A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions

Under the assumption that V∈L2([0,π];dx), we derive necessary and sufficient conditions in terms of spectral data for (non-self-adjoint) Schrödinger operators −d2/dx2+V in L2([0,π];dx) with periodic and antiperiodic boundary conditions to possess a Riesz basis of root vectors (i.e., eigenvectors and generalized eigenvectors spanning the range of the Riesz projection associated with the corresponding periodic and antiperiodic eigenvalues).We also discuss the case of a Schauder basis for periodic and antiperiodic Schrödinger operators −d2/dx2+V in Lp([0,π];dx), p∈(1,∞).