Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results

References

• Bakhvalov NS, Panasenko GP. Homogenization: Averaging processes in periodic media. Mathematical problems in mechanics of composite materials. Vol. 36. Mathematics and its Applications (Soviet Series). Dordrecht: Kluwer Academic Publishers Group; 1989.
   Google Scholar
• Bensoussan A, Lions J-L, Papanicolaou G. Asymptotic analysis for periodic structures. Vol. 5. Amsterdam: North-Holland; 1978. (Studies in Mathematics and its Applications).
   Google Scholar
• Zhikov VV, Kozlov SM, Olejnik OA. Homogenization of differential operators. Berlin: Springer-Verlag; 1994.
   Google Scholar
• Allaire G, Piatnitski A. Homogenization of the Schrödinger equation and effective mass theorems. Comm Math Phys. 2005;258:1–22.
   Web of Science ®Google Scholar
• Conca C, Orive R, Vanninathan M. Bloch approximation in homogenization and applications. SIAM J Math Anal. 2002;33(5):1166–1198.
   Web of Science ®Google Scholar
• Sevost'yanova EV. An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients. Math USSR-Sb. 1982;43(2):181–198.
   Google Scholar
• Birman MSh, Suslina TA. Homogenization with corrector term for periodic elliptic differential operators. Algebra Anal. 2005;17(6):1–104; English transl., St Petersburg Math J. 2006;17(6):897–973.
   Google Scholar
• Birman MSh, Suslina TA. Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics. In: Borichev AA, Nikolski NK, editors. Systems, approximation, singular integral operators, and related topics; 2000 Jun 13–16; Basel: Birkhäuser; 2001; p. 437–468.
   Google Scholar
• Birman MSh, Suslina TA. Second order periodic differential operators. threshold properties and homogenization. Algebra Anal. 2003;15(5):1–108; English transl., St Petersburg Math J. 2004;15(5):639–714.
   Google Scholar
• Birman MSh, Suslina TA. Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family. Algebra Anal. 2005;17(5):69–90; English transl., St Petersburg Math J. 2006;17(5):745–762.
   Google Scholar
• Birman MSh, Suslina TA. Homogenization with corrector term for periodic differential operators. Approximation of solutions in the Sobolev class H1(Rd). Algebra Anal. 2006;18(6):1–130; English transl., St Petersburg Math J. 2007;18(6):857–955.
   Google Scholar
• Vasilevskaya ES. A periodic parabolic Cauchy problem: homogenization with corrector. Algebra Anal. 2009;21(1):3–60; English transl., St Petersburg Math J. 2010;21(1):1–41.
   Google Scholar
• Vasilevskaya ES, Suslina TA. Threshold approximations for a factorized selfadjoint operator family with the first and second correctors taken into account. Algebra Anal. 2011;23(2):102–146; English transl., St Petersburg Math J. 2012;23(2):275–308.
   Google Scholar
• Vasilevskaya ES, Suslina TA. Homogenization of parabolic and elliptic periodic operators in L2(Rd) with the first and second correctors taken into account. Algebra Anal. 2012;24(2):1–103; English transl., St Petersburg Math J. 2013;24(2):185–261.
   Google Scholar
• Suslina TA. On homogenization of periodic parabolic systems. Funktsional Anal Prilozhen. 2004;38(4):86–90; English transl., Funct Anal Appl. 2004;38(4):309–312.
   Google Scholar
• Suslina TA. Homogenization of a periodic parabolic Cauchy problem. Amer Math Soc Transl. (2). 2007;220:201–233.
   Google Scholar
• Suslina TA. Homogenization of a periodic parabolic Cauchy problem in the Sobolev space H1(Rd). Math Model Nat Phenom. 2010;5(4):390–447.
   Web of Science ®Google Scholar
• Zhikov VV. On some estimates of homogenization theory. Dokl Ros Akad Nauk. 2005;406(5):597–601; English transl., Dokl Math. 2006;73:96–99.
   Google Scholar
• Zhikov VV, Pastukhova SE. On operator estimates for some problems in homogenization theory. Russ J Math Phys. 2005;12(4):515–524.
   Web of Science ®Google Scholar
• Zhikov VV, Pastukhova SE. Estimates of homogenization for a parabolic equation with periodic coefficients. Russ J Math Phys. 2006;13(2):224–237.
   Web of Science ®Google Scholar
• Zhikov VV, Pastukhova SE. Operator estimates in homogenization theory. Uspekhi Matem Nauk. 2016;71(429):27–122; English transl., Russian Math Surveys. 2016;71(3):417–511.
   Google Scholar
• Birman MSh, Suslina TA. Operator error estimates in the homogenization problem for nonstationary periodic equations. Algebra Anal. 2008;20(6):30–107; English transl., St Petersburg Math J. 2009;20(6):873–928.
   Google Scholar
• Dorodnyi MA, Suslina TA. Spectral approach to homogenization of hyperbolic equations with periodic coefficients. J Differ Equ. 2018;264(12):7463–7522.
   Web of Science ®Google Scholar
• Meshkova YM. On operator error estimates for homogenization of hyperbolic systems with periodic coefficients. J Spectr Theor. Forthcoming. Available from: https://arxiv.org/abs/1705.02531.
   Google Scholar
• Meshkova YM. On the homogenization of periodic hyperbolic systems, Matem. Zametki. 2019;105(6):937–942; English transl.: Math Notes. 2019;105(6):929–934.
   Web of Science ®Google Scholar
• Suslina TA. Spectral approach to homogenization of nonstationary Schrödinger-type equations. J Math Anal Appl.. 2017;446(2):1466–1523.
   Web of Science ®Google Scholar
• Dorodnyi MA. Homogenization of the periodic Schrödinger-type equations with the lower order terms. Algebra Anal. 2019;31(6):122–196; English transl.: St Petersburg Math J.  2020;31(6):1001–1054.
   Google Scholar
• Meshkova YM. Variations on the theme of the Trotter–Kato theorem for homogenization of periodic hyperbolic systems. arXiv:1904.02781 [Preprint]. 2019 [cited 2020 Mar 2]: [62 p.]. Available from: https://arxiv.org/abs/1904.02781.
   Google Scholar
• Kato T. Perturbation theory for linear operators. Berlin: Springer-Verlag; 1995.
   Google Scholar
• Conca C, Orive R, Vanninathan M. On Burnett coefficients in periodic media. J Math Phys. 2006;47(3). https://doi.org/10.1063/1.2179048
   Web of Science ®Google Scholar