Spectral approach to homogenization of nonstationary Schrödinger-type equations

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• 作者：Tatiana Suslina ^{t.suslina@spbu.ru" class="auth_mail" title="E-mail the corresponding author suslina@list.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Periodic differential operators ; Nonstationary Schr&ouml ; dinger-type equations ; Homogenization ; Effective operator ; Operator error estimates
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1466-1523
• 全文大小：1119 K

文摘

In stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=d42d061dcd87a65e716e5a86ad8c195c" title="Click to view the MathML source">L₂(R^d;Cⁿ), we consider selfadjoint strongly elliptic second order differential operators stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si2.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=969c11f2bc2e6e1715da9763c4539e8f" title="Click to view the MathML source">A_ε${\mathcal{A}}_{\epsilon }$ with periodic coefficients depending on stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si3.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=85156b3cb7a9f8805c4d51a546268936" title="Click to view the MathML source"><strong>xstrong>/ε. We study the behavior of the operator stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si4.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=1bbd15efb58f210660fddab3ab687e80" title="Click to view the MathML source">exp⁡(−iA_ετ), stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si365.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=46c6e9c3447192c39755654b2423aeba" title="Click to view the MathML source">τ∈R, for small ε  . Approximations for this exponential in the stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si6.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=a5ab331a6e2f5b861f114dd6729c5f4f" title="Click to view the MathML source">(H^s→L₂)-operator norm are obtained. The method is based on the scaling transformation, the Floquet–Bloch theory, and the analytic perturbation theory. The results are applied to study the behavior of the solution stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si1286.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=e5c438188906e0add42f9456d96a9a0d" title="Click to view the MathML source"><strong>ustrong>_ε${\mathbf{u}}_{\epsilon }$ of the Cauchy problem for the Schr&ouml;dinger-type equation stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305406&_mathId=si8.gif&_user=111111111&_pii=S0022247X16305406&_rdoc=1&_issn=0022247X&md5=f846d0c29ab9c09c758b651047251de5" title="Click to view the MathML source">i∂_τ<strong>ustrong>_ε=A_ε<strong>ustrong>_ε+<strong>Fstrong>$i{\partial }_{\tau }{\mathbf{u}}_{\epsilon }={\mathcal{A}}_{\epsilon }{\mathbf{u}}_{\epsilon }+\mathbf{F}$. Applications to the nonstationary Schr&ouml;dinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.