On the spectral resolution of products of Laplacian eigenfunctions

We study products of eigenfunctions of the Laplacian -Delta phi(lambda) = lambda phi(lambda) on compact manifolds. If phi(mu), phi(lambda) are two eigenfunctions and mu <= lambda, then one would perhaps expect their product phi(mu)phi(lambda) to be mostly a linear combination of eigenfunctions with eigenvalue close to lambda. This can fail quite dramatically: on T-2, we see that 2 sin (nx) sin ((n + 1)x) = cos (x) - cos ((2n + 1)x) has half of its L-2-mass at eigenvalue 1. Conversely, the product sin (nx) sin (my) lives at eigenvalue max {m(2), n(2)} <= m(2) + n(2 )<= 2 max {m(2), n(2)} and the heuristic is valid. We show that the main reason is that in the first example 'the waves point in the same direction': if the heuristic fails and multiplication carries L-2-mass to lower frequencies, then phi(mu) and phi(lambda) are strongly correlated at scale similar to lambda(-1/2) (the shorter wavelength) parallel to integral(M) p(t, x, y)(phi(lambda )(y) - phi(lambda)(x))(phi(mu)(y) - phi(mu)(x))dy parallel to(Lx2) greater than or similar to parallel to phi(mu)phi(lambda)parallel to(L2), where p(t, x, y) is the classical heat kernel and t similar to lambda(-1). This turns out to be a fairly fundamental principle and is even valid for the Hadamard product of eigenvectors of a graph Laplacian.