Microlocal Analysis and Precise Spectral Asymptotics

1. Front Matter

   Pages I-XV

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2. Introduction

   1. Introduction

      • Victor Ivrii

      Pages 1-17

3. Semiclassical Microlocal Analysis

   1. Front Matter

      Pages 19-19

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   2. Introduction to Semiclassical Microlocal Analysis

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      Pages 21-52

   3. Propagation of Singularities in the Interior of a Domain

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      Pages 53-78

   4. Propagation of Singularities near the Boundary

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      Pages 79-117

4. Local and Microlocal Semiclassical Asymptotics

   1. Front Matter

      Pages 119-119

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   2. Standard Local Semiclassical Spectral Asymptotics in the Interior of a Domain

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      Pages 121-217

   3. Standard Local Semiclassical Spectral Asymptotics near the Boundary

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      Pages 219-274

   4. Local Spectral Asymptotics for the Schrödinger Operator with Strong Magnetic Field

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      Pages 275-365

   5. Local Spectral Asymptotics for the Dirac Operator with Strong Magnetic Field

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      Pages 367-419

5. Estimates of the Spectrum

   1. Front Matter

      Pages 421-421

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   2. Estimates of the Negative Spectrum

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      Pages 423-450

   3. Estimates of the Spectrum in an Interval

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      Pages 451-478

6. Asymptotics of Spectra

   1. Front Matter

      Pages 479-479

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   2. Weylian Asymptotics of Spectra

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      Pages 481-568

   3. Asymptotics of the Spectra of Schrödinger and Dirac Operators with Strong Magnetic Field

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      Pages 569-625

   4. Miscellaneous Asymptotics

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      Pages 627-707

7. Back Matter

   Pages 709-733

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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.

• Dirac
• Microlocal analysis
• Semiclassical microlocal analysis
• Spectral asymptotics
• boundary element method
• field
• proof
• schrödinger operator
• theorem
• tool
• partial differential equations

From the reviews:

"The monograph, under review, deals with one of the central problems in the spectral theory of partial differential operators, namely with the problem of spectral asymptotics, in particular with the problem of the asymptotic distribution of eigenvalues. … Summarizing, this is an excellent monograph on microlocal analysis, propagation of singularities and spectral asymptotics, based mainly on the author’s own results. It is warmly recommended to specialists in PDE’s, theoretical physics. The monograph will be useful also for advanced students." (Jeno Hegedus, Acta Scientiarum Mathematicarum, Vol. 74, 2008)

• Department of Mathematics, University of Toronto, Toronto, Canada

  Victor Ivrii

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