Localization properties of high energy eigenfunctions on flat tori

Abstract: We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus $\mathbf{T}\Gamma:=\mathbf{R}^d/\Gamma$ are flexible enough to approximate, over the natural length scale of order $1/\sqrt\lambda$, where $\lambda\gg1$ is the eigenvalue, an arbitary solution of the Helmholtz equation $\Delta h + h=0$ on $\mathbf{R}^d$. This problem is motivated by the fact that, by the asymptotics for the local Weyl law, "approximate Laplace eigenfunctions" do have this approximation property on any compact Riemannian manifold. What we find is that the answer depends solely on the arithmetic properties of the spectrum. Specifically, recall that the eigenvalues of $\mathbf{T}\Gamma$ are of the form $\lambda_k=Q_\Gamma(k)$, where $Q_\Gamma$ is a quadratic form and $k \in \mathbf{Z}^d$. Our main result is that the eigenfunctions of $\mathbf{T}\Gamma$ have the desired approximation property if and only $Q\Gamma$ is a multiple of a quadratic form with integer coefficients. In particular, the set of lattices $\Gamma$ for which this approximation property holds has measure zero but includes all rational lattices. A consequence of this fact is that when $Q_\Gamma$ is a multiple of a quadratic form with integer coefficients, Laplace eigenfunctions exhibit an extremely flexible behavior over scales of order $1/\sqrt\lambda$. In particular, there are eigenfunctions of arbitrarily high energy that exhibit nodal components diffeomorphic to any compact hypersurface of diameter $O(1/\sqrt\lambda)$.