Spectral properties of some complex Jacobi matrices

TL;DR

This paper investigates the spectral characteristics of complex Jacobi matrices, establishing conditions for spectrum continuity, asymptotics of eigenvectors, and extensions, using generalized Turán determinants.

Contribution

It introduces new conditions for spectrum continuity and eigenvector asymptotics of complex Jacobi matrices, extending analysis to unbounded and perturbed cases.

Findings

Spectrum is continuous on certain complex subsets

Uniform asymptotics for generalized eigenvectors provided

Conditions for unique closed extensions established

Abstract

We study spectral properties of bounded and unbounded complex Jacobi matrices. In particular, we formulate conditions assuring that the spectrum of the studied operators is continuous on some subsets of the complex plane and we provide uniform asymptotics of their generalised eigenvectors. We illustrate our results by considering complex perturbations of real Jacobi matrices belonging to several classes: asymptotically periodic, periodically modulated and the blend of these two. Moreover, we provide conditions implying existence of a unique closed extension. The method of the proof is based on the analysis of a generalisation of shifted Tur\'an determinants to the complex setting.