Eigenfunction Decay For the Neumann Laplacian on Horn-Like Domains

http://dx.doi.org/10.4153/CMB-2000-007-4
Canad. Math. Bull. 43(2000), 51-59

  Published:2000-03-01
 Printed: Mar 2000

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 that pinch at an exponential rate, weaker, $L^2$ bounds are proven. A corollary is that eigenvalues can accumulate only at zero or infinity.