Quantum Entanglement and the Growth of Laplacian Eigenfunctions

Abstract

We study the growth of Laplacian eigenfunctions $ -Δϕ_k = λ_k ϕ_k$ on compact manifolds $(M,g)$. Hörmander proved sharp polynomial bounds on $\| ϕ_k\|_{L^{\infty}}$ which are attained on the sphere. On a `generic' manifold, the behavior seems to be different: both numerics and Berry's random wave model suggest $\| ϕ_k\|_{L^{\infty}} \lesssim \sqrt{\log{λ_k}}$ as the typical behavior. We propose a mechanism, centered around an $L^1-$analogue of the spectral projector, for explaining the slow growth in the generic case: for $ϕ_{n+1}(x_0)$ to be large, it is necessary that either (1) several of the first $n$ eigenfunctions were large in $x_0$ or (2) that $ϕ_{n+1}$ is strongly correlated with a suitable linear combination of the first $n$ eigenfunctions on most of the manifold or (3) both. An interesting byproduct is quantum entanglement for Laplacian eigenfunctions: the existence of two distinct points $x,y \in M$ such that the sequences $(ϕ_k(x))_{k=1}^{\infty}$ and $(ϕ_k(y))_{k=1}^{\infty}$ do not behave like independent random variables. The existence of such points is not to be expected for generic manifolds but common for the classical manifolds and subtly intertwined with eigenfunction concentration.