Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

详细信息    查看全文

• 作者：İlker Arslan ^{ilkerarslan@sabanciuniv.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Dirac operator ; Potential smoothness ; Riesz basis property
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 March 2017
• 年：2017
• 卷：447
• 期：1
• 页码：84-108
• 全文大小：472 K

文摘

The one-dimensional Dirac operator with periodic potential $V=\left(\begin{array}{cc}\hfill 0\hfill & \hfill \mathcal{P}(x)\hfill \\ \hfill \mathcal{Q}(x)\hfill & \hfill 0\hfill \end{array}\right)$, where P,Q∈L²([0,π])$\mathcal{P},\mathcal{Q}\in L^2([0,\pi ])$ subject to periodic, antiperiodic or a general strictly regular boundary condition (bc  ), has discrete spectrums. It is known that, for large enough |n|$|n|$ in the disk centered at n of radius 1/2, the operator has exactly two (periodic if n is even or antiperiodic if n   is odd) eigenvalues ${\lambda }_n^{+}$ and ${\lambda }_n^{-}$ (counted according to multiplicity) and one eigenvalue ${\mu }_n^{bc}$ corresponding to the boundary condition (bc)$(bc)$. We prove that the smoothness of the potential could be characterized by the decay rate of the sequence $|{\delta }_n^{bc}|+|{\gamma }_n|$, where ${\delta }_n^{bc}={\mu }_n^{bc}-{\lambda }_n^{+}$ and ${\gamma }_n={\lambda }_n^{+}-{\lambda }_n^{-}$. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if $\underset{{\gamma }_n\ne 0}{\sup }\frac{|{\delta }_n^{bc}|}{|{\gamma }_n|}$ is finite.