On the asymptotic spectrum of finite element matrix sequences

We derive a new formula for the asymptotic eigenvalue distribution of stiffness matrices obtained by applying Fi finite elements with standard mesh refinement to the semielliptic PDE of second order in divergence form -\u25bd(\u39a\u25bdTu) = f on \u3a9, u = g on 02\u3a9. Here \u3a9 82 \u211d2, and K is supposed to be piecewise continuous and point wise symmetric semipositive definite. The symbol describing this asymptotic eigenvalue distribution depends on the PDE, but also both on the numerical scheme for approaching the underlying bilinear form and on the geometry of triangulation of the domain. Our work is motivated by recent results on the superlinear convergence behavior of the conjugate gradient method, which requires the knowledge of such asymptotic eigenvalue distributions for sequences of matrices depending on a discretization parameter h when h \u2192 0. We compare our findings with similar results for the finite difference method which were published in recent years. In particular we observe that our sequence of stiffness matrices is part of the class of generalized locally Toeplitz sequences for which many theoretical tools are available. This enables us to derive some results on the conditioning and preconditioning of such stiffness matrices

The asymptotic lift of a completely positive map

Starting with a unit-preserving normal completely positive map L: M --> M acting on a von Neumann algebra - or more generally a dual operator system - we show that there is a unique reversible system \alpha: N --> N (i.e., a complete order automorphism \alpha of a dual operator system N) that captures all of the asymptotic behavior of L, called the {\em asymptotic lift} of L. This provides a noncommutative generalization of the Frobenius theorems that describe the asymptotic behavior of the sequence of powers of a stochastic n x n matrix. In cases where M is a von Neumann algebra, the asymptotic lift is shown to be a W*-dynamical system (N,\mathbb Z), whick we identify as the tail flow of the minimal dilation of L. We are also able to identify the Poisson boundary of L as the fixed point algebra of (N,\mathbb Z). In general, we show the action of the asymptotic lift is trivial iff L is {\em slowly oscillating} in the sense that $$\lim_{n\to\infty}\|\rho\circ L^{n+1}-\rho\circ L^n\|=0,\qquad \rho\in M_* .$$ Hence \alpha is often a nontrivial automorphism of N.Comment: New section added with an applicaton to the noncommutative Poisson boundary. Clarification of Sections 3 and 4. Additional references. 23 p

On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices

We present results on the unique reconstruction of a semi-infinite Jacobi operator from the spectra of the operator with two different boundary conditions. This is the discrete analogue of the Borg-Marchenko theorem for Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and sufficient conditions for two real sequences to be the spectra of a Jacobi operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the theorems, and we updated two reference

Inverse problems for Jacobi operators II: Mass perturbations of semi-infinite mass-spring systems

We consider an inverse spectral problem for infinite linear mass-spring systems with different configurations obtained by changing the first mass. We give results on the reconstruction of the system from the spectra of two configurations. Necessary and sufficient conditions for two real sequences to be the spectra of two modified systems are provided.Comment: 25 pages, 2 figures. Typos were corrected, two remarks were added, material added to Sec.

Right limits and reflectionless measures for CMV matrices

We study CMV matrices by focusing on their right-limit sets. We prove a CMV version of a recent result of Remling dealing with the implications of the existence of absolutely continuous spectrum, and we study some of its consequences. We further demonstrate the usefulness of right limits in the study of weak asymptotic convergence of spectral measures and ratio asymptotics for orthogonal polynomials by extending and refining earlier results of Khrushchev. To demonstrate the analogy with the Jacobi case, we recover corresponding previous results of Simon using the same approach