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Mat. Zametki, 2010, Volume 87, Issue 4, Pages 554–571 (Mi mz8699)  

This article is cited in 5 scientific papers (total in 5 papers)

Peierls Substitution and the Maslov Operator Method

V. V. Grushinab, S. Yu. Dobrokhotovac

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

Abstract: We consider a periodic Schrödinger operator in a constant magnetic field with vector potential $A(x)$. A version of adiabatic approximation for quantum mechanical equations with rapidly varying electric potentials and weak magnetic fields is the Peierls substitution which, in appropriate dimensionless variables, permits writing the pseudodifferential equation for the new auxiliary function: $\mathscr E^{\nu}(-i\mu\partial_x,x)\phi=E\phi$, where $\mathscr E^{\nu}$ is the corresponding energy level of some auxiliary Schrödinger operator, assumed to be nondegenerate, and $\mu$ is a small parameter. In the present paper, we use V. P. Maslov's operator method to show that, in the case of a constant magnetic field, such a reduction in any perturbation order leads to the equation $\mathscr{E}^{\nu}(\widehat P,\mu)\phi=E\phi$ with the operator $\mathscr{E}^{\nu}(\widehat P,\mu)$ represented as a function depending only on the operators of kinetic momenta $\widehat P_j=-i\mu\partial_{x_j}+A_j(x)$.

Keywords: Peierls substitution, pseudodifferential equation, kinetic momentum, adiabatic approximation, periodic Schrödinger operator, stationary phase method

DOI: https://doi.org/10.4213/mzm8699

Full text: PDF file (584 kB)
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English version:
Mathematical Notes, 2010, 87:4, 521–536

Bibliographic databases:

UDC: 517.9
Received: 16.10.2009

Citation: V. V. Grushin, S. Yu. Dobrokhotov, “Peierls Substitution and the Maslov Operator Method”, Mat. Zametki, 87:4 (2010), 554–571; Math. Notes, 87:4 (2010), 521–536

Citation in format AMSBIB
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\paper Peierls Substitution and the Maslov Operator Method
\jour Mat. Zametki
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\pages 554--571
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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. J. Brüning, V. V. Grushin, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Generalized Foldy–Wouthuysen transformation and pseudodifferential operators”, Theoret. and Math. Phys., 167:2 (2011), 547–566  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    2. J. Brüning, V. V. Grushin, S. Yu. Dobrokhotov, “Averaging of Linear Operators, Adiabatic Approximation, and Pseudodifferential Operators”, Math. Notes, 92:2 (2012), 151–165  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. 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
    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. 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
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