Asymptotics of eigenfunctions on plane domains
Daniel Grieser, David S Jerison · Pacific Journal of Mathematics · 2009
We consider a family of domains ( N ) N>0 obtained by attaching an N × 1 rectangle to a fixed set 0 = {(x, y) : 0 < y < 1, -φ( y) < x < 0}, for a Lipschitz function φ ≥ 0. We derive full asymptotic expansions, as N → ∞, for the m-th Dirichlet eigenvalue (for any fixed m ∈ )ގ and for the associated eigenfunction on N .The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain ∞ .We determine the first variation of this scattering phase, with respect to φ, at φ ≡ 0. This is then used to prove sharpness of results, obtained previously by the same authors, about the location of extrema and nodal line of eigenfunctions on convex domains.