Asymptotics of eigenfunctions on plane domains

TL;DR

This paper derives detailed asymptotic expansions for eigenvalues and eigenfunctions on domains extended by rectangles, revealing how scattering phases influence eigenfunction behavior and extremal properties.

Contribution

It provides new asymptotic formulas for eigenvalues and eigenfunctions on expanding domains and analyzes the scattering phase's variation, advancing understanding of eigenfunction extremal properties.

Findings

Asymptotic expansions for eigenvalues and eigenfunctions as domain size grows

First variation of scattering phase with respect to domain perturbation

Sharpness of previous results on eigenfunction extrema and nodal lines

Abstract

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