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 an id="mmlsi1" class="mathmlsrc">an class="formulatext 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si1.gif" overflow="scroll">w>Lw>w>2w>alse">(w>athvariant="double-struck">Rw>w>dw>;w>athvariant="double-struck">Cw>w>nw>alse">)ath>an>an>an>, we consider selfadjoint strongly elliptic second order differential operators an id="mmlsi2" class="mathmlsrc">an class="formulatext 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_εan>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">w>athvariant="script">Aw>w>εw>ath>an>an>an> with periodic coefficients depending on an id="mmlsi3" class="mathmlsrc">an class="formulatext 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">x/εan>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">athvariant="bold">xalse">/εath>an>an>an>. We study the behavior of the operator an id="mmlsi4" class="mathmlsrc">an class="formulatext 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">athvariant="normal">exp⁡alse">(−iw>athvariant="script">Aw>w>εw>τalse">)ath>an>an>an>, an id="mmlsi365" class="mathmlsrc">an class="formulatext 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∈Ran>an class="mathContainer hidden">an class="mathCode">ath altimg="si365.gif" overflow="scroll">τ∈athvariant="double-struck">Rath>an>an>an>, for small ε  . Approximations for this exponential in the an id="mmlsi6" class="mathmlsrc">an class="formulatext 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">alse">(w>Hw>w>sw>alse">→w>Lw>w>2w>alse">)ath>an>an>an>-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 an id="mmlsi1286" class="mathmlsrc">an class="formulatext 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">u_εan>an class="mathContainer hidden">an class="mathCode">ath altimg="si1286.gif" overflow="scroll">w>athvariant="bold">uw>w>εw>ath>an>an>an> of the Cauchy problem for the Schrödinger-type equation an id="mmlsi8" class="mathmlsrc">an class="formulatext 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∂_τu_ε=A_εu_ε+Fan>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">iw>∂w>w>τw>w>athvariant="bold">uw>w>εw>=w>athvariant="script">Aw>w>εw>w>athvariant="bold">uw>w>εw>+athvariant="bold">Fath>an>an>an>. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.