Eigenfunctions of the Laplacian of Riemannian manifolds

Preface
	0.1. Organization
	0.2. Topics which are not covered
	0.3. Topics which are double covered
	0.4. Notation
	Acknowledgments
Chapter 1. Introduction
	1.1. What are eigenfunctions and why are they useful
	1.2. Notation for eigenvalues
	1.3. Weyl\'s law for Laplacian-eigenvalues
	1.4. Quantum Mechanics
	1.5. Dynamics of the geodesic or billiard flow
	1.6. Intensity plots and excursion sets
	1.7. Nodal sets and critical point sets
	1.8. Local versus global analysis of eigenfunctions
	1.9. High frequency limits, oscillation and concentration
	1.10. Spectral projections
	1.11. Lp norms
	1.12. Matrix elements and Wigner distributions
	1.13. Egorov\'s theorem
	1.14. Eherenfest time
	1.15. Weak* limit problem
	1.16. Ergodic versus completely integrable geodesic flow
	1.17. Ergodic eigenfunctions
	1.18. Quantum unique ergodicity (QUE)
	1.19. Completely integrable eigenfunctions
	1.20. Heisenberg uncertainty principle
	1.21. Sequences of eigenfunctions and length scales
	1.22. Localization of eigenfunctions on closed geodesics
	1.23. Some remarks on the contents and on other texts
	1.24. References
Bibliography
Chapter 2. Geometric preliminaries
	2.1. Symplectic linear algebra and geometry
	2.2. Symplectic manifolds and cotangent bundles
	2.3. Lagrangian submanifolds
	2.4. Jacobi fields and Poincaré map
	2.5. Pseudo-differential operators
	2.6. Symbols
	2.7. Quantization of symbols
	2.8. Action of a pseudo-differential operator on a rapidly oscillating exponential
Bibliography
Chapter 3. Main results
	3.1. Universal Lp bounds
	3.2. Self-focal points and extremal Lp bounds for high p
	3.3. Low Lp norms and concentration of eigenfunctions around geodesics
	3.4. Kakeya-Nikodym maximal function and extremal Lp bounds for small p
	3.5. Concentration of joint eigenfunctions of quantum integrable Laplacian around closed geodesics
	3.6. Quantum ergodic restriction theorems for Cauchy data
	3.7. Quantum ergodic restriction theorems for Dirichlet data
	3.8. Counting nodal domains and nodal intersections with curves
	3.9. Intersections of nodal lines and general curves on negatively curved surfaces
	3.10. Complex zeros of eigenfunctions
Bibliography
Chapter 4. Model spaces of constant curvature
	4.1. Euclidean space
	4.2. Euclidean wave kernels
	4.3. Flat torus Tn
	4.4. Spheres Sn
	4.5. Hyperbolic space and non-Euclidean plane waves
	4.6. Dynamics and group theory of G=PSL(2,R)
	4.7. The Hyperbolic Laplacian
	4.8. Wave kernel and Poisson kernel on Hyperbolic space Hn
	4.9. Poisson kernel
	4.10. Spherical functions on H2
	4.11. The non-Euclidean Fourier transform
	4.12. Hyperbolic cylinders
	4.13. Irreducible representations of G
	4.14. Compact hyperbolic quotients X=Gamma minus H2
	4.15. Representation theory of G and spectral theory of Laplacian on compact quotients
	4.16. Appendix on the Fourier transform
Bibliography
Chapter 5. Local structure of eigenfunctions
	5.1. Local versus global eigenfunctions
	5.2. Small balls and local dilation
	5.3. Local elliptic estimates of eigenfunctions
	5.4. lambda-Poisson operators
	5.5. Bernstein estimates
	5.6. Frequency function and doubling index
	5.7. Carleman estimates
	5.8. Norm square of the Cauchy data
	5.9. Hyperbolic aspects
Bibliography
Chapter 6. Hadamard parametrices on Riemannian manifolds
	6.1. Hadamard parametrix
