Quasi-conical domains with embedded eigenvalues
Spectral Theory
2025-02-05 v2 Mathematical Physics
Analysis of PDEs
math.MP
Abstract
The spectrum of the Dirichlet Laplacian on any quasi-conical open set coincides with the non-negative semi-axis. We show that there is a connected quasi-conical open set such that the respective Dirichlet Laplacian has a positive (embedded) eigenvalue. This open set is constructed as the tower of cubes of growing size connected by windows of vanishing size. Moreover, we show that the sizes of the windows in this construction can be chosen so that the absolutely continuous spectrum of the Dirichlet Laplacian is empty.
Cite
@article{arxiv.2205.08172,
title = {Quasi-conical domains with embedded eigenvalues},
author = {David Krejcirik and Vladimir Lotoreichik},
journal= {arXiv preprint arXiv:2205.08172},
year = {2025}
}
Comments
revised version accepted for publication in Bulletin of the London Mathematical Society