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A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and antiperiodic boundary conditions. (English) Zbl 1251.34100
The authors study the generally non-self-adjoint Schrödinger operators \(H^P\) and \(H^{AP}\) in the Hilbert space \(L^2([0,\pi];dx)\) associated with the differential expression \(L:=-\frac{d^2}{dx^2}+V(x)\), \(x\in [0,\pi]\) and complex-valued potential \(V\) satisfying \(V\in L^2([0,\pi];\,dx)\), with periodic and antiperiodic boundary conditions. They give necessary and sufficient conditions in terms of spectral data for the above operators 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). The case of a Schauder basis for the above periodic and antiperiodic Schrödinger operators is also investigated.

MSC:
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
47B40 Spectral operators, decomposable operators, well-bounded operators, etc.
47A10 Spectrum, resolvent
34L05 General spectral theory of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
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