Natural Boundaries and Spectral Theory

TL;DR

This paper explores the analogy between spectral properties of Schroedinger operators and natural boundaries in power series, introducing new examples and generalizations in the context of spectral theory and complex analysis.

Contribution

It introduces novel generalizations of natural boundary examples and extends the analogy between spectral theory and power series boundaries, including cases without independence assumptions.

Findings

Generalized Hecke's example for natural boundaries

Constructed natural boundary examples for non-independent random power series

Established a new analogy between spectral theory and power series boundaries

Abstract

We present and exploit an analogy between lack of absolutely continuous spectrum for Schroedinger operators and natural boundaries for power series. Among our new results are generalizations of Hecke's example and natural boundary examples for random power series where independence is not assumed.