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Izv. Akad. Nauk SSSR Ser. Mat., 1974, Volume 38, Issue 3, Pages 625–662 (Mi izv1945)  

This article is cited in 18 scientific papers (total in 18 papers)

Asymptotics of the solution of the system $A(x,-ih\frac\partial{\partial x})$ as $h\to0$ in the case of characteristics of variable multiplicity

V. V. Kucherenko

Abstract: In this paper we develop a new asymptotic method for pseudodifferential operators in the case of characteristics of variable multiplicity; the $N$th term of the asymptotics is expressed in terms of an $N$-dimensional integral of a rapidly oscillating function of $(N+n)$ arguments, where $n$ is the dimension of the space ($x=x_1,…,x_n$).

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English version:
Mathematics of the USSR-Izvestiya, 1974, 8:3, 631–666

Bibliographic databases:

UDC: 517.9
MSC: Primary 35B40, 35S10; Secondary 35A30
Received: 04.05.1972
Revised: 26.11.1973

Citation: V. V. Kucherenko, “Asymptotics of the solution of the system $A(x,-ih\frac\partial{\partial x})$ as $h\to0$ in the case of characteristics of variable multiplicity”, Izv. Akad. Nauk SSSR Ser. Mat., 38:3 (1974), 625–662; Math. USSR-Izv., 8:3 (1974), 631–666

Citation in format AMSBIB
\by V.~V.~Kucherenko
\paper Asymptotics of the solution of the system $A(x,-ih\frac\partial{\partial x})$ as~$h\to0$ in the case of characteristics of variable multiplicity
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1974
\vol 38
\issue 3
\pages 625--662
\jour Math. USSR-Izv.
\yr 1974
\vol 8
\issue 3
\pages 631--666

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    This publication is cited in the following articles:
    1. S. A. Ivanov, B. S. Pavlov, “Carleson series of resonances in the Regge problem”, Math. USSR-Izv., 12:1 (1978), 21–51  mathnet  crossref  mathscinet  zmath
    2. V. P. Maslov, V. E. Nazaikinskii, “Asymptotics for equations with singularities in the characteristics”, Math. USSR-Izv., 19:2 (1982), 315–347  mathnet  crossref  mathscinet  zmath
    3. Yu. M. Vorob'ev, S. Yu. Dobrokhotov, “Quasiclassical asymptotic behaviors for discrete models of electron-phonon interaction: Maslov's method and the adiabatic approximation”, Theoret. and Math. Phys., 57:1 (1983), 993–1001  mathnet  crossref  mathscinet  isi
    4. V. P. Maslov, “Non-standard characteristics in asymptotic problems”, Russian Math. Surveys, 38:6 (1983), 1–42  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    5. V. V. Kucherenko, Yu. V. Osipov, “The Cauchy problem for nonstrictly hyperbolic equations”, Math. USSR-Sb., 48:1 (1984), 81–109  mathnet  crossref  mathscinet  zmath
    6. V. V. Kucherenko, Yu. V. Osipov, “Exact and asymptotic solutions of systems with turning points”, Math. USSR-Izv., 29:2 (1987), 355–370  mathnet  crossref  mathscinet  zmath
    7. V. G. Bagrov, V. V. Belov, M. F. Kondrat'eva, “The semiclassical approximation in quantum mechanics. A new approach”, Theoret. and Math. Phys., 98:1 (1994), 34–38  mathnet  crossref  mathscinet  zmath  isi
    8. V. V. Belov, M. F. Kondrat'eva, “The Hamiltonian structure of equations for quantum averages in systems with matrix Hamiltonians”, Math. Notes, 58:6 (1995), 1251–1261  mathnet  crossref  mathscinet  zmath  isi
    9. V. V. Belov, S. Yu. Dobrokhotov, T. Ya. Tudorovskii, “Asymptotic Solutions of Nonrelativistic Equations of Quantum Mechanics in Curved Nanotubes: I. Reduction to Spatially One-Dimensional Equations”, Theoret. and Math. Phys., 141:2 (2004), 1562–1592  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. V. V. Belov, S. Yu. Dobrokhotov, V. P. Maslov, T. Ya. Tudorovskii, “A generalized adiabatic principle for electron dynamics in curved nanostructures”, Phys. Usp., 48:9 (2005), 962–968  mathnet  crossref  crossref  adsnasa  isi  elib  elib
    11. T. Ya. Tudorovskii, “On the Effect of Spin on Classical and Quantum Dynamics of an Electron in Thin Twisted Tubes”, Math. Notes, 78:6 (2005), 883–889  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. A. V. Krivko, V. V. Kucherenko, “Semiclassical Asymptotics of the Matrix Sturm–Liouville Problem”, Math. Notes, 80:1 (2006), 136–140  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. A. V. Krivko, V. V. Kucherenko, “On real hyperbolic systems with characteristics of variable multiplicity”, Dokl Math, 75:1 (2007), 83  crossref  mathscinet  isi
    14. V. V. Kucherenko, A. V. Krivko, “Existence Theorem for Hyperbolic Systems with a Multiplicity Change Point of at Most the Third Order”, Math. Notes, 85:1 (2009), 128–132  mathnet  crossref  crossref  mathscinet  zmath  isi
    15. 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
    16. A. A. Tolchennikov, “Behavior of the solution of the Klein–Gordon equation with a localized initial condition”, Theoret. and Math. Phys., 199:2 (2019), 761–770  mathnet  crossref  crossref  adsnasa  isi  elib
    17. Dobrokhotov S.Yu. Tolchennikov A.A., “Solution of the Two-Dimensional Dirac Equation With a Linear Potential and a Localized Initial Condition”, Russ. J. Math. Phys., 26:2 (2019), 139–151  crossref  isi
    18. A. G. Eliseev, P. V. Kirichenko, “Reshenie singulyarno vozmuschennoi zadachi Koshi pri nalichii «slaboi» tochki povorota u predelnogo operatora”, Sib. elektron. matem. izv., 17 (2020), 51–60  mathnet  crossref
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