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Quasi-conical domains with embedded eigenvalues

David Krejcirik, Vladimir Lotoreichik

TL;DR

This paper proves that there exist connected quasi-conical domains Ω for which the Dirichlet Laplacian $- abla^2_{ m D}$ has a positive embedded eigenvalue in $[0,∞)$. The authors explicitly construct a tower of growing cubes connected by vanishing-width windows and show, via Mosco convergence and norm-resolvent perturbation, that an embedded eigenvalue persists in the limit, with energy scalable to any prescribed λ>0. Moreover, a trace-class perturbation argument (inspired by HSS91) is employed to ensure the absolutely continuous spectrum is empty for the resulting domain. The work highlights rich spectral phenomena in unbounded quasi-conical domains and provides a constructive framework to control both embedded eigenvalues and the presence of absolutely continuous spectrum through thin connections.

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.

Quasi-conical domains with embedded eigenvalues

TL;DR

This paper proves that there exist connected quasi-conical domains Ω for which the Dirichlet Laplacian has a positive embedded eigenvalue in . The authors explicitly construct a tower of growing cubes connected by vanishing-width windows and show, via Mosco convergence and norm-resolvent perturbation, that an embedded eigenvalue persists in the limit, with energy scalable to any prescribed λ>0. Moreover, a trace-class perturbation argument (inspired by HSS91) is employed to ensure the absolutely continuous spectrum is empty for the resulting domain. The work highlights rich spectral phenomena in unbounded quasi-conical domains and provides a constructive framework to control both embedded eigenvalues and the presence of absolutely continuous spectrum through thin connections.

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.
Paper Structure (8 sections, 3 theorems, 43 equations, 1 figure)

This paper contains 8 sections, 3 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dirichlet Laplacian on a (quasi-conical) open set
  4. Dirichlet Laplacian on varying domains
  5. Singular values and the trace class
  6. Embedded eigenvalues
  7. Absence of the absolutely continuous spectrum
  8. Discussion

Key Result

Theorem 1

Figures (1)

  • Figure 1: The disconnected quasi-conical open set $\Omega$ with places for the small holes to be dug indicated by circles.

Theorems & Definitions (9)

  • Theorem 1
  • Definition 2.1: Daners
  • Proposition 2.2: Daners, Daners_2008
  • Proposition 2.3: Daners
  • Definition 2.4
  • Definition 2.5
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3