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 Ly=−y″+v(x)yLy=−y″+v(x)y, considered with complex valued ππ-periodic potentials vv and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large nn, close to n2n2 there are two periodic (if nn is even) or antiperiodic (if nn is odd) eigenvalues λn−, λn+ and one Neumann eigenvalue νnνn. We study the geometry of “the spectral triangle” with vertices (λn+, λn−, νnνn), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for v∈Lp([0,π]),p>1, that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even (respectively, odd) nnsupλn+≠λn−{|λn+−νn|/|λn+−λn−|}<∞.