On the number of nodal domains of spherical harmonics

Josef Leydold

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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