On the L Norm of Spectral Clusters for Compact Manifolds with Boundary

Furthermore, the exponent of λ is sharp on every such manifold (see e.g., [15]). In the case of a sphere, the examples which prove sharpness are in fact eigenfunctions. For (1.2) the appropriate example is an eigenfunction which concentrates in a λ− 1 2 diameter tube about a geodesic. For (1.3), the example is a zonal eigenfunction of L norm λ n−1 2 which takes on value comparable to λ on a λ−1 diameter ball about each of the north and south poles. Approximate spectral clusters with similar properties can be constructed in the interior of any smooth manifold, showing that for spectral clusters (though not necessarily eigenfunctions) the exponents in (1.2) and (1.3) are also lower bounds on manifolds with boundary.