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

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• 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 844&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=496d06cac5167bb05b520c17ce6a153c">844-si1.gif">$V=\left(\begin{array}{cc}\hfill 0\hfill & \hfill \mathcal{P}(x)\hfill \\ \hfill \mathcal{Q}(x)\hfill & \hfill 0\hfill \end{array}\right)$, where 844&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=df14a243d399eb43f1480ff25ecccca2" title="Click to view the MathML source">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 844&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=b9a75d53e551fefb5817d04b7dda594a" title="Click to view the MathML source">|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 844&_mathId=si25.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=23e9d467e09d9a2f343bb1f909242af5">844-si25.gif">${\lambda }_n^{+}$ and 844&_mathId=si26.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=c305904ecec99c24a8c727f639170d95">844-si26.gif">${\lambda }_n^{-}$ (counted according to multiplicity) and one eigenvalue 844&_mathId=si115.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=6aa69c8512246b326ac5a65f85b6e25e">844-si115.gif">${\mu }_n^{bc}$ corresponding to the boundary condition 844&_mathId=si340.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=318fd7cdf3a0be8dfe085d8575d157e4" title="Click to view the MathML source">(bc)$(bc)$. We prove that the smoothness of the potential could be characterized by the decay rate of the sequence 844&_mathId=si8.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=06ff90cf8b556ea0ea1b57c4faaed053">844-si8.gif">$|{\delta }_n^{bc}|+|{\gamma }_n|$, where 844&_mathId=si9.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=f63944d2e46165be391bb6ecf0c7039c">844-si9.gif">${\delta }_n^{bc}={\mu }_n^{bc}-{\lambda }_n^{+}$ and 844&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=2bb36cf1cfc1a6a5642226307f24bbde">844-si10.gif">${\gamma }_n={\lambda }_n^{+}-{\lambda }_n^{-}$. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if 844&_mathId=si11.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=77476112631f955ea858993842f7b805">844-si11.gif">$\underset{{\gamma }_n\ne 0}{\sup }\frac{|{\delta }_n^{bc}|}{|{\gamma }_n|}$ is finite.