Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 200

On complex Bloch-spaces of periodic Schroedinger operators

M. U. Schmidt

We generalize a result of Kn"orrer and Trubowitz, that the infinite energy limit of the complex Bloch-variety of a two-dimensional periodic Schr"odinger operator contains the complex Bloch-varieties of the averaged one-dimensional potentials to arbitrary dimensions. In fact, a completion of the complex Bloch-variety contains in the infinite energy limit all Bloch-varieties of the non-trivial averaged potentials. Finally we give a characterization of all complex Bloch-spaces, whose completions are compact complex spaces. The corresponding periodic Schr"odinger operators are exactly the so-called algebraically integrable Schr"odinger operators, which roughly speaking have a maximal number of commuting differential operators.

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