Geodesic
Boundary-Value Problems for the Schr\"odinger Equation with Rapidly Oscillating and Delta-Liked Potentials
Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 842-852.

Voir la notice de l'article provenant de la source Math-Net.Ru

This paper deals with boundary-value problems on the closed interval $[a,b]$ for the Schrödinger equation with potential of the form $q(x,\mu^{-1}x)+\varepsilon^{-1}Q(\varepsilon^{-1}x)$, where $q(x,\zeta)$ is a $1$-periodic (in $\zeta$) function, $Q(\xi)$ is a compactly supported function, $0\in(a,b)$, and $\mu,\varepsilon$ are small positive parameters. The solutions of these boundary-value problems up to $O(\varepsilon+\mu)$ are constructed by combining the homogenization method and the method of matching asymptotic expansions.
Keywords: Schrödinger equation, boundary-value problem, $1$-periodic function, homogenization method, matching method, rapidly oscillating potential, delta-liked potential. 
@article{MZM_2015_98_6_a4,
     author = {T. R. Gadylshin},
     title = {Boundary-Value {Problems} for the {Schr\"odinger} {Equation} with {Rapidly} {Oscillating} and {Delta-Liked} {Potentials}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {842--852},
     publisher = {mathdoc},
     volume = {98},
     number = {6},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a4/}
}
TY  - JOUR
AU  - T. R. Gadylshin
TI  - Boundary-Value Problems for the Schr\"odinger Equation with Rapidly Oscillating and Delta-Liked Potentials
JO  - Matematičeskie zametki
PY  - 2015
SP  - 842
EP  - 852
VL  - 98
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a4/
LA  - ru
ID  - MZM_2015_98_6_a4
ER  - 
%0 Journal Article
%A T. R. Gadylshin
%T Boundary-Value Problems for the Schr\"odinger Equation with Rapidly Oscillating and Delta-Liked Potentials
%J Matematičeskie zametki
%D 2015
%P 842-852
%V 98
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a4/
%G ru
%F MZM_2015_98_6_a4
T. R. Gadylshin. Boundary-Value Problems for the Schr\"odinger Equation with Rapidly Oscillating and Delta-Liked Potentials. Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 842-852. http://geodesic.mathdoc.fr/item/MZM_2015_98_6_a4/

[1] A. M. Ilin, A. R. Danilin, Asimptoticheskie metody v analize, Fizmatlit, M., 2009 | Zbl

[2] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh. Matematicheskie zadachi mekhaniki kompozitsionnykh materialov, Nauka, M., 1984 | MR | Zbl

[3] O. A. Oleinik, G. A. Iosifyan, A. S. Shamaev, Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Izd-vo Mosk. un-ta, M., 1990 | MR | Zbl

[4] A. L. Pyatnitskii, G. A. Chechkin, A. S. Shamaev, Usrednenie. Metody i prilozheniya. Belaya seriya v matematike i fizike, Tamara Rozhkovskaya, Novosibirsk, 2004

[5] Yu. D. Golovatyi, S. A. Nazarov, O. A. Oleinik, T. S. Soboleva, “O sobstvennykh kolebaniyakh struny s prisoedinennoi massoi”, Sib. matem. zhurn., 29:5 (1988), 71–91 | MR | Zbl

[6] V. G. Mazya, S. A. Nazarov, B. A. Plamenevskii, Asimptotika reshenii ellipticheskikh kraevykh zadach pri singulyarnom vozmuschenii oblasti, Izd-vo Tbilissk. un-ta, Tbilisi, 1981 | MR | Zbl

[7] A. M. Ilin, Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR | Zbl

[8] S. Albeverio, F. Gestezi, R. Kheeg-Kron, Kh. Kholden, Reshaemye modeli v kvantovoi mekhanike, Mir, M., 1991 | MR | Zbl

[9] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, W. Kirch, “On point interactions one dimension”, J. Operator Theory, 12:1 (1984), 101–126 | MR | Zbl

[10] A. M. Savchuk, A. A. Shkalikov, “Operatory Shturma–Liuvillya s singulyarnymi potentsialami”, Matem. zametki, 66:6 (1999), 897–912 | DOI | MR | Zbl

[11] V. P. Mikhailov, Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976 | MR | Zbl