	6.2. Hadamard-Riesz parametrix
	6.3. The Hadamard-Feynman fundamental solution and Hadamard\'s parametrix
	6.4. Sketch of proof of Hadamard\'s construction
	6.5. Convergence in the real analytic case
	6.6. Away from CR
	6.7. Hadamard parametrix on a manifold without conjugate points
	6.8. Dimension 3
	6.9. Appendix on Homogeneous distributions
Bibliography
Chapter 7. Lagrangian distributions and Fourier integral operators
	7.1. Introduction
	7.2. Homogeneous Fourier integral operators
	7.3. Semi-classical Fourier integral operators
	7.4. Principal symbol, testing and matrix elements
	7.5. Composition of half-densities on canonical relations in cotangent bundles
Bibliography
Chapter 8. Small time wave group and Weyl asymptotics
	8.1. Hörmander parametrix
	8.2. Wave group and spectral projections
	8.3. Small-time asymptotics for microlocal wave operators
	8.4. Weyl law and local Weyl law
	8.5. Fourier Tauberian approach
	8.6. Tauberian Lemmas
Bibliography
Chapter 9. Matrix elements
	9.1. Invariance properties
	9.2. Proof of Egorov\'s theorem
	9.3. Weak* limit problem
	9.4. Matrix elements of spherical harmonics
	9.5. Quantum ergodicity and mixing of eigenfunctions
	9.6. Hassell\'s scarring result for stadia
	9.7.  Appendix on Duhamel\'s formula
Bibliography
Chapter 10. Lp norms
	10.1. Discrete Restriction theorems
	10.2. Random spherical harmonics and extremal spherical harmonics
	10.3. Sketch of proof of the Sogge Lp estimates
	10.4. Maximal eigenfunction growth
	10.5. Geometry of loops and return maps.
	10.6. Proof of Theorem 10.21. Step 1: Safarov\'s pre-trace formula
	10.7. Proof of Theorem 10.29. Step 2: Estimates of remainders at L-points
	10.8. Completion of the proof of Proposition 10.30 and Theorem 10.29: study of tilde Rj1
	10.9. Infinitely many twisted self-focal points
	10.10. Dynamics of the first return map at a self-focal point
	10.11. Proof of Proposition 10.20
	10.12. Uniformly bounded orthonormal basis
	10.13. Appendix: Integrated Weyl laws in the real domain
Bibliography
Chapter 11. Quantum Integrable systems
	11.1. Classical integrable systems
	11.2. Normal forms of integrable Hamiltonians near non-degenerate singular orbits
	11.3. Joint eigenfunctions
	11.4. Quantum toral integrable systems
	11.5. Lagrangian torus fibration and classical moment map
	11.6. Lp norms of Quantum integrable eigenfunctions
	11.7. Sketch of proof of Theorem 11.8
	11.8. Mass concentration of special eigenfunctions on hyperbolic orbits in the quantum integrable case
	11.9. Details on Mh
	11.10. Concentration of quantum integrable eigenfunctions on submanifolds
Bibliography
Chapter 12. Restriction theorems
	12.1. Null restrictions, degenerate restrictions and `goodness\'
	12.2. L2 upper bounds on Dirichlet or Neumann data of eigenfunctions
	12.3. Cauchy data of Dirichlet eigenfunctions for manifolds with boundary
	12.4. Restriction bounds for Neumann eigenfunctions
	12.5. Periods and Fourier coefficients of eigenfunctions on a closed geodesic
	12.6. Kuznecov sum formula: Proofs of Theorems 12.8 and 12.10
	12.7. Restricted Weyl laws
	12.8. Relating matrix elements of restrictions to global matrix elements
	12.9. Geodesic geometry of hypersurfaces
	12.10. Tangential cutoffs
	12.11. Canonical relation of gammaH
	12.12. The canonical relation of
	12.13. The canonical relation
	12.14. The pullback
	12.15. The pushforward
	12.16. The symbol of
