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 Schrö ; 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

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In mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si1.gif" overflow="scroll">L2(mathvariant="double-struck">Rd;mathvariant="double-struck">Cn)math>, we consider selfadjoint strongly elliptic second order differential operators mathmlsrc">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_εmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">mathvariant="script">Aεmath> with periodic coefficients depending on mathmlsrc">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">x/εmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">mathvariant="bold">x/εmath>. We study the behavior of the operator mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">mathvariant="normal">exp⁡(−imathvariant="script">Aετ)math>, mathmlsrc">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∈RmathContainer hidden">mathCode"><math altimg="si365.gif" overflow="scroll">τ∈mathvariant="double-struck">Rmath>, for small ε  . Approximations for this exponential in the mathmlsrc">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)mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll">(Hs→L2)math>-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 mathmlsrc">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">u_εmathContainer hidden">mathCode"><math altimg="si1286.gif" overflow="scroll">mathvariant="bold">uεmath> of the Cauchy problem for the Schrödinger-type equation mathmlsrc">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∂_τu_ε=A_εu_ε+FmathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll">i∂τmathvariant="bold">uε=mathvariant="script">Aεmathvariant="bold">uε+mathvariant="bold">Fmath>. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.