(4), 179 (1979; Zbl 316.14018)), because they are the zero sets of homogeneous polynomials, and the spherical harmonics are the restrictions of spherical harmonic homogeneous polynomials in the space to the plane.
Item Type:  Paper 

Additional Information:  In: Topology 35(2), pp. 301321, 1996 
Keywords:  spherical harmonics / algebraic curves 
Classification Codes:  MSC 33C35 
Divisions:  Departments > Finance, Accounting and Statistics > Statistics and Mathematics 
Depositing User:  Repository Administrator 
Date Deposited:  03 May 2004 20:04 
Last Modified:  22 Oct 2019 00:41 
URI:  https://epub.wu.ac.at/id/eprint/1050 
Actions
View Item 
Downloads
Downloads per month over past year