Characterization of potential smoothness and the Riesz basis property of the Hill-Schr?dinger operator in terms of periodic, antiperiodic and Neumann spectra

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• 作者：Ahmet Batal ^{ahmetbatal@sabanciuniv.edu}
• 关键词：Hill operator ; Potential smoothness ; Riesz bases
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2013
• 出版时间：15 September, 2013
• 年：2013
• 卷：405
• 期：2
• 页码：453-465
• 全文大小：457 K

文摘

The Hill operators , considered with complex valued -periodic potentials and subject to periodic, antiperiodic or Neumann boundary conditions have discrete spectra. For sufficiently large , close to there are two periodic (if is even) or antiperiodic (if is odd) eigenvalues , and one Neumann eigenvalue . We study the geometry of ¡°the spectral triangle¡± with vertices (, , ), and show that the rate of decay of triangle size characterizes the potential smoothness. Moreover, it is proved, for , that the set of periodic (antiperiodic) root functions contains a Riesz basis if and only if for even (respectively, odd) .