Characterization of potential smoothness and Riesz basis property of Hill-Scr\"odinger operators with singular periodic potentials in terms of periodic, antiperiodic and Neumann spectra

The Hill operators Ly=-y''+v(x)y, considered with singular complex valued \pi-periodic potentials v of the form v=Q' with Q in L^2([0,\pi]), and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large n, the disc {z: |z-n^2|<n} contains two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues \lambda_n^-, \lambda_n^+ and one Neumann eigenvalue \nu_n. We show that rate of decay of the sequence |\lambda_n^+-\lambda_n^-|+|\lambda_n^+ - \nu_n| determines the potential smoothness, and there is a basis consisting of periodic (or antiperiodic) root functions if and only if for even (respectively, odd) n, \sup_{\lambda_n^+\neq \lambda_n^-}{|\lambda_n^+-\nu_n|/|\lambda_n^+-\lambda_n^-|} < \infty.