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Characterization of potential smoothness and Riesz basis property of Hill-Scrödinger 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 π-periodic potentials v of the form v=Q' with Q in L^2([0,π]), 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 λ_n^-, λ_n^+ and one Neumann eigenvalue ν_n. We show that rate of decay of the sequence |λ_n^+-λ_n^-|+|λ_n^+ - ν_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_{λ_n^+\neq λ_n^-}{|λ_n^+-ν_n|/|λ_n^+-λ_n^-|} < \infty.