On point spectrum of Jacobi matrices generated by iterations of quadratic polynomials

TL;DR

This paper investigates the point spectrum of a class of limit-periodic Jacobi matrices associated with quadratic polynomials, showing that all elements in their hull have empty point spectrum, advancing understanding in spectral theory of ergodic operators.

Contribution

It proves that for Jacobi matrices generated by quadratic polynomial iterations, the entire hull has no point spectrum, a novel result in the spectral analysis of such operators.

Findings

All elements of the hull have empty point spectrum.

The result applies to matrices associated with the Julia set of $z^2-\lambda$ for large $\lambda$.

Advances the spectral theory of ergodic operators with zero measure spectrum.

Abstract

In general, point spectrum of an almost periodic Jacobi matrix can depend on the element of the hull. In this paper, we study the hull of the limit-periodic Jacobi matrix corresponding to the equilibrium measure of the Julia set of the polynomial $z^2-\lambda$ with large enough $\lambda$; this is the leading model in inverse spectral theory of ergodic operators with zero measure spectrum. We prove that every element of the hull has empty point spectrum.