JournalsrmiVol. 32, No. 3pp. 971–994

Focal points and sup-norms of eigenfunctions

  • Christopher D. Sogge

    The Johns Hopkins University, Baltimore, USA
  • Steve Zelditch

    Northwestern University, Evanston, USA
Focal points and sup-norms of eigenfunctions cover
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If (M,g)(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)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 x0Mx_0 \in M for the geodesic flow at which the associated Perron–Frobenius operator Ux0 ⁣:L2(Sx0M)L2(Sx0M)U_{x_0}\colon L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M) has a nontrivial invariant L2L^2 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