Related papers: The $L^2$ essential spectrum of the 2D Euler opera…
Emission spectrum is calculated for a weak axisymmetric pulsar. Also calculated are the observed spectrum, efficiency, and the observed efficiency. The underlying flow of electrons and positrons turns out to be curiously intricate.
The spectral theorem of the linear 2D Euler operator in Sobolev spaces is presented as a corollary of the spectral theorem in $\ell_2$ space in [Li,00]. Study on the (dashed) line model introduced in [Li,01] is continued. Specifically,…
We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations…
This article concerns the global-in-time existence of smooth solutions with small amplitude to two space dimensional Euler-Poisson system. The main difficulty lies in the slow time decay $(1+t)^{-1}$ of the linear system. Inspired by Ozawa,…
We study the spectrum of a one-dimensional Schroedinger operator perturbed by a fast oscillating potential. The oscillation period is a small parameter. The essential spectrum is found in an explicit form. The existence and multiplicity of…
In this paper, we establish a condition on the coefficients of differential operators generated in the space of square-integrable functions on the entire real line by an ordinary differential expression with periodic, complex-valued…
The 2D Euler equations are a simple but rich set of non-linear PDEs that describe the evolution of an ideal inviscid fluid, for which one dimension is negligible. Solving numerically these equations can be extremely demanding. Several…
The problem of finding eigenvalue estimates for the Schr\"odinger operator turns out to be most complicated for the dimension 2. Some important results for this case have been obtained recently. We discuss these results and establish their…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations in an infinite cylinder are considered. It is shown that if the inflow-outflow is highly oscillating in time, the corresponding Euler flow cannot keep the…
For 2D periodic Jacobi operators we obtain the estimate of the Lebesgue measure of the spectrum and estimates of edges of spectral bands.
We characterize the spectrum and essential spectrum of "essentially linear fractional" composition operators acting on the Hardy space H-two of the open unit disc U. When the symbols of these composition operators have Denjoy-Wolff point on…
We investigate solutions of the 2d incompressible Euler equations, linearized around steady states which are radially decreasing vortices. Our main goal is to understand the smoothness of what we call the spectral density function…
In this note we deal with the problem of determining the essential spectrum of a Toeplitz operator $T_{f}:A^{2}(\mathbb{D}^{2})\rightarrow A^{2}(\mathbb{D}^{2})$ acting on the Bergman space $A^{2}(\mathbb{D}^{2})$ of the bi-disc whose…
We consider steady solutions to the incompressible Euler equations in a two-dimensional channel with rigid walls. The flow consists of two periodic layers of constant vorticity separated by an unknown interface. Using global bifurcation…
The goal of this note is to study the spectrum of a self-adjoint convolution operator in $L^2(\mathbb R^d)$ with an integrable kernel that is perturbed by an essentially bounded real-valued potential tending to zero at infinity. We show…
For a wide class of 2D periodic elliptic operators, we show that the global extrema of all spectral band functions are isolated.
We consider steady states of the incompressible Euler equation on two-dimensional domains. For non-radial analytic steady states on bounded simply connected domains, it was shown previously that there must be a global functional…
For a sequence of self--adjoint operators, which converges in the norm resolvent sense, the formula is derived, which expresses the essential spectrum of the limit through the essential spectrum of the elements of the sequence.
A simple and efficient algorithm to numerically compute the genus of surfaces of three-dimensional objects using the Euler characteristic formula is presented. The algorithm applies to objects obtained by thresholding a scalar field in a…
We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that…