TITLE: Literature on behaviour of eigenfunctions under multiplication?
QUESTION [6 upvotes]: Dear community,
I would be happy about any literature or comments on the behaviour of the pointwise product of eigenfunctions of a self-adjoint operator with discrete spectrum, acting on a separable Hilbert space which is closed under pointwise multiplication. The operator I'm actually looking at is a symmetric Markov operator acting on $L^2(\mathcal{A},\mu)$, where $\mathcal{A}$ is some function algebra and $\mu$ the invariant measure.
Some questions I'm especially interested in are:

If you multiply two eigenfunctions, can it happen that the product has an infinite eigenfunction expansion? By "infinite eigenfunction expansion" I mean that it can not be expressed as a finite sum of eigenfunctions. 
Somewhat related: If the squares of two eigenfunctions have a finite expansion, respectively, can it happen that the square of the sum of these two eigenfunctions has an infinite expansion?  
In the above Markov setting: Is the following "projected Cauchy-Schwarz inequality" always true?
$$ 
\int \operatorname{proj}(fg \mid E)^2 \ \text{d} \mu \leq 
\sqrt{\int \operatorname{proj}(f^2 \mid E)^2 \ \text{d} \mu}
\sqrt{\int \operatorname{proj}(g^2 \mid E)^2 \ \text{d} \mu}
$$
Here, $f$ and $g$ are eigenfunctions lying in some common eigenspace, $E$ is another eigenspace and $\operatorname{proj}(f \mid E)$ denotes the projection of $f$ on $E$.

Note that the answer to questions 1 and 2 is no, if the eigenfunctions are orthogonal polynomials.
Thanks for your help,
Simon

REPLY [3 votes]: For question 1, one example of interest comes from the energy eigenfunctions of the one dimensional quantum harmonic oscillator.  The Hilbert space is separable, and the Hamiltonian satisfies your conditions.  Under suitable normalization, the eigenfunctions have the form $p(x)e^{-\pi x^2}$ for $p$ a Hermite polynomial, and the product of any two then has the form $q(x)e^{-2\pi x^2}$ for $q$ a nonzero polynomial.  These products are not given by finite linear combinations of eigenfunctions.