Continuity of the asymptotic spectra for Toeplitz matrices

En este Trabajo de Grado, consideramos los espacios de Hilbert L^2(T) y l^2(Z), y los relacionamos mediante un isomorfismo isométrico el cual llamamos la transformada de Fourier. Es importante conocer las matrices de Toeplitz y el Lema de Coburn que nos indica cómo determinar el espectro de un operador de Toeplitz. Con estos preliminares iniciamos el estudio del artículo Asymptotic spectra of dense Toeplitz matrices are unstable , para un símbolo continuo en el círculo unitario complejo construimos su matriz de Toeplitz. Luego, diferentes truncamientos de esta matriz infinita nos permiten hallar los respectivos valores propios mediante cálculos computacionales. La convergencia de los espectros asintóticos varía haciendo pequeñas perturbaciones al símbolo, esto demuestra que no hay convergencia en la métrica de Hausdorff.In this bachelor thesis, we consider the Hilbert spaces L^2(T) and l^2(Z), and we relate them through an isometric isomorphism which we call the Fourier transform. It is important to know the Toeplitz matrices and Coburn s Lemma that indicates how to compute the spectrum of a Toeplitz operator. With these preliminaries we begin the study of the paper Asymptotic spectra of dense Toeplitz matrices are unstable , for a continuous symbol on the unit circle in the complex plane we construct its Toeplitz matrix. Then, various truncations of this infinite matrix allow us to find the respective eigenvalues by using computacional calculations. The convergence of the asymptotic spectra differ by doing small perturbations to the symbol, this shows that there is no convergence in the Hausdorff metric.Matemático (a)Pregrad

On the singular values and eigenvalues of the Fox–Li and related operators

The Fox–Li operator is a convolution operator over a finite interval with a special highly oscillatory kernel. It plays an important role in laser engineering. However, the mathematical analysis of its spectrum is still rather incomplete. In this expository paper we survey part of the state of the art, and our emphasis is on showing how standard Wiener–Hopf theory can be used to obtain insight into the behaviour of the singular values of the Fox–Li operator. In addition, several approximations to the spectrum of the Fox–Li operator are discussed and results on the singular values and eigenvalues of certain related operators are derived

Semiclassical inverse spectral theory for singularities of focus-focus type

We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.Comment: 14 pages, 2 figure

Spectral Properties of Random Non-self-adjoint Matrices and Operators

We describe some numerical experiments which determine the degree of spectral instability of medium size randomly generated matrices which are far from self-adjoint. The conclusion is that the eigenvalues are likely to be intrinsically uncomputable for similar matrices of a larger size. We also describe a stochastic family of bounded operators in infinite dimensions for almost all of which the eigenvectors generate a dense linear subspace, but the eigenvalues do not determine the spectrum. Our results imply that the spectrum of the non-self-adjoint Anderson model changes suddenly as one passes to the infinite volume limit.Comment: keywords: eigenvalues, spectral instability, matrices, computability, pseudospectrum, Schroedinger operator, Anderson mode

A geometric approach to approximating the limit set of eigenvalues for banded Toeplitz matrices

This article is about finding the limit set for banded Toeplitz matrices. Our main result is a new approach to approximate the limit set $\Lambda(b)$ where $b$ is the symbol of the banded Toeplitz matrix. The new approach is geometrical and based on the formula $\Lambda(b) = \cap_{\rho \in (0, \infty)} \text{sp } T(b_\rho)$. We show that the full intersection can be approximated by the intersection for a finite number of $\rho$'s, and that the intersection of polygon approximations for $\text{sp } T(b_\rho)$ yields an approximating polygon for $\Lambda(b)$ that converges to $\Lambda(b)$ in the Hausdorff metric. Further, we show that one can slightly expand the polygon approximations for $\text{sp } T(b_\rho)$ to ensure that they contain $\text{sp } T(b_\rho)$. Then, taking the intersection yields an approximating superset of $\Lambda(b)$ which converges to $\Lambda(b)$ in the Hausdorff metric, and is guaranteed to contain $\Lambda(b)$. We implement the algorithm in Python and test it. It performs on par to and better in some cases than existing algorithms. We argue, but do not prove, that the average time complexity of the algorithm is $O(n^2 + mn\log m)$, where $n$ is the number of $\rho$'s and $m$ is the number of vertices for the polygons approximating $\text{sp } T(b_\rho)$. Further, we argue that the distance from $\Lambda(b)$ to both the approximating polygon and the approximating superset decreases as $O(1/\sqrt{k})$ for most of $\Lambda(b)$, where $k$ is the number of elementary operations required by the algorithm.Comment: 20 pages, 8 figures. Submitted to SIAM Journal on Matrix Analysis and Application

Spectral Theory of Pseudo-Ergodic Operators

We define a class of pseudo-ergodic non-self-adjoint Schr\"odinger operators acting in spaces $l^2(X)$ and prove some general theorems about their spectral properties. We then apply these to study the spectrum of a non-self-adjoint Anderson model acting on $l^2(\Z)$, and find the precise condition for 0 to lie in the spectrum of the operator. We also introduce the notion of localized spectrum for such operators.Comment: 22 page