RUS  ENG
Full version
JOURNALS // Zapiski Nauchnykh Seminarov POMI

Zap. Nauchn. Sem. POMI, 2022, Volume 519, Pages 114–151 (Mi znsl7304)

Homogenization of a one-dimensional periodic elliptic operator at the edge of a spectral gap: operator estimates in the energy norm
A. A. Mishulovich, V. A. Sloushch, T. A. Suslina

References

1. A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., 5, North-Holland Publishing Co., 1978  mathscinet
2. N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984  mathscinet
3. V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Nauka, M., 1993  mathscinet
4. M. Sh. Birman, T. A. Suslina, “Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics”, Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Oper. Theory Adv. Appl., 129, Birkhäuser, Basel, 2001, 71–107  mathscinet  adsnasa
5. M. Sh. Birman, T. A. Suslina, “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva i usredneniya”, Algebra i analiz, 15:5 (2003), 1–108  mathnet
6. M. Sh. Birman, T. A. Suslina, “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora”, Algebra i analiz, 17:6 (2005), 1–104  mathnet
7. M. Sh. Birman, T. A. Suslina, “Usrednenie periodicheskikh differentsialnykh operatorov s uchetom korrektora. Priblizhenie reshenii v klasse Soboleva $H^1(\mathbb{R}^d)$”, Algebra i analiz, 18:6 (2006), 1–130  mathnet
8. V. V. Zhikov, “Ob operatornykh otsenkakh v teorii usredneniya”, Dokl. RAN, 403:3 (2005), 305–308  mathnet  mathscinet
9. V. V. Zhikov, S. E. Pastukhova, “On operator estimates for some problems in homogenization theory”, Russ. J. Math. Phys., 12:4 (2005), 515–524  mathscinet  elib
10. V. V. Zhikov, S. E. Pastukhova, “Ob operatornykh otsenkakh v teorii usredneniya”, Uspekhi matem. nauk, 71:3 (2016), 27–122  mathnet  mathscinet
11. M. Sh. Birman, “O protsedure usredneniya dlya periodicheskikh operatorov v okrestnosti kraya vnutrennei lakuny”, Algebra i analiz, 15:4 (2003), 61–71  mathnet
12. T. A. Suslina, A. A. Kharin, “Usrednenie s uchetom korrektora dlya periodicheskogo ellipticheskogo operatora vblizi kra vnutrennei lakuny”, Problemy matematicheskogo analiza, 41 (2009), 127–141
13. M. Sh. Birman, T. A. Suslina, “Usrednenie mnogomernogo periodicheskogo ellipticheskogo operatora v okrestnosti kraya vnutrennei lakuny”, Zapiski nauchnykh seminarov POMI, 318, 2004, 60–74  mathnet
14. T. A. Suslina, A. A. Kharin, “Usrednenie s uchetom korrektora dlya mnogomernogo periodicheskogo ellipticheskogo operatora vblizi kraya vnutrennei lakuny”, Problemy matematicheskogo analiza, 59 (2011), 177–193
15. A. R. Akhmatova, E. S. Aksenova, V. A. Sloushch, T. A. Suslina, “Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap”, Complex Variables and Elliptic Equations, 67:3 (2022), 523–555  crossref  mathscinet
16. W. Kirsch, B. Simon, “Comparison theorems for the gap of Schrödinger operators”, J. Funct. Anal., 75:2 (1987), 396–410  crossref  mathscinet
17. M. Sh. Birman, T. A. Suslina, “Two-dimensional periodic Pauli operator. The effective masses at the lower edge of the spectrum”, Oper. Theory Adv. Appl., 108, Birkhäuser, Basel, 1999, 13–31  mathscinet
18. M. Sh. Birman, “The discrete spectrum in gaps of the perturbed periodic Schrödinger operator. I. Regular perturbations”, Boundary Value Problems, Schrödinger Operators, Deformation Quantization: Adv. Partial Differential Equations, Math. Top., 8, Akademie Verlag, Berlin, 1995, 334–352  mathscinet
19. M. Sh. Birman, “Diskretnyi spektr v lakunakh vozmuschennogo periodicheskogo operatora Shredingera. II. Neregulyarnye vozmuscheniya”, Algebra i analiz, 9:6 (1997), 62–89  mathnet
20. E. Ch. Titchmarsh, Razlozheniya po sobstvennym funktsiyam, svyazannye s differentsialnymi uravneniyami vtorogo poryadka, v. 2, Izd-vo IL, M., 1961
21. T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972  mathscinet
22. M. Sh. Birman, M. Z. Solomyak, “Otsenki singulyarnykh chisel integralnykh operatorov”, Uspekhi mat. nauk, 32:1 (1977), 17–84  mathnet  mathscinet


© Steklov Math. Inst. of RAS, 2025