	12.17. Proof of the restricted local Weyl law: Proposition 12.14
	12.18. Asymptotic completeness and orthogonality of Cauchy data
	12.19. Expansions in Cauchy data of eigenfunctions
	12.20. Bochner-Riesz means for Cauchy data
	12.21. Quantum ergodic restriction theorems
	12.22. Rellich approach to QER: Proof of Theorem 12.33
	12.23. Proof of Theorem 12.36 and Corollary 12.37
	12.24. Quantum ergodic restriction (QER) theorems for Dirichlet data
	12.25.  Time averaging
	12.26. Completion of the proofs of Theorems 12.39 and 12.40
Bibliography
Chapter 13. Nodal sets: Real domain
	13.1. Fundamental existence theorem for nodal sets
	13.2. Curvature of nodal lines and level lines
	13.3. Sub-level sets of eigenfunctions
	13.4. Nodal sets of real homogeneous polynomials
	13.5. Rectifiability of the nodal set
	13.6. Doubling estimates
	13.7. Lower bounds for  for Cinfty metrics
	13.8. Counting nodal domains
Bibliography
Chapter 14. Eigenfunctions in the complex domain
	14.1. Grauert tubes and complex geodesic flow
	14.2. Analytic continuation of the exponential map
	14.3. Maximal Grauert tubes
	14.4. Model examples
	14.5. Analytic continuation of eigenfunctions
	14.6. Maximal holomorphic extension
	14.7. Husimi functions
	14.8. Poisson wave operator and Szego projector on Grauert tubes
	14.9. Poisson operator and analytic Continuation of eigenfunctions
	14.10. Analytic continuation of the Poisson wave group
	14.11. Complexified spectral projections
	14.12. Poisson operator as a complex Fourier integral operator
	14.13. Complexified Poisson kernel as a complex Fourier integral operator
	14.14. Analytic continuation of the Poisson wave kernel
	14.15. Hörmander parametrix for the Poisson wave kernel
	14.16. Subordination to the heat kernel
	14.17. Fourier integral distributions with complex phase
	14.18. Analytic continuation of the Hadamard parametrix
	14.19. Analytic continuation of the Hörmander parametrix
	14.20. Delta g box g and characteristics
	14.21. Characteristic variety and characteristic conoid
	14.22. Hadamard parametrix for the Poisson wave kernel
	14.23. Hadamard parametrix as an oscillatory integral with complex phase
	14.24. Tempered spectral projector and Poisson semi-group as complex Fourier integral operators
	14.25. Complexified wave group and Szego kernels
	14.26. Growth of complexified eigenfunctions
	14.27. Siciak extremal functions: Proof of Theorem 14.14 (1)
	14.28. Pointwise phase space Weyl laws on Grauert tubes
	14.29. Proof of Corollary 14.16
	14.30. Complex nodal sets and sequences of logarithms
	14.31. Real zeros and complex analysis
	14.32. Background on hypersurfaces and geodesics
	14.33. Proof of the Donnelly-Fefferman lower bound (A. Brudnyi)
	14.34. Properties of eigenfunctions in good balls
	14.35. Background on good-ness
	14.36. A. Brudnyi\'s proof of Proposition 14.38
	14.37. Equidistribution of complex nodal sets of real ergodic eigenfunctions
	14.38. Sketch of the proof
	14.39. Growth properties of complexified eigenfunctions
	14.40. Proof of Lemma 14.48
	14.41. Proof of Lemma 14.47
	14.42. Intersections of nodal sets and analytic curves on real analytic surfaces
	14.43. Counting nodal lines which touch the boundary in analytic plane domains
	14.44. Application to Pleijel\'s conjecture
	14.45. Equidistribution of intersections of nodal lines and geodesics on surfaces
Bibliography
Index