Concentration of the first eigenfunction for a second order elliptic operator

TL;DR

This paper investigates the semi-classical limit behavior of the first eigenfunction of a second order elliptic operator on a compact manifold, revealing concentration phenomena on recurrent sets where topological pressure is maximized.

Contribution

It characterizes the limiting measures of eigenfunctions concentrating on recurrent sets with maximal topological pressure, extending understanding of eigenfunction behavior in the semi-classical limit.

Findings

Eigenfunctions concentrate on recurrent sets of maximal dimension.

Limit measures are absolutely continuous on cycles and torii.

The analysis uses blow-up techniques to determine these measures.

Abstract

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. We assume that the first order term is given by a vector field $b$, whose recurrent components are either hyperbolic points or cycles or two dimensional torii. The limits of the normalized eigenfunctions concentrate on the recurrent sets of maximal dimension where the topological pressure \cite{Kifer90} is attained. On the cycles and torii, the limit measures are absolutely continuous with respect to the invariant probability measure on these sets. We have determined these limit measures, using a blow-up analysis.