Semi-classical limits of the first eigenfunction and concentration on the recurrent sets of a dynamical system

TL;DR

This paper investigates how the first eigenfunction of a second order elliptic operator behaves as the diffusion parameter approaches zero, showing concentration on recurrent sets of a Morse-Smale vector field.

Contribution

It provides a detailed analysis of the semi-classical limit of eigenfunctions, especially their concentration on recurrent sets, extending previous partial results with rigorous proofs.

Findings

Eigenfunctions concentrate on recurrent sets as diffusion vanishes

Blow-up analysis reveals properties of limit measures

Results clarify the semi-classical behavior of eigenfunctions

Abstract

Dear Reader, please find the third and last part of a series of papers on the singular perturbation of the first eigenfunction associated to a non self-adjoint second order elliptic operators. This series started in 1999 and we presented the early results in 2000 at Columbia University. We published two notes in CRAS in 2001 and 2005 summarizing our results. The present paper contains the proofs of the announced theorems and many open questions. We tried to publish these results in the the top tier of mathematical journals (Annals, Acta, Duke...) but our results were not deemed sufficiently interesting for them and probably not trendy enough. Some of you may like this work, so here it is. Best Regards, Ivan and David. We study the semi-classical limits of the first eigenfunction of a positive second order operator on a compact Riemannian manifold, when the diffusion constant $\epsilon$ goes to zero. If the drift of the diffusion is given by a Morse-Smale vector field $b$, the limits of the eigenfunctions concentrate on the recurrent set of $b$. A blow-up analysis enables us to find the main properties of the limit measures on a recurrent set.