Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators

TL;DR

This paper establishes necessary and sufficient conditions for the root function systems of Hill and 1D Dirac operators to form Riesz bases, using spectral gap and deviation analysis, with implications for various function spaces.

Contribution

It provides the first comprehensive criteria for Riesz basis property of root functions of Hill and Dirac operators with singular potentials.

Findings

Derived conditions based on Fourier coefficients and spectral gaps

Proved equivalence of basis properties in different function spaces

Applied equiconvergence theorems to analyze basis properties

Abstract

We study the system of root functions (SRF) of Hill operator $Ly = -y^{\prime \prime} +vy $ with a singular potential $v \in H^{-1}_{per}$ and SRF of 1D Dirac operator $ Ly = i {pmatrix} 1 & 0 0 & -1 {pmatrix} \frac{dy}{dx} + vy $ with matrix $L^2$-potential $v={pmatrix} 0 & P Q & 0 {pmatrix},$ subject to periodic or anti-periodic boundary conditions. Series of necessary and sufficient conditions (in terms of Fourier coefficients of the potentials and related spectral gaps and deviations) for SRF to contain a Riesz basis are proven. Equiconvergence theorems are used to explain basis property of SRF in $L^p$-spaces and other rearrangement invariant function spaces.