Asymptotics of eigenfunctions on plane domains

Abstract

We consider a family of domains $({\Omega }_N{)}_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set ${\Omega }_0=\{(x,y):0<y<1,-\varphi (y)<x<0\}$, for a Lipschitz function $\varphi \ge 0$. We derive full asymptotic expansions, as $N\to \infty$, for the $m$th Dirichlet eigenvalue (for any fixed $m$) and for the associated eigenfunction on ${\Omega }_N$. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain ${\Omega }_{\infty }$. We determine the first variation of this scattering phase, with respect to $\varphi$, at $\varphi \equiv 0$. 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.

19 pages, 2 figures