Eigenfunction Decay For the Neumann Laplacian on Horn-Like Domains

The growth properties at infinity for eigenfunctions corresponding to embedded eigenvalues of the Neumann Laplacian on horn-like domains are studied. For domains that pinch at polynomial rate, it is shown that the eigenfunctions vanish at infinity faster than the reciprocal of any polynomial. For a class of domains thatpinchat an exponentialrate, weaker, L 2 boundsareproven. A corollaryis thateigenvaluescan accumulate only at zero or infinity.