Focal points and sup-norms of eigenfunctions
By Christopher D. Sogge, Steve Zelditch. 2013
53 citations recorded by the source corpus.
If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(\lambda)$ 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 $x_0\in M$ for the geodesic flow at which the associated Perron-Frobenius operator $U_{x_0}: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$ has a nontrivial invariant $L^2$ function. The proof is based on an explict Duistermaat-Guillemin-Safarov pre-trace formula and von Neumann's ergodic theorem.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/396deaf6323638dbf665d34d012483d30287ce2e
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In Applied Mathematics
In Mathematical Physics
Citations in the corpus
- Scaling Limit for the Kernel of the Spectral Projector and Remainder Estimates in the Pointwise Weyl Law (2014, methodology): “We recall the following definition from [Safarov 1988; Sogge and Zelditch 2013]: Definition 5.”