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Characterization of potential smoothness and Riesz basis property of the Hill-Scrödinger operator in terms of periodic, antiperiodic and Neumann spectra

The Hill operators $Ly=-y"+v(x)y$, considered with complex valued $π$-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 $λ_n^-$, $λ_n^+$ and one Neumann eigenvalue $ν_n$. We study the geometry of "the spectral triangle" with vertices ($λ_n^+$,$λ_n^-$,$ν_n$), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for $v\in L^p ([0,π]), \; p>1,$ that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even $n$ (respectively, odd $n$) $ \; \sup_{λ_n^+\neq λ_n^-}\{|λ_n^+-ν_n|/|λ_n^+-λ_n^-| \} < \infty. $