English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The Hill operators $Ly=-y"+v(x)y$, considered with complex valued$\pi$-periodic potentials $v$ and subject to periodic, antiperiodic or Neumannboundary 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 $\nu_n$. Westudy the geometry of "the spectral triangle" with vertices($\lambda_n^+$,$\lambda_n^-$,$\nu_n$), and show that the rate of decay oftriangle size characterizes the potential smoothness. Moreover, it is proved,for $v\in L^p ([0,\pi]), \; p>1,$ that the set of periodic (antiperiodic) rootfunctions 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. $

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