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 hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=496d06cac5167bb05b520c17ce6a153c">height="39" width="152" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si1.gif">hContainer hidden">hCode">h altimg="si1.gif" overflow="scroll">V=w>(0hvariant="script">Phy="false">(xhy="false">)hvariant="script">Qhy="false">(xhy="false">)0)w>h>, where hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_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])hContainer hidden">hCode">h altimg="si2.gif" overflow="scroll">hvariant="script">P,hvariant="script">Q∈w>Lw>w>2w>hy="false">(hy="false">[0,πhy="false">]hy="false">)h> subject to periodic, antiperiodic or a general strictly regular boundary condition (bc  ), has discrete spectrums. It is known that, for large enough hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=b9a75d53e551fefb5817d04b7dda594a" title="Click to view the MathML source">|n|hContainer hidden">hCode">h altimg="si3.gif" overflow="scroll">hy="false">|nhy="false">|h> 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 hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si25.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=23e9d467e09d9a2f343bb1f909242af5">height="17" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si25.gif">hContainer hidden">hCode">h altimg="si25.gif" overflow="scroll">w>λw>w>nw>w>+w>h> and hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si26.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=c305904ecec99c24a8c727f639170d95">height="16" width="20" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si26.gif">hContainer hidden">hCode">h altimg="si26.gif" overflow="scroll">w>λw>w>nw>w>−w>h> (counted according to multiplicity) and one eigenvalue hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si115.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=6aa69c8512246b326ac5a65f85b6e25e">height="18" width="24" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si115.gif">hContainer hidden">hCode">h altimg="si115.gif" overflow="scroll">w>μw>w>nw>w>bcw>h> corresponding to the boundary condition hmlsrc">hImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si340.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=318fd7cdf3a0be8dfe085d8575d157e4" title="Click to view the MathML source">(bc)hContainer hidden">hCode">h altimg="si340.gif" overflow="scroll">hy="false">(bchy="false">)h>. We prove that the smoothness of the potential could be characterized by the decay rate of the sequence hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si8.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=06ff90cf8b556ea0ea1b57c4faaed053">height="18" width="74" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si8.gif">hContainer hidden">hCode">h altimg="si8.gif" overflow="scroll">hy="false">|w>δw>w>nw>w>bcw>hy="false">|+hy="false">|w>γw>w>nw>hy="false">|h>, where hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si9.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=f63944d2e46165be391bb6ecf0c7039c">height="18" width="104" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si9.gif">hContainer hidden">hCode">h altimg="si9.gif" overflow="scroll">w>δw>w>nw>w>bcw>=w>μw>w>nw>w>bcw>−w>λw>w>nw>w>+w>h> and hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si10.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=2bb36cf1cfc1a6a5642226307f24bbde">height="17" width="98" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si10.gif">hContainer hidden">hCode">h altimg="si10.gif" overflow="scroll">w>γw>w>nw>=w>λw>w>nw>w>+w>−w>λw>w>nw>w>−w>h>. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if hmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305844&_mathId=si11.gif&_user=111111111&_pii=S0022247X16305844&_rdoc=1&_issn=0022247X&md5=77476112631f955ea858993842f7b805">height="35" width="63" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16305844-si11.gif">hContainer hidden">hCode">h altimg="si11.gif" overflow="scroll">hvariant="normal">supw>w>γw>w>nw>≠0w>⁡w>hy="false">|w>δw>w>nw>w>bcw>hy="false">|w>w>hy="false">|w>γw>w>nw>hy="false">|w>h> is finite.