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Register a new userEigenvalue continuity and Gerv{s}gorins theorem
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Two types of eigenvalue continuity are commonly used in the literature. However, their meanings and the conditions under which continuities are used are not always stated clearly. This can lead to some confusion and needs to be addressed. In this note, we revisit the Gerv{s}gorin disk theorem and clarify the issue concerning the proofs of the theorem by continuity.
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We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincare operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.
The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutati
We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.
We demonstrate lower bounds for the eigenvalues of compact Bakry-Emery manifolds with and without boundary. The lower bounds for the first eigenvalue rely on a generalised maximum principle which allows gradient estimates in the Riemannian setting to be directly applied to the Bakry-Emery setting. Lower bounds for all eigenvalues are demonstrated using heat kernel estimates and a suitable Sobolev inequality.
Let $Lambdasubset mathbb{R}^d$ be a domain consisting of several cylinders attached to a bounded center. One says that $Lambda$ admits a threshold resonance if there exists a non-trivial bounded function $u$ solving $-Delta u= u u$ in $Lambda$ and vanishing at the boundary, where $ u$ is the bottom of the essential spectrum of the Dirichlet Laplacian in $Lambda$. We derive a sufficient condition for the absence of threshold resonances in terms of the Laplacian eigenvalues on the center. The proof is elementary and is based on the min-max principle. Some two- and three-dimensional examples and applications to the study of Laplacians on thin networks are discussed.
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