A Schauder and Riesz Basis Criterion for Non-Self-Adjoint Schr\"odinger   Operators with Periodic and Antiperiodic Boundary Conditions
  Under the assumption that $V \in L^2([0,\pi]; dx)$, we derive necessary and sufficient conditions for (non-self-adjoint) Schr\"odinger operators $-d^2/dx^2+V$ in $L^2([0,\pi]; dx)$ 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 Schr\"odinger operators $-d^2/dx^2+V$ in $L^p([0,\pi]; dx)$, $p \in (1,\infty)$. 
Fritz Gesztesy and Vadim Tkachenko
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