##
**Microlocal analysis and precise spectral asymptotics.**
*(English)*
Zbl 0906.35003

Springer Monographs in Mathematics. Berlin: Springer. xv, 731 p. (1998).

Microlocal analysis and spectral asymptotics are considered as active topics of research in partial differential equations in the last twenty years. Historically, the problem of spectral asymptotics originated in the 1911 paper of Herman Weyl on eigenvalue asymptotics for the Laplace operator in a bounded domain with regular boundary. Subsequently, considerable amount of work on spectral asymptotics was published by many prominent mathematicians and mathematical physicists. The problem of spectral asymptotics and the problem of the asymptotic distribution of eigenvalues are the major problems of the spectral theory of partial differential operators. The author is known for his own important contributions to the subjet and his active interests have naturally influenced his choice of the topics and level of the book. This book essentially deals with recent developments of the theory of spectral asymptotics with considerable attention to the asymptotic distribution of eigenvalues and propagation of singularities. It gives an excellent and useful information of the theory and applications of microlocal analysis and spectral asymptotics to quantum mechanics, continuum mechanics, differential geometry, dynamical systems theory and ergodic theory, and has twelve chapters.

The first chapter introduces basic ideas and results of the theory of \(h\)-pseudo-differential operators and the Fourier integral operators. Chapter two deals with the propagation of singularities of the solutions in the interior of a domain, while chapter three studies the propagation of singularities near the boundary on which boundary conditions are given. The proof of the main theorem concerning the propagation of singularities is given.

Chapters 4 and 5 deal with the detailed treatment of local and microlocal semiclassical spectral asymptotics near the boundary. The standard spectral asymptotics with respect to the Planck constant \(h\to +0\) is also discussed in some detail. Some general basic theorems are developed with particular attention to partial differential operators with unbounded coefficients. The author also gives a special treatment of the local spectral asymptotics for the Schrödinger and Dirac operators, and operators with periodic Hamiltonian flow. The local spectral asymptotics is studied for the Schrödinger and Dirac operators with strong magnetic field with respect to the small parameter \(h\) and the large coupling parameter \(\mu\) which is responsible for the interaction of the particle with the magnetic field. An analysis is made of the long-time propagation of singularities in order to extend the time domain and to improve the estimate in the Weyl-type estimates.

Chapters 8 and 9 dealing with estimates of the spectrum, are essentially based on the author’s own research work. The subject of these chapters deals with local semiclassical spectral asymptotics with variational estimates of Rozenbloum type. It is shown that the Birman-Schrödinger principle plays a major role on the analysis of estimates of the spectrum in a given interval. Applications of the results of earlier chapters to eigenvalue asymptotics are discussed in Chapter 10. Chapter 11 is devoted to the problem of distribution of spectra for the Schrödinger and Dirac operators in the presence of strong magnetic fields.

Miscellaneous asymptotics is the main content of the closing Chapter 12 which describes nonclassical asymptotics in domains with cusp. However, there is little or no accurate remainder estimates available. This chapter has more open questions and unsolved problems of spectral asymptotics which require future study. According to the author: The theory of operators with strong degeneration of the potential also is not very well developed.

In summary, this is an excellent research monograph for all with only research interest in microlocal analysis, spectral asymptotics and propagation of singularities. In the opinion of the reviewer, most graduate students first learning microlocal analysis and spectral asymptotics may find this book difficult to digest. However, it would be an invaluable contribution to the literature of partial differential equations.

The first chapter introduces basic ideas and results of the theory of \(h\)-pseudo-differential operators and the Fourier integral operators. Chapter two deals with the propagation of singularities of the solutions in the interior of a domain, while chapter three studies the propagation of singularities near the boundary on which boundary conditions are given. The proof of the main theorem concerning the propagation of singularities is given.

Chapters 4 and 5 deal with the detailed treatment of local and microlocal semiclassical spectral asymptotics near the boundary. The standard spectral asymptotics with respect to the Planck constant \(h\to +0\) is also discussed in some detail. Some general basic theorems are developed with particular attention to partial differential operators with unbounded coefficients. The author also gives a special treatment of the local spectral asymptotics for the Schrödinger and Dirac operators, and operators with periodic Hamiltonian flow. The local spectral asymptotics is studied for the Schrödinger and Dirac operators with strong magnetic field with respect to the small parameter \(h\) and the large coupling parameter \(\mu\) which is responsible for the interaction of the particle with the magnetic field. An analysis is made of the long-time propagation of singularities in order to extend the time domain and to improve the estimate in the Weyl-type estimates.

Chapters 8 and 9 dealing with estimates of the spectrum, are essentially based on the author’s own research work. The subject of these chapters deals with local semiclassical spectral asymptotics with variational estimates of Rozenbloum type. It is shown that the Birman-Schrödinger principle plays a major role on the analysis of estimates of the spectrum in a given interval. Applications of the results of earlier chapters to eigenvalue asymptotics are discussed in Chapter 10. Chapter 11 is devoted to the problem of distribution of spectra for the Schrödinger and Dirac operators in the presence of strong magnetic fields.

Miscellaneous asymptotics is the main content of the closing Chapter 12 which describes nonclassical asymptotics in domains with cusp. However, there is little or no accurate remainder estimates available. This chapter has more open questions and unsolved problems of spectral asymptotics which require future study. According to the author: The theory of operators with strong degeneration of the potential also is not very well developed.

In summary, this is an excellent research monograph for all with only research interest in microlocal analysis, spectral asymptotics and propagation of singularities. In the opinion of the reviewer, most graduate students first learning microlocal analysis and spectral asymptotics may find this book difficult to digest. However, it would be an invaluable contribution to the literature of partial differential equations.

Reviewer: L.Debnath (Orlando)

### MSC:

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

35P15 | Estimates of eigenvalues in context of PDEs |

81Qxx | General mathematical topics and methods in quantum theory |

35Jxx | Elliptic equations and elliptic systems |

35Lxx | Hyperbolic equations and hyperbolic systems |

58Jxx | Partial differential equations on manifolds; differential operators |