相关论文: The property $(E_A)$ and local spectral theory

Neural networks represent more features than they have dimensions via superposition, forcing features to share representational space. Current methods decompose activations into sparse linear features but discard geometric structure. We…

In this paper, the asymptotics of the spectral data (eigenvalues and weight numbers) are obtained for the higher-order differential operators with distribution coefficients and separated boundary conditions. Additionally, we consider the…

We show how two level atoms can be used to determine the local time dependent spectrum. The method is applied to a one dimensional cavity. The spectrum obtained is compared with the mode spectrum determined using spatially filtered second…

Spectral properties of 1-D Schr\"odinger operators $\mathrm{H}_{X,\alpha}:=-\frac{\mathrm{d}^2}{\mathrm{d} x^2} + \sum_{x_{n}\in X}\alpha_n\delta(x-x_n)$ with local point interactions on a discrete set $X=\{x_n\}_{n=1}^\infty$ are well…

The spectral problem for self-adjoint extensions is studied using the machinery of boundary triplets. For a class of symmetric operators having Weyl functions of a special type we calculate explicitly the spectral projections in the form of…

Let $X,Y$ be Banach spaces, $A:X \longrightarrow Y$ and $B,C:Y \longrightarrow X$ be bounded linear operators satisfying operator equation $ABA=ACA$. Recently, as extensions of Jacobson's lemma, Corach, Duggal and Harte studied common…

This paper deals with spectral inequalities for one-dimensional Schr\"odinger operators with potentials bounded between two increasing functions (weights). The spectral inequality allows one to estimate the norm of a function with bounded…

Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial…

For a positive element $A$ of a $C^*$-algebra $\mathfrak{A}$, let ${\|X\|}_{A}$ denote the $A$-operator semi-norm of $X\in\mathfrak{A}$. In this paper, we aim to introduce and study the notion of $A$-spectrum for $X$ such that…

We show that, apart from the usual area operator of non-perturbative quantum gravity, there exists another, closely related, operator that measures areas of surfaces. Both corresponding classical expressions yield the area. Quantum…

The aim of this article is to present a brief overview of spectral perturbation theory for matrices, bounded linear operators and holomorphic operator-valued functions. We focus on bounds for perturbed eigenvalues, eigenvectors and…

In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability…

We compare the bottom of the spectrum of discrete and continuous Schr\"odinger operators with periodic potentials with barriers at the boundaries of their fundamental domains. Our results show that these energy levels coincide in the…

A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of…

Lecture notes for one of the courses at the OPSFA Summerschool 6, July 11-15, 2016. All the results in these notes have appeared in the literature. Many special functions are eigenfunctions to explicit operators, such as difference and…

We establish a semiclassical trace formula in a general framework of microhyperbolic hermitian systems of $h$-pseudodifferential operators, and apply it to the study of the spectral shift function associated to a pair of selfadjoint…

For nonautonomous linear difference equations with bounded coefficients on $\mathbb{N}$ which have a bounded inverse, we introduce two different notions of spectra and discuss their relation to the well-known exponential dichotomy spectrum.…

An operator $A$ on an $l^p$-space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of $A$ in the Calkin algebra determines, for example, the Fredholmness…

A quasi-product on the normed space is defined. In addition, the notions of the eigenvectors of a linear operator can be extended for the nonlinear operator. Based on the quasi-product and the generalized eigenvectors, the spectral theorems…

Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the…