Focal points and sup-norms of eigenfunctions
Christopher D. Sogge
The Johns Hopkins University, Baltimore, USASteve Zelditch
Northwestern University, Evanston, USA
![Focal points and sup-norms of eigenfunctions cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-rmi-volume-32-issue-3.png&w=3840&q=90)
Abstract
If is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point for the geodesic flow at which the associated Perron–Frobenius operator has a nontrivial invariant function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.
Cite this article
Christopher D. Sogge, Steve Zelditch, Focal points and sup-norms of eigenfunctions. Rev. Mat. Iberoam. 32 (2016), no. 3, pp. 971–994
DOI 10.4171/RMI/904