Asymptotics of eigenfunctions on plane domains

Daniel Grieser and David Jerison

Vol. 240 (2009), No. 1, 109–133

DOI: 10.2140/pjm.2009.240.109

Abstract

We consider a family of domains (Ω_N)_N>0 obtained by attaching an N × 1 rectangle to a fixed set Ω₀ = {(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 ∈ ℕ) and for the associated eigenfunction on Ω_N. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain Ω_∞. 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.

Keywords

nodal line, matched asymptotic expansion, scattering phase, quantum graph, thick graph

Mathematical Subject Classification 2000

Primary: 35B25, 35P99

Secondary: 81Q10

Milestones

Received: 29 April 2008

Accepted: 26 November 2008

Published: 2 March 2009