A. Stern’s analysis of the nodal sets of some families of spherical harmonics revisited

Abstract

In this paper, we revisit the analyses of Stern (Bemerkungen über asymptotisches Verhalten von Eigenwerten und Eigenfunktionen, W. Fr. Kaestner, Göttingen, 1925) and Lewy (Commun Partial Differ Equ 2(12):1233–1244, 1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets. This paper is a natural continuation of our critical reading of A. Stern’s results for Dirichlet eigenfunctions in the square, see arXiv:14026054.

Appendices

Appendix

Some simulations with Maple

In this appendix, we provide some pictures issued from numerical computations with Maple. The nodal sets are viewed in the exponential map at the north pole. The outer circle, at distance $\pi$, is the cut-locus of $p_{+}$ and corresponds to the south pole.

Figures 11 resp. 12 illustrate Proposition 1 in the cases $\ell = 3$ (left) and $\ell = 4$ (right), resp. $\ell = 5$ (left) and $\ell = 6$ (right). They display the nodal set of $H^{\mu ,\ell }$ (black thick line), with $\mu = 5\cdot 10^{-3}$ when $\ell$ is odd, and $\mu = 2\cdot 10^{-3}$ when $\ell$ even. The figures in the top line also display the checkerboards associated with $Z_{\ell }$ and $W_{\ell }$.

Fig. 11

Example 1: $\ell = 3$ and $\ell = 4$

Fig. 12

Example 1: $\ell = 5$ and $\ell = 6$

Figure 13 illustrates the occurrence of critical zeros in Stern’s Example 1, with $\ell = 3$. The corresponding Legendre polynomial is $P_3(t) = \frac{1}{2} \, t \, (5 t^2-3)\,$. The polynomial $P_2(t) = \frac{1}{2}\, (3t^2-1)$ has two roots $\pm \, \frac{1}{\sqrt{3}}\,$. According to Sect. 3.2.1, there are twelve possible critical zeros, given in spherical coordinates by the points $\left( \arccos (\pm \frac{1}{\sqrt{3}}),j\frac{\pi }{6}\right)$ with $j \in \{1, 3, 5, 7, 9, 11\}$, and exactly two critical values of the parameter, $\mu = \pm \,\sqrt{2}\,$. For $\mu > 0$, there is exactly one critical value $\mu = \sqrt{2}$, which is associated with six critical zeros.

Figure 13 shows the nodal set $N(H^{\mu ,3})$ for $\mu < \sqrt{2}$ (left), for $\mu = \sqrt{2}$ (center) and for $\mu > \sqrt{2}$ (right).

Fig. 13

Appearance and disappearance of critical zeros

Figure 14 illustrates Proposition 2. The figures display the nodal set of the function $H^{\mu }$ (thick lines), with $\mu = 10^{-3}$ and $\varepsilon = 0.4$. The figures in the top line also display the checkerboards associated with $W, V_{\alpha }$. The great circle $M'_0 \cup M'_1$ divides the sphere into two closed hemispheres. Each one contains a simple closed nodal curve tangent to the great circle at one of the poles. As usual, the south pole is represented by the outer circle (dotted line), the cut-locus of 0 in the tangent space at $p_{+}$.

Fig. 14

Example 2: $\ell = 4$ and $\ell = 6$

Translation of citations from Stern’s thesis

We provide below a rough translation of the citations from Stern’s thesis in Sect. 1

[E1] ...one can for example easily show that on the sphere the numbers 2 or 3 occur as number of nodal domains for each eigenvalue, and that for the square, if we arrange the eigenvalues in increasing order, the number 2 always reappears as number of nodal domains. [K1] We shall then show that for each eigenvalue there exist eigenfunctions on the sphere whose nodal lines divide the sphere into 2 or 3 nodal domains only ...The number of nodal domains 2 occurs for the eigenvalues $\lambda _n = (2r+1)(2r+2)\, ~r=1,2,\ldots$; [K2] similarly, we shall now show that 3 occurs as number of nodal domains for all eigenvalues $\lambda _n =2r (2r+1), ~r=1,2,\ldots$.

[I1] Superimpose the systems of nodal lines of the two functions, and hatch the domains in which the functions have the same sign, then the nodal lines of the spherical function

$$\begin{aligned} P^{2r+1}_{2r+1} (\cos \vartheta )\cos (2r+1) \varphi + \mu P_{2r+1}(\cos \vartheta )\,,\quad \mu >0 \end{aligned}$$

can only pass in the non-hatched domains [I3]  and hence for values of $\mu$ which are small enough, they stay in a neighborhood of the nodal lines of

$$\begin{aligned} P^{2r+1}_{2r+1} (\cos \vartheta )\cos (2r+1) \varphi , \end{aligned}$$

i.e.  the $2r+1$ meridians, so that the system of nodal lines varies continuously with $\mu$ .... [I2] Furthermore, the nodal lines must pass through the $2(2r+1)^2$ intersection points of the system of nodal lines of the two above spherical functions ...