Courant nodal domain Theorem for sums of eigenfunctions?

Question

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.

Answer 1

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.