**CBMS Regional Conference Series in Mathematics**

Volume: 125;
2017;
394 pp;
Softcover

MSC: Primary 34; 35; 53; 58;

**Print ISBN: 978-1-4704-1037-7
Product Code: CBMS/125**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

**Electronic ISBN: 978-1-4704-4344-3
Product Code: CBMS/125.E**

List Price: $79.00

AMS Member Price: $63.20

MAA Member Price: $71.10

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#### Supplemental Materials

# Eigenfunctions of the Laplacian on a Riemannian Manifold

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*Steve Zelditch*

A co-publication of the AMS and CBMS

Eigenfunctions of the Laplacian of a
Riemannian manifold can be described in terms of vibrating membranes
as well as quantum energy eigenstates. This book is an introduction to
both the local and global analysis of eigenfunctions. The local
analysis of eigenfunctions pertains to the behavior of the
eigenfunctions on wavelength scale balls. After re-scaling to a unit
ball, the eigenfunctions resemble almost-harmonic functions. Global
analysis refers to the use of wave equation methods to relate
properties of eigenfunctions to properties of the geodesic flow.

The emphasis is on the global methods and the use of Fourier
integral operator methods to analyze norms and nodal sets of
eigenfunctions. A somewhat unusual topic is the analytic continuation
of eigenfunctions to Grauert tubes in the real analytic case, and the
study of nodal sets in the complex domain.

The book, which grew out of lectures given by the author at a CBMS
conference in 2011, provides complete proofs of some model results,
but more often it gives informal and intuitive explanations of proofs
of fairly recent results. It conveys inter-related themes and results
and offers an up-to-date comprehensive treatment of this important
active area of research.

#### Readership

Graduate students and researchers interested in analysis related to spectral theory and eigenfunctions of Laplacians on Riemannian manifolds.

#### Reviews & Endorsements

The exposition is very clear and elegant, although it reaches rather technical results.

-- G. V. Rozenblum, Mathematical Reviews

We have a very serious work of scholarship, covering (if you'll pardon the pun) quite a spectrum of analysis (largely of the hard kind, as opposed to the soft kind), and useful to many audiences, from advanced students to experienced insiders. It's also distinctive as a springboard to any number of deeper studies flowing from the material in the book's individual chapters. It promises to be a very valuable resource.

-- Michael Berg, MAA Reviews

#### Table of Contents

# Table of Contents

## Eigenfunctions of the Laplacian on a Riemannian Manifold

Table of Contents pages: 1 2

- Cover Cover11
- Title page iii4
- Contents v6
- Preface xi12
- Chapter 1. Introduction 116
- 1.1. What are eigenfunctions and why are they useful 116
- 1.2. Notation for eigenvalues 318
- 1.3. Weyl’s law for Laplacian-eigenvalues 318
- 1.4. Quantum mechanics 419
- 1.5. Dynamics of the geodesic or billiard flow 621
- 1.6. Intensity plots and excursion sets 722
- 1.7. Nodal sets and critical point sets 924
- 1.8. Local versus global analysis of eigenfunctions 1025
- 1.9. High frequency limits, oscillation and concentration 1025
- 1.10. Spectral projections 1126
- 1.11. Lp norms 1227
- 1.12. Matrix elements and Wigner distributions 1328
- 1.13. Egorov’s theorem 1429
- 1.14. Eherenfest time 1429
- 1.15. Weak* limit problem 1429
- 1.16. Ergodic versus completely integrable geodesic flow 1732
- 1.17. Ergodic eigenfunctions 1833
- 1.18. Quantum unique ergodicity (QUE) 1833
- 1.19. Completely integrable eigenfunctions 1833
- 1.20. Heisenberg uncertainty principle 1934
- 1.21. Sequences of eigenfunctions and length scales 2035
- 1.22. Localization of eigenfunctions on closed geodesics 2136
- 1.23. Some remarks on the contents and on other texts 2237
- 1.24. References 2338

- Bibliography 2540
- Chapter 2. Geometric preliminaries 2944
- 2.1. Symplectic linear algebra and geometry 2944
- 2.2. Symplectic manifolds and cotangent bundles 3146
- 2.3. Lagrangian submanifolds 3247
- 2.4. Jacobi fields and Poincaré map 3348
- 2.5. Pseudo-differential operators 3449
- 2.6. Symbols 3550
- 2.7. Quantization of symbols 3651
- 2.8. Action of a pseudo-differential operator on a rapidly oscillating exponential 3752

