TITLE: Courant nodal domain Theorem for sums of eigenfunctions?
QUESTION [7 upvotes]: Courant's nodal domain theorem gives a bound on the number of nodal domains for an eigenfunction of the Laplacian.  Namely, if $M$ is a smooth compact Riemannian manifold, and $f$ is an eigenfunction for the $n$th eigenvalue, then the number of nodal domains is bounded by $n$.
Is there a bound on the number of nodal domains for a sum of eigenfunctions (with different eigenvalues)?
Note that in the case $M$ is a sphere, any linear combination of eigenfunctions (spherical harmonics) is a restriction of a homogeneous polynomial of degree $d$ (which is roughly the square root of the largest eigenvalue) and Harnack's bound says that there is a bound of order $d^2$ (same order as Courant's bound for the top eigenfunction).  Similarly, higher dimensional spheres and flat tori admit bounds with the same order as Courant's that apply for a sum of eigenfunctions.

REPLY [3 votes]: On a Riemann surface $\Sigma$, consider the space $H_\lambda$ spanned by eiigenfunctions corresponding to eigenvalues $\leq \lambda$.   By Weyl's asymptotic formula we know that
$$ \dim H_\lambda  \sim const \lambda $$
as $\lambda \to \infty$.  Denote  by $S_\lambda$ the unit sphere in $H_\lambda$ with respect to the $L^2$-norm.  Equip with with the unique rotationally invariant measure of total volume $1$ so now  you can think of $S_\lambda$ as a probability space. Thus, for any $f\in S_\lambda$, the number $N_f$ of zonal regions of $f$ is a  random   variable.  We denote by $N_\lambda$ its expectation, i.e., the average number of  zonal domains  of a function $f\in S_\lambda$.  One can show  that there exists a constant  $C>0$ such that
$$ N_\lambda \leq C\lambda $$
for $\lambda \gg 0$.     For a proof see this preprint.
I actually believe that 
$$ N_\lambda \sim C\lambda  $$
as $\lambda \to \infty$, but I have no promising idea how to approach this.