Pseudodifferential operators and spectral theory. Transl. from the Russian by Stig I. Andersson. 2nd ed.

*(English)*Zbl 0980.35180
Berlin: Springer. xii, 288 p. (2001).

This book is the second edition of the well-known book by Shubin (1st English ed., Springer (1987; Zbl 0616.47040); Russian original, Nauka, Moskva (1978; Zbl 0451.47064)). The monograph occupies an exceptional place among books devoted to the theory of pseudodifferential operators, because it provides a rather simple presentation of main facts and deep results of the theory of pseudodifferential operators as well as Fourier integral operators.

The applications included in the books are: 1) Complex powers of elliptic operators, 2) HĂ¶rmander asymptotics of the spectral function and eigenfunctions, 3) Method of approximate spectral projection for asymptotics of eigenfunctions of elliptic pseudodifferential operators on \( \mathbb R ^n.\)

Appendix 1 is devoted to the wave fronts and propagation of singularities and Appendix 2 to quasiclassical asymptotic behavior of eigenfunctions of elliptic operators with parameter.

The applications included in the books are: 1) Complex powers of elliptic operators, 2) HĂ¶rmander asymptotics of the spectral function and eigenfunctions, 3) Method of approximate spectral projection for asymptotics of eigenfunctions of elliptic pseudodifferential operators on \( \mathbb R ^n.\)

Appendix 1 is devoted to the wave fronts and propagation of singularities and Appendix 2 to quasiclassical asymptotic behavior of eigenfunctions of elliptic operators with parameter.

Reviewer: Vladimir S.Rabinovich (Mexico)

##### MSC:

35S05 | Pseudodifferential operators as generalizations of partial differential operators |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

47G30 | Pseudodifferential operators |

35S30 | Fourier integral operators applied to PDEs |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

35P20 | Asymptotic distributions of eigenvalues in context of PDEs |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |