Equiconvergence of spectral decompositions of Hill–Schrödinger operators

Official URL: http://dx.doi.org/10.1016/j.jde.2013.07.030

We study in various functional spaces the equiconvergence of spectral decompositions of the Hill operator L = d(2)/dx(2) + v(x), x is an element of [0, pi], with H-per(-1)-potential and the free operator L-0 = -d(2)/dx(2), subject to periodic, antiperiodic or Dirichlet boundary conditions. In particular, we prove that parallel to S-N - S-N(0) : L-a -> L-b parallel to -> 0 if 1 < a <= b < infinity, 1/a - 1/b < 1/2, where S-N and S-N(0) are the N-th partial sums of the spectral decompositions of L and L-0. Moreover, if v is an element of H-alpha with 1/2 < alpha < 1 and 1/a = 3/2 - alpha, then we obtain uniform equiconvergence: parallel to S-N - S-N(0) : L-a -> L-infinity parallel to -> 0 as N -> infinity.