On the Riesz basis property of root vectors system for 2 × 2 Dirac type operators

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• 作者：Anton A. Lunyov ; ^{A.A.Lunyov@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Mark M. Malamud ^{mmm@telenet.dn.ua" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Systems of ordinary differential equations ; Regular boundary conditions ; Transformation operators ; Riesz basis property ; Timoshenko beam model
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 September 2016
• 年：2016
• 卷：441
• 期：1
• 页码：57-103
• 全文大小：904 K

文摘

The paper is concerned with the Riesz basis property of a boundary value problem associated in 257&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=a3b2a860d42f9086d452b21f1fdde5a3" title="Click to view the MathML source">L²[0,1]⊗C²$L^2[0,1]\otimes {\mathbb{C}}^2$ with the following 257&_mathId=si2.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=c7172983214de82c4f6672630723f9a1" title="Click to view the MathML source">2×2$2\times 2$ Dirac type equation with a summable potential matrix 257&_mathId=si391.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=06f35a8d7e90557c774febb46f669ddd" title="Click to view the MathML source">Q∈L¹[0,1]⊗C^2×2$Q\in L^1[0,1]\otimes {\mathbb{C}}^{2\times 2}$ and 257&_mathId=si5.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=166e3270bc6cffd498094e5ecad71c5e" title="Click to view the MathML source">b₁<0<b₂$b_1<0<b_2$. If 257&_mathId=si6.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=8b11ede2472a9c916d7eb8d0faf0e863" title="Click to view the MathML source">b₂=−b₁=1$b_2=-b_1=1$ this equation is equivalent to one dimensional Dirac equation. It is proved that the system of root functions of a linear boundary value problem constitutes a Riesz basis in 257&_mathId=si1.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=a3b2a860d42f9086d452b21f1fdde5a3" title="Click to view the MathML source">L²[0,1]⊗C²$L^2[0,1]\otimes {\mathbb{C}}^2$ provided that the boundary conditions are strictly regular  . By analogy with the case of ordinary differential equations, boundary conditions are called strictly regular if the eigenvalues of the corresponding unperturbed 257&_mathId=si8.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=dbc8c931a850e02efa4375bdd798413c" title="Click to view the MathML source">(Q=0)$(Q=0)$ operator are asymptotically simple and separated. In opposite to the Dirac case there is no simple algebraic criterion of the strict regularity whenever 257&_mathId=si692.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=955b3125697b452f7facfcc57e53f5de" title="Click to view the MathML source">b₁+b₂≠0$b_1+b_2\ne 0$. However under certain restrictions on coefficients of the boundary linear forms we present certain algebraic criteria of the strict regularity in the latter case. In particular, it is shown that regular separated boundary conditions are always strictly regular while antiperiodic boundary conditions are strictly regular if 257&_mathId=si10.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=eba3293af14a241d5ccd3b3f0fdaa06e" title="Click to view the MathML source">b₁$b_1$, 257&_mathId=si11.gif&_user=111111111&_pii=S0022247X16300257&_rdoc=1&_issn=0022247X&md5=f9637def38862d509d367e8466ef796e" title="Click to view the MathML source">b₂$b_2$ are coprime integers of different parity. The proof of the main result is based on existence of triangular transformation operators for system (0.1). Their existence is also established here in the case of a summable Q. In the case of regular (but not strictly regular) boundary conditions we prove the Riesz basis property with parentheses. The main results are applied to establish the Riesz basis property of the dynamic generator of spatially non-homogeneous damped Timoshenko beam model.