Sturm-Liouville operators with periodically modulated parameters. Part I: Regular case

TL;DR

This paper introduces a new class of Sturm-Liouville operators with periodic parameter modulation, analyzing their spectral properties and proving the spectral density's continuity and positivity across the real line.

Contribution

It presents a novel class of operators and investigates their spectral characteristics, including the behavior of the spectral density and monodromy matrix dependence.

Findings

Spectral density is continuous and positive everywhere.

Spectral properties depend on the monodromy matrix at zero.

Asymptotic analysis of Christoffel functions and density of states.

Abstract

We introduce a new class of Sturm-Liouville operators with periodically modulated parameters. Their spectral properties depend on the monodromy matrix of the underlying periodic problem computed for the spectral parameter equal to $0$. Under certain assumptions, by studying the asymptotic behavior of Christoffel functions and density of states, we prove that the spectral density is a continuous positive everywhere function on the real line.