93-08-01.ley
On the number of nodal domains of spherical harmonics
Review from Zentralblatt für Mathematik
It is well known that the $n$-th eigenfunction to one-dimensional
Sturm-Liouville eigenvalue problems has exactly $n - 1$ nodes, i.e.
non-degenerate 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 $S^2\setminus
N(u)$ 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 {\it S. Y. Cheng}, Comment. Math. Helv. 51,
43-55 (1976; Zbl 334.35022). The proof uses facts on real projective
plane algebraic curves
(see {\it D. A. Gudkov}, Usp. Mat. Nauk 29(4), 3-79, Russian
Math. Surveys 29(4), 1-79 (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.
Mathematics Subject Classification:
33C35 (Spherical functions),
14H05 (Algebraic functions)
Key Words:
spherical harmonics, algebraic curves
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Josef.Leydold@statistik.wu-wien.ac.at