文摘
In thmlsrc">text 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)thContainer hidden">thCode">th altimg="si1.gif" overflow="scroll">L2tretchy="false">(thvariant="double-struck">Rd;thvariant="double-struck">Cntretchy="false">)th>, we consider selfadjoint strongly elliptic second order differential operators thmlsrc">text 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">AεthContainer hidden">thCode">th altimg="si2.gif" overflow="scroll">thvariant="script">Aεth> with periodic coefficients depending on thmlsrc">text 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">trong>xtrong>/εthContainer hidden">thCode">th altimg="si3.gif" overflow="scroll">thvariant="bold">xtretchy="false">/εth>. We study the behavior of the operator thmlsrc">text 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)thContainer hidden">thCode">th altimg="si4.gif" overflow="scroll">thvariant="normal">exptretchy="false">(−ithvariant="script">Aετtretchy="false">)th>, thmlsrc">text 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∈RthContainer hidden">thCode">th altimg="si365.gif" overflow="scroll">τ∈thvariant="double-struck">Rth>, for small ε . Approximations for this exponential in the thmlsrc">text 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)thContainer hidden">thCode">th altimg="si6.gif" overflow="scroll">tretchy="false">(Hstretchy="false">→L2tretchy="false">)th>-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 thmlsrc">text 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">trong>utrong>εthContainer hidden">thCode">th altimg="si1286.gif" overflow="scroll">thvariant="bold">uεth> of the Cauchy problem for the Schrödinger-type equation thmlsrc">text 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∂τtrong>utrong>ε=Aεtrong>utrong>ε+trong>Ftrong>thContainer hidden">thCode">th altimg="si8.gif" overflow="scroll">i∂τthvariant="bold">uε=thvariant="script">Aεthvariant="bold">uε+thvariant="bold">Fth>. Applications to the nonstationary Schrödinger equation and the two-dimensional Pauli equation with singular rapidly oscillating potentials are given.