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Under the assumption that $V \in L^2([0,\pi]; dx)$, we derive necessary andsufficient conditions for (non-self-adjoint) Schr\"odinger operators$-d^2/dx^2+V$ in $L^2([0,\pi]; dx)$ with periodic and antiperiodic boundaryconditions to possess a Riesz basis of root vectors (i.e., eigenvectors andgeneralized eigenvectors spanning the range of the Riesz projection associatedwith the corresponding periodic and antiperiodic eigenvalues). We also discussthe case of a Schauder basis for periodic and antiperiodic Schr\"odingeroperators $-d^2/dx^2+V$ in $L^p([0,\pi]; dx)$, $p \in (1,\infty)$.

A Schauder and Riesz Basis Criterion for Non-Self-Adjoint Schr\"odinger Operators with Periodic and Antiperiodic Boundary Conditions