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

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Matematicheskie Zametki, 2012, Volume 92, Issue 2, Pages 163–180
DOI: https://doi.org/10.4213/mzm9749 (Mi mzm9749)

This article is cited in 15 scientific papers (total in 15 papers)

Averaging of Linear Operators, Adiabatic Approximation, and Pseudodifferential Operators

J. Brüning^a, V. V. Grushin^bc, S. Yu. Dobrokhotov^dc

^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

Full-text PDF (627 kB) Citations (15)

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

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.

Received: 28.12.2011

English version:
Mathematical Notes, 2012, Volume 92, Issue 2, Pages 151–165
DOI: https://doi.org/10.1134/S0001434612070188

Bibliographic databases:

Document Type: Article

UDC: 517.9

Language: Russian

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

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This publication is cited in the following 15 articles:

S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Metod osredneniya dlya zadach o kvaziklassicheskikh asimptotikakh”, Funktsionalnye prostranstva. Differentsialnye operatory. Problemy matematicheskogo obrazovaniya, SMFN, 70, no. 1, Rossiiskii universitet druzhby narodov, M., 2024, 53–76
S. Yu. Dobrokhotov, V. E. Nazaikinskii, “Homogenization Method for Problems on Quasiclassical Asymptotics”, J Math Sci, 2024
D. I. Borisov, “Homogenization of Operators with Perturbations of General Form in the Lower-Order Terms”, Math. Notes, 113:1 (2023), 138–142
S. Yu. Dobrokhotov, “Asymptotics of the Cauchy Problem for the One-Dimensional Schrödinger Equation with Rapidly Oscillating Initial Data and Small Addition to the Smooth Potential”, Russ. J. Math. Phys., 30:4 (2023), 466
D.S. Minenkov, S.A. Sergeev, “Asymptotics of the Whispering Gallery-Type in the Eigenproblem for the Laplacian in a Domain of Revolution Diffeomorphic To a Solid Torus”, Russ. J. Math. Phys., 30:4 (2023), 599
S. A. Sergeev, “Asymptotic Solution of the Cauchy Problem with Localized Initial Data for a Wave Equation with Small Dispersion Effects”, Diff Equat, 58:10 (2022), 1376
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
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
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
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
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
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
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
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
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

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