For small range of p > 2, we improve the L p bounds of eigenfunctions of the Laplacian on negatively curved manifolds. Our improvement is by a power of logarithm for a full density sequence of eigenfunctions. We also derive improvements on the size of the nodal sets. Our proof is based on a quantum ergodicity property of independent interest, which holds for families of symbols supported in balls whose radius shrinks at a logarithmic rate.

-3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices.APDE peer review and production are managed by EditFLOW ™ from Mathematical Sciences Publishers.PUBLISHED BY mathematical sciences publishers

No abstract

We prove that bounded real analytic domains in R n , with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded real analytic domains with the same symmetries and axis length. Some non-degeneracy conditions are also imposed on the class of domains. It follows that bounded, convex analytic domains are determined by their spectra among other such domains. This seems to be the first positive result for the well-known Kac problem, "Can one hear the shape of a drum?", in higher dimensions.

No abstract