Resumen de Asymptotics of eigenfunctions on plane domains

Daniel Grieser, David Jerison

• We consider a family of domains (ON)N>0 obtained by attaching an N × 1 rectangle to a fixed set O0 = {(x,y) : 0 < y < 1, - ?(y) < x < 0}, for a Lipschitz function ? = 0. We derive full asymptotic expansions, as N ?8, for the m-th Dirichlet eigenvalue (for any fixed m in N) and for the associated eigenfunction on ON. The second term involves a scattering phase arising in the Dirichlet problem on the infinite domain O8. We determine the first variation of this scattering phase, with respect to ?, at ? = 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.