Literature on behaviour of eigenfunctions under multiplication?

Question

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:

1. 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.

2. 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?

3. 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

Answer 1

It can easily happen that the product of two eigenfunctions has an infinite eigenfunction expansion. Probably this is more typical (in natural problems) than not. For example, the Laplace-Beltrami operator on compact Riemannian manifolds (or suitable values $(\Delta+c)^{-1}$ of its resolvent if one must have a bounded operator) has discrete spectrum, but products of eigenfunctions are rarely eigenfunctions.

For example, on the sphere, (restrictions of) homogeneous harmonic polynomials are the eigenfunctions for the natural Laplacian. Products or squares of harmonic polynomials are rarely harmonic, but do have finite expressions as sums of eigenfunctions. (From my viewpoint, this finiteness is predictable because the irreducibles of the rotation/orthogonal group are all finite-dimensional, because it is compact.)

Either from the a viewpoint of geometric analysis, or from a viewpoint of repn theory, in generality such re-expressions of products would rarely be finite, but should exist under mild hypotheses on the situation. Explicit examples where the decomposition/Fourier coefficients $\langle fg,h\rangle$ are explicitly expressible for triples of eigenfunctions are not so easy to manufacture, and the ones I know (having to do with automorphic forms, zonal spherical harmonics, etc.) I suspect are not in the direction of your interest.

It is also true that there seems to be fairly skimpy literature on such issues.

About your question 2: I don't have an easy example. About question 3: no, there are easy examples of the failure: on $[0,2\pi]$, with $f(x)=e^{ix}$ and $g(x)=e^{-ix}$ (or cosines and sines), and $E$ the space spanned by constants, $fg$ projects to $1$, while both products $f^2$ and $g^2$ project to $0$.

Edit: in response to the questioner's further comment/questions... First, the latter counter-example, with exponential functions, refers to $d^2/dx^2$.

Second, as @Matt Y. noted, someone like me is thinking of automorphic things, which may not be what everyone wants to see as examples... But, further, I think it is not so easy to make any such examples explicit. Even without the conjectured (and probable) typical-non-vanishing and so on, we can find definitive examples of infinite, discrete decompositions in the pseudo-Laplacians Y. Colin de Verdiere used c. 1981 to give another proof of meromorphic continuation of Eisenstein series: these are self-adjoint operators with compact resolvents (so discrete spectrum) on Hilbert spaces of automorphic functions which have, among their eigenfunctions certain truncated Eisenstein series. The integral of three truncated Eisenstein series is not completely trivial to compute, but turns out (e.g., Zagier an others, circa 1978) to be a product/quotient of values of zeta at points that can be controlled directly... so can be made not to vanish. Thus, provably, an infinite-not-finite re-expression of product of two eigenfunctions (for a pseudo-Laplacian...) as sum of eigenfunctions. The "pseudo-Laplacian" is not wildly different from a/the genuine Laplacian...

Answer 2

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.