Some remarks on spherical harmonics

Abstract

Abstract. Several observations on spherical harmonics and their nodal sets are presented: a construction for harmonics with prescribed zeros; a natural representation for harmonics on S 2 ; upper and lower bounds for the nodal length and the inner radius (the upper bounds are sharp); the sharp upper bound for the number of common zeros of two spherical harmonics on S 2 ; the mean Hausdorff measure of the intersection of k nodal sets for harmonics of different degrees on S m , where k ≤ m (in particular, the mean number of common zeros of m harmonics).

Level Spacings and Nodal Sets at Infinity for Radial Perturbations of the Harmonic Oscillator

“…provided K = K(n) and J = J(n) satisfy The approximation (34) is the source of the O( ∞ n ) error in (9). The function E V K ,n (ε) whose jets appear in Proposition 7 is formally defined in the same way as E V ,n (ε).…”

Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

“…The nth eigenspace E λn = H m n corresponds to the eigenvalue λ n = n(n + m − 1) and consists of traces of harmonic homogeneous of degree n polynomials on S m . This case was considered in the papers [27], [11], [39], [40], [24], [25], where u was subject either to the Gaussian distribution in E λ or to the uniform one in S. Remark 1. Both distributions mentioned above are rotation invariant.…”

Nodal lengths of eigenfunctions in the disc

“…One then deduces (1.4) by observing that λ = k(k +n−1). In the same paper [G,Page 563], Gichev conjectured that on S 2 ,…”

“…In §2.1, we discuss finding H 1 and H 2 in the rectangles and the tori. Gichev [G,Theorem 3] proved that on the n-dim unit sphere S n ,…”