- Bibliography 3954
- Chapter 3. Main results 4156
- 3.1. Universal Lp bounds 4156
- 3.2. Self-focal points and extremal Lp bounds for high p 4257
- 3.3. Low Lp norms and concentration of eigenfunctions around geodesics 4358
- 3.4. Kakeya-Nikodym maximal function and extremal Lp bounds for small p 4459
- 3.5. Concentration of joint eigenfunctions of quantum integrable Laplacian around closed geodesics 4560
- 3.6. Quantum ergodic restriction theorems for Cauchy data 4863
- 3.7. Quantum ergodic restriction theorems for Dirichlet data 5065
- 3.8. Counting nodal domains and nodal intersections with curves 5267
- 3.9. Intersections of nodal lines and general curves on negatively curved surfaces 5671
- 3.10. Complex zeros of eigenfunctions 5671

- Bibliography 5974
- Chapter 4. Model spaces of constant curvature 6176
- 4.1. Euclidean space 6176
- 4.2. Euclidean wave kernels 6580
- 4.3. Flat torus Tn 7388
- 4.4. Spheres Sn 7489
- 4.5. Hyperbolic space and non-Euclidean plane waves 8095
- 4.6. Dynamics and group theory of G=PSL(2,R) 8196
- 4.7. The Hyperbolic Laplacian 8297
- 4.8. Wave kernel and Poisson kernel on Hyperbolic space Hn 8398
- 4.9. Poisson kernel 86101
- 4.10. Spherical functions on H2 87102
- 4.11. The non-Euclidean Fourier transform 87102
- 4.12. Hyperbolic cylinders 87102
- 4.13. Irreducible representations of G 88103
- 4.14. Compact hyperbolic quotients X=Gamma minus H2 88103
- 4.15. Representation theory of G and spectral theory of Laplacian on compact quotients 89104
- 4.16. Appendix on the Fourier transform 89104

- Bibliography 93108
- Chapter 5. Local structure of eigenfunctions 95110
- 5.1. Local versus global eigenfunctions 95110
- 5.2. Small balls and local dilation 96111
- 5.3. Local elliptic estimates of eigenfunctions 98113
- 5.4. lambda-Poisson operators 102117
- 5.5. Bernstein estimates 104119
- 5.6. Frequency function and doubling index 105120
- 5.7. Carleman estimates 107122
- 5.8. Norm square of the Cauchy data 109124
- 5.9. Hyperbolic aspects 113128

- Bibliography 115130
- Chapter 6. Hadamard parametrices on Riemannian manifolds 119134
- 6.1. Hadamard parametrix 119134
- 6.2. Hadamard-Riesz parametrix 121136
- 6.3. The Hadamard-Feynman fundamental solution and Hadamard’s parametrix 122137
- 6.4. Sketch of proof of Hadamard’s construction 123138
- 6.5. Convergence in the real analytic case 126141
- 6.6. Away from CR 126141
- 6.7. Hadamard parametrix on a manifold without conjugate points 127142
- 6.8. Dimension 3 127142
- 6.9. Appendix on homogeneous distributions 131146

- Bibliography 133148
- Chapter 7. Lagrangian distributions and Fourier integral operators 135150
- Bibliography 159174
- Chapter 8. Small time wave group and Weyl asymptotics 161176
- Bibliography 173188
- Chapter 9. Matrix elements 175190
- 9.1. Invariance properties 176191
- 9.2. Proof of Egorov’s theorem 176191
- 9.3. Weak* limit problem 178193
- 9.4. Matrix elements of spherical harmonics 179194
- 9.5. Quantum ergodicity and mixing of eigenfunctions 180195
- 9.6. Hassell’s scarring result for stadia 188203
- 9.7. Appendix on Duhamel’s formula 192207

- Bibliography 195210
- Chapter 10. Lp norms 197212
- 10.1. Discrete Restriction theorems 199214
- 10.2. Random spherical harmonics and extremal spherical harmonics 200215
- 10.3. Sketch of proof of the Sogge Lp estimates 201216
- 10.4. Maximal eigenfunction growth 203218
- 10.5. Geometry of loops and return maps. 210225
- 10.6. Proof of Theorem 10.21. Step 1: Safarov’s pre-trace formula 216231
- 10.7. Proof of Theorem 10.29. Step 2: Estimates of remainders at L-points 222237
- 10.8. Completion of the proof of Proposition 10.30 and Theorem 10.29: study of tilde Rj1 223238
- 10.9. Infinitely many twisted self-focal points 227242
- 10.10. Dynamics of the first return map at a self-focal point 228243
- 10.11. Proof of Proposition 10.20 229244
- 10.12. Uniformly bounded orthonormal basis 231246
- 10.13. Appendix: Integrated Weyl laws in the real domain 232247

