Focal points and sup-norms of eigenfunctions

• Christopher D. Sogge ^[1] ; Steve Zelditch ^[2]
  1. [1] Johns Hopkins University

     Johns Hopkins University

     Estados Unidos

     
  2. [2] Northwestern University

     Northwestern University

     Township of Evanston, Estados Unidos

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 32, Nº 3, 2016, págs. 971-994
• Idioma: inglés
• DOI: 10.4171/RMI/904
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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(λ) 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 x0∈M for the geodesic flow at which the associated Perron–Frobenius operator Ux0:L2(S∗x0M)→L2(S∗x0M) has a nontrivial invariant L2 function. The proof is based on an explicit Duistermaat–Guillemin–Safarov pre-trace formula and von Neumann's ergodic theorem.