Spectral approach to homogenization of nonstationary Schrödinger-type equations
详细信息    查看全文
文摘
In class="mathmlsrc">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(Rd;Cn)class="mathContainer hidden">class="mathCode">croll">L2chy="false">(ck">Rd;ck">Cnchy="false">), we consider selfadjoint strongly elliptic second order differential operators class="mathmlsrc">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">Aεclass="mathContainer hidden">class="mathCode">croll">cript">Aε with periodic coefficients depending on class="mathmlsrc">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">ng>xng>/εclass="mathContainer hidden">class="mathCode">croll">xchy="false">/ε. We study the behavior of the operator class="mathmlsrc">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⁡(−iAετ)class="mathContainer hidden">class="mathCode">croll">expchy="false">(icript">Aετchy="false">), class="mathmlsrc">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∈Rclass="mathContainer hidden">class="mathCode">croll">τck">R, for small ε  . Approximations for this exponential in the class="mathmlsrc">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">(Hs→L2)class="mathContainer hidden">class="mathCode">croll">chy="false">(Hschy="false">→L2chy="false">)-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 class="mathmlsrc">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">ng>ung>εclass="mathContainer hidden">class="mathCode">croll">uε of the Cauchy problem for the Schrödinger-type equation class="mathmlsrc">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∂τng>ung>ε=Aεng>ung>ε+ng>Fng>class="mathContainer hidden">class="mathCode">croll">iτuε=cript">Aεuε+F. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.

© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号

地址:北京市海淀区学院路29号 邮编:100083

电话:办公室:(+86 10)66554848;文献借阅、咨询服务、科技查新:66554700