Jacobi matrices: continued fractions, approximation, spectrum

Eduard Ianovich

Home/Archive/arXiv

arXiv · 1707.04695

Jacobi matrices: continued fractions, approximation, spectrum

Eduard Ianovich

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this work the spectral theory of self-adjoint operator $A$ represented by Jacobi matrix is considered. The approach is based on the continued fraction representation of the resolvent matrix element of $A$. Different criteria of absolute continuity of a spectrum are found. For the analysis of the absolutely continuous spectrum it is used an approximation of $A$ by a sequence of operators $A_n$ with absolutely continuous spectrum on a given interval $[a,b\,]$ which converges to $A$ in a strong sense on a dense set. In the case when $[a,b\,]\subseteq\sigma(A)$ it was found the sufficient condition of absolute continuity of the operator $A$ spectrum on $[a,b\,]$. This condition uses the notion of equi-absolute continuity. It is constructed the system of functions converging to the distribution function of the operator. In the case of the absolutely continuous spectrum, the system of continuous functions converging to the spectral weight of the operator on a given interval is also constructed and was analyzed. The conditions when the derivative of the distribution function of $A$ belongs to the class $C[a,b\,]$ are also obtained.

Explore related subjects

math.SP cs.NA math.FA math.NA

Keep this discovery

BibTeX RIS

Eduard Ianovich. 2017-07-15. Jacobi matrices: continued fractions, approximation, spectrum. https://arxiv.org/abs/1707.04695

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Regular hyperbolic tilings have no $\ell^2$ eigenfunctions

We show that the adjacency operator of the $1$-skeleton of any regular tiling of the hyperbolic plane has no nonzero square-integrable eigenfunctions. As a consequence, the same holds for every infinite connected regular graph admitting a proper planar embedding with regular dual.

math.SP↗

arXiv2026

Inverse Heat Source Problems from Boundary Flux and Interior Observations on Sets of Low Hausdorff Dimension

This paper investigates conditional stability for inverse source problems for the heat equation with a known temporal factor and an unknown spatial component in a bounded $C^{1,1}$ domain. We focus on observations supported on sets of low Hausdorff dimension and establish conditional stability in this setting. For boundary observations on compact sets of positive $q$-dimensional Hausdorff content, we establish logarithmic stability from full-time boundary flux observations and double-logarithmic stability from delayed-time boundary flux observations. The admissible dimensional ranges are $q>d-2$ when the observation set is contained in a flat boundary patch and $q>d-1-c_{d+1}$ on a general $C^{1,1}$ boundary, where $c_{d+1}>0$ depends only on the dimension. A key ingredient in deriving these results is a new boundary spectral inequality for the Dirichlet Laplacian, which controls a finite Dirichlet spectral sum through observations of the normal derivative of its elliptic extension on such a boundary set. Our results also cover inverse heat source problems with interior observations on sets of positive $q$-dimensional Hausdorff content for some $q>d-1$, yielding logarithmic stability from full-time observations for general sources in $H_0^1(\Omega)$ and H\"older stability from terminal-time observations for sources in a suitable spectral Gevrey class.

math.SP↗

arXiv2026

Resolvent bounds and eigenvalue estimates of generalized Schr\"odinger operators with complex potentials on compact manifolds

We extend Cuenin's compact-manifold spectral bounds for Schr\"odinger operators with complex potentials to a general pseudodifferential setting. More precisely, we study operators $P+V$, where $P$ is a positive self-adjoint elliptic classical pseudodifferential operator of positive order and $V$ is complex-valued. The main analytic input is a resolvent principle showing that spectral cluster estimates for $P$ imply $L^p$-$L^{p'}$ resolvent estimates along suitable complex curves. Combined with Sogge's spectral cluster bounds, this yields exterior-region resolvent estimates extending those of Krupchyk and Uhlmann; we also prove direct resolvent bounds in the interior region. On Zoll manifolds, we discuss the sharpness of the resulting spectral bounds.

math.SP↗