93-01-01.ley
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Nodal Properties of Spherical Harmonics

PhD Thesis

### Abstract

In the following we study the properties of nodal lines and nodal
domains of spherical harmonics. The motivation for dealing with the
solutions of this special elliptic PDE is on the one hand that this
problem is relatively simple
(the spherical harmonics are harmonic homogeneous
polynomials restricted onto to the sphere; the sphere is compact;
etc.), on the other hand the importance of these functions for
arbitrary PDE. In spite of of this a lot of problems are open till
now.
In the first chapter we summarize some well-known properties of
spherical harmonics and make a conjecture on a sharp estimate for the
number of nodal domains.

In the second part we apply theorems about plane real algebraic curves
on nodal lines of spherical harmonics. It turns out, that we arrive at
the same results achieved by Courant's theorem.
By using further properties (e.g. by Bers' theorem) we get better
results. In detail: It is possible to make a full classification of
spherical harmonics, which decomposes into linear factors only; we can
derive a sharp upper bound for the number of nodal domains for a large
class of spherical harmonics, which consists of linear and quadratic
factors only. Moreover some further properties of spherical harmonics
can be found, such that we can proof the above conjecture for
degree less or equal to six.

In the third part an upper bound for the number of nodal domains of
non-singular spherical harmonics is derived, which is better than
Harnack's theorem.
Hence there are no harmonic (Harnack) M-curves.
Furthermore it is shown by construction, that this estimate is
of right order.

In the forth part graphs are used to describe and define the
conception of ``nodal pattern''.

In the last part the space of spherical harmonics -
especially the components of the set of normalized non-singular curves
- are studied.
Moreover a few inequalities about spherical harmonics are deduced.

**Key Words:**
spherical harmonics, algebraic curves, nodal sets, Courant's nodal domain theorem, M-curves

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Josef.Leydold@statistik.wu-wien.ac.at