Microlocal Analysis and Precise Spectral Asymptotics

Victor Ivrii

Research Monography

Springer-Verlag, May 98 (published!)

I am very grateful to Ed Doolittle for for editorial assistance!

• Preface
• Table of Contents
• Introduction

  Part I. SEMICLASSICAL MICROLOCAL ANALYSIS

• Chapter 1 Introduction to semiclassical microlocal analysis
  • 1.1. h-pseudo-differential operators
  • 1.2. Lagrangian distributions and Fourier integral operators
  • 1.3. Oscillations front sets and related topics
  • 1.4. Functional calculus of h-pseudo-differential operators
• Chapter 2 Propagation of singularities in the interior of the domain
  • 2.1. Basic theorems
  • 2.2. Geometrical interpretation
  • 2.3. Propagation of singularities along long bicharacteristics
• Chapter 3 Propagation of singularities near the boundary
  • 3.1. Basic theorems
  • 3.2. Related topics
  • 3.3. Propagation of singularities along long transversally reflecting bicharacteristics
  • Appendix A. On the solution of linear matrix equations

  Part II. LOCAL AND MICROLOCAL SEMICLASSICAL SPECTRAL ASYMPTOTICS (LSSA)

• Chapter 4 Chapter 4. Standard LSSA in the interior of the domain
  • 4.1. Preliminary analysis
  • 4.2. Construction of the fundamental solution u(x,y,t) by the method of the successive approximations
  • Appendix B. On the traces of the almost analytic functions
  • 4.3. Basic theorems
  • 4.4. Spectral asymptotics for scalar operators
  • 4.5. LSSA for the Schrödinger and Dirac operators
  • Appendix C. On the Dirac matrices
  • 4.6. Spectral asymptotics for the families of commuting operators
  • 4.7.Scalar operators with the periodic Hamiltonian flow : propagation of singularities and LSSA
• Chapter 5 LSSA near the boundary
  • 5.1. Preliminary analysis
  • 5.2. Method of successive approximations
  • 5.3. Basic theorems
  • 5.4. LSSA for the Schrödinger operator
  • Appendix D. On the structure of certain sets connected with the microhyperbolicity condition
  • Appendix E. Semiclassical asymptotics of the Riesz means with respect to spectral parameter
• Chapter 6 LSSA for the Schrödinger and Dirac operators with the strong magnetic field
  • 6.1. Preliminary analysis
  • 6.2. Microlocal canonical form
  • 6.3. LSSA in the case of the weak enough magnetic field
  • 6.4. LSSA in the case of the strong enough magnetic field
  • 6.5. Improved LSSA in the three-dimensional case
  • 6.6. Lacunas in the semiclassical approximation to the spectrum
  • 6.7. Certain generalizations
  • Appendix F. On two integral formulas
  • Appendix G. Trajectories of the classical particles in the magnetic field
• Chapter 7 LSSA for the Dirac operator with the strong magnetic field
  • 7.1. Preliminary analysis
  • 7.2. Microlocal canonical form
  • 7.3. LSSA in the case of the weak enough magnetic field
  • 7.4. LSSA in the case of the strong enough magnetic field
  • 7.5. LSSA in the case of the very strong magnetic field
  • 7.6. Improved asymptotics in the three-dimensional case
  • 7.7. Lacunas in the semiclassical approximation to the spectrum
  • 7.8. Certain generalizations
  • Appendix H. An important remark to Chapters 4-7

  Part III. SPECTRAL ESTIMATES

• Chapter 8 Estimates of the negative spectrum
  • 8.1. Statement of the main theorems
  • 8.2. Proof of the main theorems
  • 8.3. Final theorems
  • Appendix I. A remark on the Riesz means
• Chapter 9 Estimates of the spectrum in the interval
  • 9.1. Statement of the main theorems
  • 9.2. Some functional analytical results and the proof of the main theorems

  Part IV. ASYMPTOTICS OF SPECTRA

• Chapter 10 Weylian asymptotics of spectra
  • 10.1. Semiclassical asymptotics of spectra
  • 10.2. Asymptotics of large eigenvalues: the case of weakly singular potentials
  • 10.3. Asymptotics of large eigenvalues: the case of strongly singular potentials
  • 10.4. Asymptotics of large eigenvalues: the case of essentially non-bounded domain
  • 10.5. Asymptotics of eigenvalues tending to -0
  • 10.6. Multiparametrical asymptotics
  • Appendix J. Distributions of the spectra for negative-orde pseudo-differential operators
• Chapter 11 Distributions of the spectra for the Schrödinger and Dirac operators with the strong magnetic field
  • 11.1. The case d=2: a fixed spectral parameter
  • 11.2. The case d=3: a fixed spectral parameter
  • 11.3. The case d=2: asymptotics with the spectral parameter for the Schrödinger operator
  • 11.4. The case d=2: asymptotics with the spectral parameter for the Dirac operator
  • 11.5. The case d=3: asymptotics with the spectral parameter for the Schrödinger and Dirac operators
  • Appendix L. Examples of the admissible vector potentials
• Chapter 12 Miscellaneous asymptotics
  • 12.1. Eigenvalue asymptotics for operators in the domains with the thick cusps
  • 12.2. Eigenvalue asymptotics for operators with potentialsdegenerating at the infinity
  • 12.3. Eigenvalue asymptotics for maximally hypoelliptic operators
  • 12.4. Asymptotics of the Riesz means for a class of operators singular at few points
  • 12.5. Asymptotics of the Riesz means for singular at point Schrödinger operator with the strong magnetic field
  • 12.6. Asymptotics of eigenvalues tending to the bottom of essential spectrum for three-dimensional Schrödinger and Dirac operators with the constant magnetic field and with electric potential quickly decreasing at infinity
• References
• Subject index
• Index of notations