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Uspekhi Mat. Nauk, 1971, Volume 26, Issue 1(157), Pages 67–112 (Mi umn5166)  

This article is cited in 27 scientific papers (total in 27 papers)

The stationary phase method and pseudodifferential operators

M. V. Fedoryuk


Abstract: In the paper asymptotic expansions are calculated for integrals
$$ \int f(x)\exp(i\lambda g(x))dx,\qquad\lambda\to+\infty, $$
of rapidly oscillating functions, in which $x\in R^n$, $f$ and $g$ are smooth functions, and $g$ is real-valued. The results obtained serve to develop a calculus of pseudodifferential operators and generalizations of them, the Fourier integral operators.

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English version:
Russian Mathematical Surveys, 1971, 6:1, 65–115

Bibliographic databases:

UDC: 517.43
MSC: 47G30, 58J40, 41A60, 35S30
Received: 11.06.1970

Citation: M. V. Fedoryuk, “The stationary phase method and pseudodifferential operators”, Uspekhi Mat. Nauk, 26:1(157) (1971), 67–112; Russian Math. Surveys, 6:1 (1971), 65–115

Citation in format AMSBIB
\Bibitem{Fed71}
\by M.~V.~Fedoryuk
\paper The stationary phase method and pseudodifferential operators
\jour Uspekhi Mat. Nauk
\yr 1971
\vol 26
\issue 1(157)
\pages 67--112
\mathnet{http://mi.mathnet.ru/umn5166}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=467849}
\zmath{https://zbmath.org/?q=an:0221.47036}
\transl
\jour Russian Math. Surveys
\yr 1971
\vol 6
\issue 1
\pages 65--115
\crossref{https://doi.org/10.1070/RM1971v026n01ABEH003813}


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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. V. A. Fateev, A. S. Schwarz, “On axiomatic scattering theory”, Theoret. and Math. Phys., 14:2 (1973), 112–124  mathnet  crossref  mathscinet  zmath
    2. 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”, Math. USSR-Izv., 8:3 (1974), 631–666  mathnet  crossref  mathscinet  zmath
    3. Yu. V. Egorov, “Subelliptic operators”, Russian Math. Surveys, 30:2 (1975), 59–118  mathnet  crossref  mathscinet  zmath
    4. M. V. Fedoryuk, “Of Fourier integral operators and the asymptotic behaviour of the solution of the mixed problem”, Russian Math. Surveys, 32:6 (1977), 67–120  mathnet  crossref  mathscinet  zmath
    5. M. V. Karasev, V. E. Nazaikinskii, “On the quantization of rapidly oscillating symbols”, Math. USSR-Sb., 34:6 (1978), 737–764  mathnet  crossref  mathscinet  zmath
    6. Daisuke Fujiwara, “A construction of the fundamental solution for the Schrödinger equation”, J Anal Math, 35:1 (1979), 41  crossref  mathscinet  zmath  isi
    7. V. E. Nazaikinskii, V. G. Oshmyan, B. Yu. Sternin, V. E. Shatalov, “Fourier integral operators and the canonical operator”, Russian Math. Surveys, 36:2 (1981), 93–161  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    8. M. V. Karasev, V. P. Maslov, “Asymptotic and geometric quantization”, Russian Math. Surveys, 39:6 (1984), 133–205  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. C. J. Thomson, C. H. Chapman, “An introduction to Maslov's asymptotic method”, Geophys J Int, 83:1 (1985), 143  crossref  zmath  adsnasa  isi
    10. Carlos Moreno, “Invariant star products and representations of compact semisimple Lie groups”, Lett Math Phys, 12:3 (1986), 217  crossref  mathscinet  zmath  isi
    11. Daniel Barlet, Jean-Louis Clerc, “Le comportement à l'infini des fonctions de Bessel généralisées, I”, Advances in Mathematics, 61:2 (1986), 165  crossref
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    13. D. Bollé, C. Danneels, T. A. Osborn, “Local and global spectral shift functions in R2”, J Math Phys (N Y ), 30:2 (1989), 420  crossref  mathscinet  zmath  adsnasa  isi
    14. R. A. Corns, T. A. Osborn, “Propagators for relativistic systems with non-Abelian interactions”, J Math Phys (N Y ), 31:4 (1990), 901  crossref  mathscinet  zmath  adsnasa  isi
    15. F.H Molzahn, T.A Osborn, S.A Fulling, “Gauge invariant asymptotic expansion of Schrödinger propagators on manifolds”, Annals of Physics, 204:1 (1990), 64  crossref
    16. J.P. McClure, R. Wong, “Multidimensional stationary phase approximation: boundary stationary point”, Journal of Computational and Applied Mathematics, 30:2 (1990), 213  crossref
    17. A. Saksena, T. A. Osborn, F. H. Molzahn, “An asymptotic analysis of quantum evolution with electromagnetic fields”, J Math Phys (N Y ), 32:4 (1991), 938  crossref  mathscinet  zmath  adsnasa  isi
    18. T. V. Dudnikova, A. I. Komech, H. Spohn, “On the convergence to statistical equilibrium for harmonic crystals”, J Math Phys (N Y ), 44:6 (2003), 2596  crossref  mathscinet  zmath  adsnasa  isi
    19. M V Karasev, T A Osborn, “Cotangent bundle quantization: entangling of metric and magnetic field”, J Phys A Math Gen, 38:40 (2005), 8549  crossref  mathscinet  zmath  adsnasa  isi  elib
    20. V. P. Maslov, “Dependence of the Superfluidity Criterion on the Capillary Radius”, Theoret. and Math. Phys., 143:3 (2005), 741–759  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. T. V. Dudnikova, A. I. Komech, “On a two-temperature problem for Klein–Gordon equation”, Theory Probab. Appl., 50:4 (2006), 582–611  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    22. E. Micheli, G. A. Viano, “Geometrical theory of diffracted rays, orbiting and complex rays”, Russ J Math Phys, 13:3 (2006), 253  crossref  mathscinet  zmath  isi
    23. F Cardin, T Gramchev, A Lovison, “Exponential estimates for oscillatory integrals with degenerate phase functions”, Nonlinearity, 21:3 (2008), 409  crossref  mathscinet  zmath  adsnasa  isi  elib
    24. Fatih Ecevit, Fernando Reitich, “Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case”, Numer Math, 2009  crossref  mathscinet  isi
    25. S. V. Talalov, “Ob effekte perenosa massy vdol kosmicheskoi struny”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(31) (2013), 259–266  mathnet  crossref  elib
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