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Mat. Zametki, 2012, Volume 92, Issue 2, Pages 163–180 (Mi mz9749)  

This article is cited in 9 scientific papers (total in 9 papers)

Averaging of Linear Operators, Adiabatic Approximation, and Pseudodifferential Operators

J. Brüninga, V. V. Grushinbc, S. Yu. Dobrokhotovdc

a Humboldt University, Germany
b Moscow State Institute of Electronics and Mathematics
c Moscow Institute of Physics and Technology
d A. Ishlinsky Institite for Problems in Mechanics, Russian Academy of Sciences

Abstract: An example of Schrödinger and Klein–Gordon equations with fast oscillating coefficients is used to show that they can be averaged by an adiabatic approximation based on V. P. Maslov's operator method.

Keywords: Klein–Gordon equation, Schrödinger equation, adiabatic approximation, asymptotic solution, pseudodifferential operator, adiabatic principle, perturbation theory

DOI: https://doi.org/10.4213/mzm9749

Full text: PDF file (627 kB)
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English version:
Mathematical Notes, 2012, 92:2, 151–165

Bibliographic databases:

UDC: 517.9
Received: 28.12.2011

Citation: J. Brüning, V. V. Grushin, S. Yu. Dobrokhotov, “Averaging of Linear Operators, Adiabatic Approximation, and Pseudodifferential Operators”, Mat. Zametki, 92:2 (2012), 163–180; Math. Notes, 92:2 (2012), 151–165

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Brüning J., Grushin V.V., Dobrokhotov S.Yu., “Approximate formulas for eigenvalues of the Laplace operator on a torus arising in linear problems with oscillating coefficients”, Russ. J. Math. Phys., 19:3 (2012), 261–272  crossref  mathscinet  zmath  isi  elib
    2. V. V. Grushin, S. Yu. Dobrokhotov, S. A. Sergeev, “Homogenization and dispersion effects in the problem of propagation of waves generated by a localized source”, Proc. Steklov Inst. Math., 281 (2013), 161–178  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. Dobrokhotov S.Yu., Sergeev S.A., Tirozzi B., “Asymptotic Solutions of the Cauchy Problem with Localized Initial Conditions for Linearized Two-Dimensional Boussinesq-Type Equations with Variable Coefficients”, Russ. J. Math. Phys., 20:2 (2013), 155–171  crossref  mathscinet  zmath  isi  elib
    4. V. V. Grushin, S. Yu. Dobrokhotov, “Homogenization in the Problem of Long Water Waves over a Bottom Site with Fast Oscillations”, Math. Notes, 95:3 (2014), 324–337  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    5. Dobrokhotov S.Yu., Nazaikinskii V.E., Tirozzi B., “on a Homogenization Method For Differential Operators With Oscillating Coefficients”, Dokl. Math., 91:2 (2015), 227–231  crossref  mathscinet  zmath  isi  elib
    6. Dobrokhotov S.Yu., Grushin V.V., Sergeev S.A., Tirozzi B., “Asymptotic theory of linear water waves in a domain with nonuniform bottom with rapidly oscillating sections”, Russ. J. Math. Phys., 23:4 (2016), 455–474  crossref  mathscinet  zmath  isi  elib  scopus
    7. S. Yu. Dobrokhotov, V. E. Nazaikinskii, A. V. Tsvetkova, “One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions”, Math. Notes, 103:5 (2018), 33–43  mathnet  crossref  crossref  isi  elib
    8. D. A. Karaeva, A. D. Karaev, V. E. Nazaikinskii, “Homogenization method in the problem of long wave propagation from a localized source in a basin over an uneven bottom”, Differ. Equ., 54:8 (2018), 1057–1072  crossref  crossref  isi  elib  elib  scopus
    9. Dobrokhotov S.Yu. Nazaikinskii V.E., “Asymptotic Localized Solutions of the Shallow Water Equations Over a Nonuniform Bottom”, AIP Conference Proceedings, 2048, ed. Pasheva V. Popivanov N. Venkov G., Amer Inst Physics, 2018, 040026  crossref  isi
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