Asymptotics of eigenfunctions on plane domains

We consider a family of domains $(\Omega_N)_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set $\Omega_0 = \{(x,y): 0<y<1, -\phi(y)<x<0\}$, for a Lipschitz function $\phi\geq 0$. 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 $\phi$, at $\phi\equiv 0$. 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.