- Bibliography 235250
- Chapter 11. Quantum Integrable systems 239254
- 11.1. Classical integrable systems 239254
- 11.2. Normal forms of integrable Hamiltonians near non-degenerate singular orbits 242257
- 11.3. Joint eigenfunctions 243258
- 11.4. Quantum toral integrable systems 243258
- 11.5. Lagrangian torus fibration and classical moment map 246261
- 11.6. Lp norms of Quantum integrable eigenfunctions 246261
- 11.7. Sketch of proof of Theorem 11.8 247262
- 11.8. Mass concentration of special eigenfunctions on hyperbolic orbits in the quantum integrable case 249264
- 11.9. Details on Mh 250265
- 11.10. Concentration of quantum integrable eigenfunctions on submanifolds 251266

- Bibliography 253268
- Chapter 12. Restriction theorems 255270
- 12.1. Null restrictions, degenerate restrictions and ‘goodness’ 256271
- 12.2. L2 upper bounds on Dirichlet or Neumann data of eigenfunctions 258273
- 12.3. Cauchy data of Dirichlet eigenfunctions for manifolds with boundary 259274
- 12.4. Restriction bounds for Neumann eigenfunctions 260275
- 12.5. Periods and Fourier coefficients of eigenfunctions on a closed geodesic 260275
- 12.6. Kuznecov sum formula: Proofs of Theorems 12.8 and 12.10 262277
- 12.7. Restricted Weyl laws 263278
- 12.8. Relating matrix elements of restrictions to global matrix elements 265280
- 12.9. Geodesic geometry of hypersurfaces 266281
- 12.10. Tangential cutoffs 268283
- 12.11. Canonical relation of gammaH 268283
- 12.12. The canonical relation of 269284
- 12.13. The canonical relation 271286
- 12.14. The pullback 272287
- 12.15. The pushforward 272287
- 12.16. The symbol of 274289
- 12.17. Proof of the restricted local Weyl law: Proposition 12.14 275290
- 12.18. Asymptotic completeness and orthogonality of Cauchy data 276291
- 12.19. Expansions in Cauchy data of eigenfunctions 278293
- 12.20. Bochner-Riesz means for Cauchy data 280295
- 12.21. Quantum ergodic restriction theorems 281296
- 12.22. Rellich approach to QER: Proof of Theorem 12.33 283298
- 12.23. Proof of Theorem 12.36 and Corollary 12.37 286301
- 12.24. Quantum ergodic restriction (QER) theorems for Dirichlet data 287302
- 12.25. Time averaging 289304
- 12.26. Completion of the proofs of Theorems 12.39 and 12.40 292307

- Bibliography 295310
- Chapter 13. Nodal sets: Real domain 299314
- 13.1. Fundamental existence theorem for nodal sets 300315
- 13.2. Curvature of nodal lines and level lines 301316
- 13.3. Sub-level sets of eigenfunctions 302317
- 13.4. Nodal sets of real homogeneous polynomials 304319
- 13.5. Rectifiability of the nodal set 305320
- 13.6. Doubling estimates 307322
- 13.7. Lower bounds for for Cinfty metrics 309324
- 13.8. Counting nodal domains 315330

- Bibliography 327342
- Chapter 14. Eigenfunctions in the complex domain 333348
- 14.1. Grauert tubes and complex geodesic flow 334349
- 14.2. Analytic continuation of the exponential map 335350
- 14.3. Maximal Grauert tubes 335350
- 14.4. Model examples 336351
- 14.5. Analytic continuation of eigenfunctions 337352
- 14.6. Maximal holomorphic extension 338353
- 14.7. Husimi functions 339354
- 14.8. Poisson wave operator and Szego projector on Grauert tubes 339354
- 14.9. Poisson operator and analytic continuation of eigenfunctions 339354
- 14.10. Analytic continuation of the Poisson wave group 340355
- 14.11. Complexified spectral projections 340355
- 14.12. Poisson operator as a complex Fourier integral operator 341356
- 14.13. Complexified Poisson kernel as a complex Fourier integral operator 342357

Table of Contents pages: 1 2