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

The Hill operators L y = − 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 ∈ L p ( [ 0 , π ] ) , p > 1 , that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even (respectively, odd) n sup λ n + ≠ λ n − { | λ n + − ν n | / | λ n + − λ n − | } < ∞ .