ASYMPTOTICS OF EIGENFUNCTIONS ON PLANE DOMAINS

作者：Grieser Daniel^*; Jerison David

来源：Pacific Journal of Mathematics, 2009, 240(1): 109-133.

DOI：10.2140/pjm.2009.240.109

• nodal line
• matched asymptotic expansion
• scattering phase
• quantum graph
• thick graph

We consider a family of domains. (Omega(N))(N>0) obtained by attaching an N x 1 rectangle to a fixed set Omega(0) = {(x,y) : 0 < y < 1; - phi (y) < x < 0}, for a Lipschitz function phi >= 0. We derive full asymptotic expansions, as N -> infinity, for the m-th Dirichlet eigenvalue ( for any fixed m 2 N) 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(infinity). We determine the first variation of this scattering phase, with respect to phi, at phi equivalent to 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.

• 出版日期2009-3