Microlocal Analysis and Precise Spectral Asymptotics

The problem of spectral asymptotics, in particular the problem of the asymptotic dis tribution of eigenvalues, is one of the central problems in the spectral theory of partial differential operators; moreover, it is very important for the general theory of partial differential operators. I started working in this domain in 1979 after R. Seeley found a remainder estimate of the same order as the then hypothetical second term for the Laplacian in domains with boundary, and M. Shubin and B. M. Levitan suggested that I should try to prove Weyl's conjecture. During the past fifteen years I have not left the topic, although I had such intentions in 1985 when the methods I invented seemed to fai! to provide furt her progress and only a couple of not very exciting problems remained to be solved. However, at that time I made the step toward local semiclassical spectral asymptotics and rescaling, and new horizons opened.

The pathbreaking work of Victor Ivrii in the last ten years represents very difficult mathematics. Because of its technical difficulty and its size, it cannot be expected to sell to a large audience. However for all those working on this subject,it will be a indispensable reference.

Inhalt
0. Introduction.- I. Semiclassical Microlocal Analysis.- 1. Introduction to Semiclassical Microlocal Analysis.- 2. Propagation of Singularities in the Interior of a Domain.- 3. Propagation of Singularities near the Boundary.- II. Local and Microlocal Semiclassical Asymptotics.- 4. LSSA in the Interior of a Domain.- 5. Standard LSSA near the Boundary.- 6. Schrödinger Operators with Strong Magnetic Field.- 7. Dirac Operators with Strong Magnetic Field.- III. Estimates of the Spectrum.- 8. Estimates of the Negative Spectrum.- 9. Estimates of the Spectrum in an Interval.- IV. Asymptotics of Spectra.- 10. Weylian Asymptotics of Spectra.- 11. Schrödinger, Dirac Operators with Strong Magnetic Field.- 12. Miscellaneous Asymptotics.- References.