2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140360We consider a family of domains $(Ω_N)_{N>0}$ obtained by attaching an $N\times 1$ rectangle to a fixed set $Ω_0 = \{(x,y): 0<y<1, -ϕ(y)<x<0\}$, for a Lipschitz function $ϕ\geq 0$. 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 $Ω_N$. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain $Ω_\infty$. We determine the first variation of this scattering phase, with respect to $ϕ$, at $ϕ\equiv 0$. 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 figuresSpectral TheoryAnalysis of PDEs35B25, 35P99, 81Q10text