Some applications of heat flow to Laplace eigenfunctions

Abstract

We consider mass concentration properties of Laplace eigenfunctions ϕ λ , that is, smooth functions satisfying the equation −∆ϕ λ = λϕ λ , on a smooth closed Riemannian manifold. Using a heat diffusion technique, we first discuss mass concentration/localization properties of eigenfunctions around their nodal sets. Second, we discuss the problem of avoided crossings and (non)existence of nodal domains which continue to be thin over relatively long distances. Further, using the above techniques, we discuss the decay of Laplace eigenfunctions on Euclidean domains which have a central "thick" part and "thin" elongated branches representing tunnels of sub-wavelength opening. Finally, in an Appendix, we record some new observations regarding sub-level sets of the eigenfunctions and interactions of different level sets.

Heat profile, level sets and hot spots of Laplace eigenfunctions

“…We also prove an analogous result for the Neumann boundary condition, using a somewhat novel approach involving Doob's optional stopping theorem (see Theorem 4.7 below). As a further application of Doob's optional stopping theorem, domains with narrow horns/tentacles/narrow connectors as in [GM2] make an appearance again, and we prove that for the Neumann boundary condition, the eigenfunction does not need to decay sharply in the narrow connector at all, in stark contrast to the Dirichlet case which was addressed in [GM2].…”

“…For the purposes of the present note, we are in particular interested in phenomena of heat diffusion and how a deterministic diffusion process can be expressed as an expectation over the behavior of some random variables in terms of Brownian motion. In [GM2], the authors study this diffusion process and some of its implications on high energy eigenfunctions. Here we continue the discussion by concerning ourselves more with ground state and/or low energy eigenfunctions.…”

“…The expression Θ n r 2 /t will appear in many places in our paper. It seems that there is no nice closed formula for Θ n in the literature, and we quickly jot down a few facts from [GM2].…”

“…Here we extend the result to arbitrary level sets at levels µ and ν. The Dirichlet version of the following theorem has already been proved in [GM2], we start by proving the Neumann version here.…”

“…The following proof appears implicitly in [GM2], but since it seems quite useful we still include the main ideas here for the sake of completeness.…”