Asymptotics of eigenfunctions on plane domains

@article{Grieser2007AsymptoticsOE,
  title={Asymptotics of eigenfunctions on plane domains},
  author={Daniel Grieser and David Jerison},
  journal={Pacific Journal of Mathematics},
  year={2007},
  volume={240},
  pages={109-133}
}
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… 
View PDF on arXiv
19 Citations
Highly Influential Citations
3
Background Citations
6
Methods Citations
5
Results Citations
3

Figures from this paper

19 Citations

Spectra of graph neighborhoods and scattering

Let (Gε)ε>0 be a family of ‘ε‐thin’ Riemannian manifolds modeled on a finite metric graph G, for example, the ε‐neighborhood of an embedding of G in some Euclidean space with straight edges. We study…

Nodal Set Openings on Perturbed Rectangular Domains

. We study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio N , we identify the first…

On the number of critical points of the second eigenfunction of the laplacian in convex planar domains

Nodal line estimates for the second Dirichlet eigenfunction

We study the nodal curves of low energy Dirichlet eigenfunctions in generalized curvilinear quadrilaterals. The techniques can be seen as a generalization of the tools developed by Grieser-Jerison in…

Convergence of Dirichlet Eigenvalues for Elliptic Systems on Perturbed Domains

We consider the eigenvalues of an elliptic operator for systems with bounded, measurable, and symmetric coefficients. We assume we have two non-empty, open, disjoint, and bounded sets and add a set…

Nodal sets of thin curved layers

Eigenvalues of Elliptic Systems for the Mixed Problem in Perturbations of Lipschitz Domains with Nonhomogeneous Neumann Boundary Conditions

We study eigenvalues of an elliptic operator with mixed boundary conditions on very general decompositions of the boundary. We impose nonhomogeneous conditions on the part of the boundary where the…

Manifolds with cylindrical ends having a finite and positive number of embedded eigenvalues

We construct a surface with a cylindrical end which has a finite number of Laplace eigenvalues embedded in its continuous spectrum. The surface is obtained by attaching a cylindrical end to a…

The Fundamental Gap and One-Dimensional Collapse

The fundamental gap of a bounded, connected domain in Rn is the difference between the first two (positive) eigenvalues of the Euclidean Laplacian with Dirichlet boundary condition. The one…

Scales, blow-up and quasimode constructions

In this expository article we show how the concepts of manifolds with corners, blow-ups and resolutions can be used effectively for the construction of quasimodes, i.e. approximate eigenfunctions of…

References

SHOWING 1-10 OF 16 REFERENCES

Spectra of graph neighborhoods and scattering

Let (Gε)ε>0 be a family of ‘ε‐thin’ Riemannian manifolds modeled on a finite metric graph G, for example, the ε‐neighborhood of an embedding of G in some Euclidean space with straight edges. We study…

Variational Problems¶on Multiply Connected Thin Strips I:¶Basic Estimates and Convergence¶of the Laplacian Spectrum

Abstract Let M be a planar embedded graph whose arcs meet transversally at the vertices. Let ?(ɛ) be a strip-shaped domain around M, of width ɛ except in a neighborhood of the singular points. Assume…

Analytic surgery and the accumulation of eigenvalues

where e > 0 is a parameter. The limiting metric, go, is an exact b-metric on the disjoint union M = M+ U M_, i.e. it gives M± asymptotically cylindrical ends with cross-section H. In an earlier…

The diameter of the first nodal line of a convex domains

The main goal of this paper is to prove that the first nodal line for the Dirichlet problem in a convex planar domain has diameter less than an absolute constant times the inradius of the domain.…

The size of the first eigenfunction of a convex planar domain

The goal of this paper is to estimate the size of the first eigenfunction u uniformly for all convex domains. In particular, we will locate the place where u achieves its maximum to within a distance…

Convergence of spectra of graph-like thin manifolds

Location of the nodal set for thin curved tubes

The Dirichlet Laplacian in curved tubes of arbitrary constant cross-section rotating together with the Tang frame along a bounded curve in Euclidean spaces of arbitrary dimension is investigated in…

Scattering Theory and Adiabatic Decomposition of the $\z$-determinant of the Dirac Laplacian

In this note we announce the adiabatic decomposition formula for the ζ-determinant of the Dirac Laplacian. Theorem 1.1 of this paper extends the re- sult of our earlier work (see (8) and (9)), which…

Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph

Abstract Let M be a finite graph in the plane and let Me be a domain that looks like the e-fattened graph M (exact conditions on the domain are given). It is shown that the spectrum of the Neumann…

Thin tubes in mathematical physics, global analysis and spectral geometry

A thin tube is an $n$-dimensional space which is very thin in $n-1$ directions, compared to the remaining direction, for example the $\epsilon$-neighborhood of a curve or an embedded graph in $\R^n$…