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We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus T Γ := R d /Γ are flexible enough to approximate, over the natural length scale of order 1/ √ λ where λ ≫ 1 is the eigenvalue, an arbitary solution of the Helmholtz equation ∆h + h = 0 on 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 T Γ are of the form λ k = Q Γ (k), where Q Γ is a quadratic form and k ∈ Z d . Our main result is that the eigenfunctions of T Γ have the desired approximation property if and only Q Γ is a multiple of a quadratic form with integer coefficients. In particular, the set of lattices Γ for which this approximation property holds has measure zero but includes all rational lattices. A consequence of this fact is that when Q Γ is a multiple of a quadratic form with integer coefficients, Laplace eigenfunctions exhibit an extremely flexible behavior over scales of order 1/ √ λ. In particular, there are eigenfunctions of arbitrarily high energy that exhibit nodal components diffeomorphic to any compact hypersurface of diameter O(1/ √ λ).

We consider the question of whether the high-energy eigenfunctions of certain Schrödinger operators on the d-dimensional hyperbolic space of constant curvature −κ 2 are flexible enough to approximate an arbitrary solution of the Helmholtz equation ∆h + h = 0 on R d , over the natural length scale O(λ −1/2 ) determined by the eigenvalue λ ≫ 1. This problem is motivated by the fact that, by the asymptotics of the local Weyl law, approximate Laplace eigenfunctions do have this approximation property on any compact Riemannian manifold. In this paper we are specifically interested in the Coulomb and harmonic oscillator operators on the hyperbolic spaces H d (κ). As the dimension of the space of bound states of these operators tends to infinity as κ ց 0, one can hope to approximate solutions to the Helmholtz equation by eigenfunctions for some κ > 0 that is not fixed a priori. Our main result shows that this is indeed the case, under suitable hypotheses. We also prove a global approximation theorem with decay for the Helmholtz equation on manifolds that are isometric to the hyperbolic space outside a compact set, and consider an application to the study of the heat equation on H d (κ). Although global approximation and inverse approximation results are heuristically related in that both theorems explore flexibility properties of solutions to elliptic equations on hyperbolic spaces, we will see that the underlying ideas behind these theorems are very different.

We consider the question of when the Laplace eigenfunctions on an arbitrary flat torus ${\mathbb {T}}_\Gamma :={\mathbb {R}}^d/\Gamma $ are flexible enough to approximate, over the natural length scale of order $1/\sqrt \lambda $ where $\lambda \gg 1$ is the eigenvalue, an arbitary solution of the Helmholtz equation $\Delta h + h=0$ on ${\mathbb {R}}^d$. 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 ${\mathbb {T}}_\Gamma $ are of the form $\lambda _k=Q_\Gamma (k)$, where $Q_\Gamma $ is a quadratic form and $k\in {\mathbb {Z}}^d$. Our main result is that the eigenfunctions of ${\mathbb {T}}_\Gamma $ have the desired approximation property if and only if $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 $O(1/\sqrt \lambda )$.