Quantum entanglement and the growth of Laplacian eigenfunctions

Abstract We study the growth of Laplacian eigenfunctions on compact manifolds (M, g). Hörmander proved sharp polynomial bounds on 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 as the typical behavior. We propose a mechanism, centered around an analog of the spectral projector, for explaining the slow growth in the generic case: for to be large, it is necessary that either (1) several of the first n eigenfunctions were large in x 0 or (2) that 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 such that the sequences and 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.