On complex Fermi curves of two-dimensional periodic Schrödinger operators [Elektronische Ressource] / vorgelegt von Alexander Klauer

ON COMPLEX FERMICURVES OFTWO-DIMENSIONALPERIODIC SCHRODINGEROPERATORSInauguraldissertationzur Erlangung des akademischen Gradeseines Doktors der Naturwissenschaftender Universitat Mannheimvorgelegt vonDipl.-Phys. Alexander Klaueraus HeidelbergMannheim, 2011Dekan: Professor Dr. Wolfgang E elsberg, Universit at MannheimReferent: Professor Dr. Martin Schmidt, Universitat MannheimKorreferent: Professor Dr. Andreas Knauf, Universitat Erlangen-NurnbergTag der mundlichen Prufung: 3. Juni 2011 AbstractIn dimensionsd 2, the complex Bloch varieties and the associated Fermi curvesof periodic Schrodinger operators with quasi-periodic boundary conditions arede ned as complex analytic varieties. The Schr odinger potentials are takend=2from the Lebesgue space L in the case d> 2, and from the Lorentz{Fourier1;1spaceF‘ in the case d = 2. Then, an asymptotic analysis of the Fermicurves in the cased = 2 is performed. The decomposition of a Fermi curve intoa compact part, an asymptotically free part, and thin handles, is recovered asexpected. Furthermore, it is shown that the set of potentials whose associated1;1Fermi curve has nite geometric genus is a dense subset of F‘ . Moreover, theFourier transforms of the potentials are locally isomorphic to perturbed Fouriertransforms induced by the handles. Finally, an asymptotic family of parametersdescribing the sizes of the handles is introduced.