Characterization of potential smoothness and Riesz basis property of the Hill-Scr\"odinger operator in terms of periodic, antiperiodic and Neumann spectra

TL;DR

This paper investigates the spectral properties of Hill operators with complex potentials, linking the geometry of spectral triangles to potential smoothness and establishing conditions for the root functions to form a Riesz basis.

Contribution

It provides a new characterization of potential smoothness via spectral triangle decay and criteria for Riesz basis property based on spectral ratios.

Findings

Spectral triangle size decay characterizes potential smoothness.

Riesz basis property depends on bounded spectral ratios for certain eigenvalues.

Spectral geometry offers insights into the operator's spectral and basis properties.

Abstract

The Hill operators $Ly=-y"+v(x)y$, considered with complex valued $\pi$-periodic potentials $v$ and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large $n,$ close to $n^2$ there are two periodic (if $n$ is even) or antiperiodic (if $n$ is odd) eigenvalues $\lambda_n^-$, $\lambda_n^+$ and one Neumann eigenvalue $\nu_n$. We study the geometry of "the spectral triangle" with vertices ($\lambda_n^+$,$\lambda_n^-$,$\nu_n$), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for $v\in L^p ([0,\pi]), \; p>1,$ that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even $n$ (respectively, odd $n$) $ \; \sup_{\lambda_n^+\neq \lambda_n^-}\{|\lambda_n^+-\nu_n|/|\lambda_n^+-\lambda_n^-| \} < \infty. $