The essential numerical range and a theorem of Simon on the absorption of eigenvalues

Plasmonic eigenvalue problem for corners: limiting absorption principle and absolute continuity in the essential spectrum

We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincaré 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.

Inequalities for the eigenvalues of non-selfadjoint Jacobi operators

We prove Lieb-Thirring-type bounds on eigenvalues of non-selfadjoint Jacobi operators, which are nearly as strong as those proven previously for the case of selfadjoint operators by Hundertmark and Simon. We use a method based on determinants of operators and on complex function theory, extending and sharpening earlier work of Borichev, Golinskii and Kupin.

离散三角晶格中长程扰动的极限吸收原理

Limiting absorption principle for long-range perturbation in the discrete triangular lattice setting

本文在离散三角晶格上对度量和势引入长程扰动，利用正交换子技术建立了远离嵌入特征值的极限吸收原理。

On the eigenvalue counting function for Schrödinger operator: some upper bounds

The aim of this work is to provide an upper bound on the eigenvalues counting function $N(\mathbb{R}^n,-Δ+V,e)$ of a Schödinger operator $-Δ+V$ on $\mathbb{R}^n$ corresponding to a potential $V\in L^{\frac{n}{2}+\varepsilon}(\mathbb{R}^n,dx)$, in terms of the sum of the eigenvalues counting function of the Dirichlet integral $\mathcal{D}$ with Dirichlet boundary conditions on the subpotential domain $\{V< e\}$, endowed with weighted Lebesgue measure $(V-e)_-\cdot dx$ and the eigenvalues counting function of the absorption-to-reflection operator on the equipotential surface $\{V=e\}$.

Limiting Absorption Principle for Schrödinger Operators with Oscillating Potential

Making use of the weighted Mourre theory developed in [GJ1], we show the limiting absorption principle for Schr{ö}dinger operators with perturbed oscillating potential on appropriate energy intervals. We focus on a certain class of oscillating potentials (larger than the one in [GJ2]) that was already studied in [BD, MU, ReT1, ReT2]. We allow long-range and short-range components and local singularities in the perturbation. We improve known results, the main novelty being the presence of a long-range perturbation. A subclass of the considered potentials actually cannot be treated by the usual …

Non-self-adjoint Jacobi matrices with rank one imaginary part

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank one imaginary part. It is shown that given a set of $n$ not necessarily distinct non-real numbers in the open upper (lower) half-plane uniquely determines a n x n Jacobi matrix with a rank one imaginary part having those numbers as its eigenvalues counting multiplicity. An algorithm for reconstruction for such finite Jacobi matrices is presented. A new model complementing the well known Livsic triangular model for bounded linear operators with rank one imaginary part is obt…