Geodesic
Homogenization in the Scattering Problem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 2-13 Cet article a éte moissonné depuis la source Math-Net.Ru

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The scattering problem is studied, which is described by the equation $(-\Delta_x+q(x,x/\varepsilon)-E)\psi=f(x)$, where $\psi=\psi(x,\varepsilon)\in\mathbb{C}$, $x\in\mathbb{R}^d$, $\varepsilon>0$, $E>0$, the function $q(x,y)$ is periodic with respect to $y$, and the function $f$ is compactly supported. The solution satisfying radiation conditions at infinity is considered, and its asymptotic behavior as $\varepsilon\to0$ is described. The asymptotic behavior of the scattering amplitude of a plane wave is also considered. It is shown that in principal order both the solution and the scattering amplitude are described by the homogenized equation with potential $$ \hat{q}(x)=\frac1{|\Omega|}\int_\Omega q(x,y)\,dy. $$
Keywords: scattering problem for the Schoedinger equation, two-scale dependence of potential on coordinates, homogenization, static load model. 
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V. S. Buslaev; A. A. Pozharskii. Homogenization in the Scattering Problem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 4, pp. 2-13. http://geodesic.mathdoc.fr/item/FAA_2010_44_4_a0/

[1] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[2] M. Sh. Birman, T. A. Suslina, “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva i usredneniya”, Algebra i analiz, 15:5 (2003), 1–108 | MR | Zbl

[3] M. Sh. Birman, T. A. Suslina, “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora”, Algebra i analiz, 17:6 (2005), 1–104 | MR | Zbl

[4] 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 | MR | Zbl

[5] D. I. Borisov, “Asimptotiki dlya reshenii ellipticheskikh sistem s bystro ostsilliruyuschimi koeffitsientami”, Algebra i analiz, 20:2 (2008), 19–42 | MR

[6] B. R. Vainberg, Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki, Izd-vo MGU, M., 1982 | MR | Zbl

[7] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993 | MR | Zbl

[8] O. A. Oleinik, G. A. Iosifyan, A. S. Shamaev, Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Izd-vo MGU, M., 1990 | MR

[9] E. Sanches-Palensiya, Neodnorodnye sredy i teoriya kolebanii, Mir, M., 1984 | MR

[10] L. Khërmander, Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, v. 2, Mir, M., 1986