Quasi-conical domains with embedded eigenvalues

TL;DR

This paper constructs a specific quasi-conical domain where the Dirichlet Laplacian has embedded eigenvalues, demonstrating new spectral properties and the possibility of empty absolutely continuous spectrum in such domains.

Contribution

It introduces a novel construction of a connected quasi-conical domain with embedded eigenvalues and an empty absolutely continuous spectrum for the Dirichlet Laplacian.

Findings

Existence of a connected quasi-conical domain with embedded eigenvalues.

Construction method using towers of cubes connected by vanishing windows.

Possibility to eliminate absolutely continuous spectrum in such domains.

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.