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

Abstract

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 n2 there are two periodic (if n is even) or antiperiodic (if n is odd) eigenvalues λn−, λn+ and one Neumann eigenvalue νn. We study the geometry of “the spectral triangle” with vertices (λn+, λn−, νn), 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) nsupλn+≠λn−{|λn+−νn|/|λn+−λn−|}<∞.