Eigenfunctions of the Laplacian on a Riemannian manifold:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2017]
|
| Series: | Regional conference series in mathematics
number 125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 394 Seiten |
| ISBN: | 9781470410377 |
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| 100 | 1 | |a Zelditch, Steven |d 1953- |e Verfasser |0 (DE-588)1145810098 |4 aut | |
| 245 | 1 | 0 | |a Eigenfunctions of the Laplacian on a Riemannian manifold |c Steve Zelditch |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2017] | |
| 264 | 4 | |c © 2017 | |
| 300 | |a xiv, 394 Seiten | ||
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Record in the Search Index
| _version_ | 1804178440311537664 |
|---|---|
| adam_text | Contents
Preface xi
0.1. Organization xii
0.2. Topics which are not covered xiii
0.3. Topics which are double covered xiv
0.4. Notation xiv
Acknowledgments xiv
Chapter 1. Introduction 1
1.1. What are eigenfunctions and why are the}^ useful 1
1.2. Notation for eigenvalues 3
1.3. Weyl’s law for ( — A)-eigenvalues 3
1.4. Quantum mechanics 4
1.5. Dynamics of the geodesic or billiard flow 6
1.6. Intensity plots and excursion sets 7
1.7. Nodal sets and critical point sets 9
1.8. Local versus global analysis of eigenfunctions 10
1.9. High frequency limits, oscillation and concentration 10
1.10. Spectral projections 11
1.11. Lp norms 12
1.12. Matrix elements and Wigner distributions 13
1.13. Egorov’s theorem 14
1.14. Eherenfest time 14
1.15. Weak* limit problem 14
1.16. Ergodic versus completely integrable geodesic flow 17
1.17. Ergodic eigenfunctions 18
1.18. Quantum unique ergodicity (QUE) 18
1.19. Completely integrable eigenfunctions 18
1.20. Heisenberg uncertainty principle 19
1.21. Sequences of eigenfunctions and length scales 20
1.22. Localization of eigenfunctions on closed geodesics 21
1.23. Some remarks on the contents and on other texts 22
1.24. References 23
Bibliography 25
Chapter 2. Geometric preliminaries 29
2.1. Symplectic linear algebra and geometry 29
2.2. Symplectic manifolds and cotangent bundles 31
2.3. Lagrangiari submanifolds 32
2.4. Jacobi fields and Poincare map 33
V
VI
CONTENTS
2.5. Pseudo-differential operators 34
2.6. Symbols 35
2.7. Quantization of symbols 36
2.8. Action of a pseudo-differential operator on a rapidly
oscillating exponential 37
Bibliography 39
Chapter 3. Main results 41
3.1. Universal Lp bounds 41
3.2. Self-focal points and extremal Lv bounds for high p 42
3.3. Low LP norms and concentration of eigenfunctions around geodesics 43
3.4. Kakeya-Nikodym maximal function and extremal Lv bounds
for small p 44
3.5. Concentration of joint eigenfunctions of quantum integrable A around
closed geodesics 45
3.6. Quantum ergo die restriction theorems for Cauchy data 48
3.7. Quantum ergodic restriction theorems for Dirichlet data 50
3.8. Counting nodal domains and nodal intersections with curves 52
3.9. Intersections of nodal lines and general curves on negatively
curved surfaces 56
3.10. Complex zeros of eigenfunctions 56
Bibliography 59
Chapter 4. Model spaces of constant curvature 61
4.1. Euclidean space 61
4.2. Euclidean wave kernels 65
4.3. Flat torus Tn 73
4.4. Spheres Sn 74
4.5. Hyperbolic space and non-Euclidean plane waves 80
4.6. Dynamics and group theory of G — PSL(2, R) 81
4.7. The Hyperbolic Laplacian 82
4.8. Wave kernel and Poisson kernel on Hyperbolic space Hn 83
4.9. Poisson kernel 86
4.10. Spherical functions on H2 87
4.11. The non-Euclidean Fourier transform 87
4.12. Hyperbolic cylinders 87
4.13. Irreducible representations of G 88
4.14. Compact hyperbolic quotients Xp = T H2 88
4.15. Representation theory of G and spectral theory of A on
compact quotients 89
4.16. Appendix on the Fourier transform 89
Bibliography 93
Chapter 5. Local structure of eigenfunctions 95
5.1. Local versus global eigenfunctions 95
5.2. Small balls and local dilation 93
5.3. Local elliptic estimates of eigenfunctions 98
CONTENTS
vii
5.4. A-Poisson operators 102
5.5. Bernstein estimates 104
5.6. Frequency function and doubling index :: 105
5.7. Carleman estimates ? •* 107
5.8. Norm square of the Cauchy data ; ;?r * - - 1 109
5.9. Hyperbolic aspects i4113
Bibliography 115
Chapter 6. Hadamard parametrices on Riemannian manifolds T 119
6.1. Hadamard parametrix 119
6.2. Hadamard-Riesz parametrix o 121
6.3. The Hadamard-Feynman fundamental solution and
Hadamard’s parametrix , 122
6.4. Sketch of proof of Hadamard’s construction 123
6.5. Convergence in the real analytic case 126
6.6. Away from Cu 126
6.7. Hadamard parametrix on a manifold without conjugate points 127
6.8. Dimension 3 127
6.9. Appendix on homogeneous distributions 131
Bibliography 133
Chapter 7. Lagrangian distributions and Fourier integral operators 135
7.1. Introduction 135
7.2. Homogeneous Fourier integral operators 137
7.3. Semi-classical Fourier integral operators 146
7.4. Principal symbol, testing and matrix elements 150
7.5. Composition of half-densities on canonical relations
in cotangent bundles ^ 157
Bibliography 159
Chapter 8. Small time wave group and Weyl asymptotics 161
8.1. Hormander parametrix ?v 161
8.2. Wave group and spectral projections 162
8.3. Small-time asymptotics for microlocal wave operators 163
8.4. Weyl law and local Weyl law 165
8.5. Fourier Tauberian approach , 167
8.6. Tauberian lemmas 171
Bibliography ’ 173
Chapter 9. Matrix elements 175
9.1. Invariance properties 176
9.2. Proof of Egorov’s theorem 176
9.3. Weak* limit problem 178
9.4. Matrix elements of spherical harmonics 179
9.5. Quantum ergo dicity and mixing of eigenfunctions 180
9.6. Hassell’s scarring result for stadia 188
9.7. Appendix on Duhamel’s formula y-* 192
CONTENTS
viii
Bibliography
195
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 £-points
10.8. Completion of the proof of Proposition 10.30 and Theorem 10.29:
study of Rj i
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
197
199
200
201
203
210
216
222
223
227
228
229
231
232
Bibliography
235
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 M^
11.10. Concentration of quantum integrable eigenfunctions
on submanifolds
239
239
242
243
243
246
246
247
249
250
251
Bibliography 253
Chapter 12. Restriction theorems 255
12.1. Null restrictions, degenerate restrictions and goodness’ 256
12.2. L2 upper bounds on Dirichlet or Neumann data of eigenfunctions 258
12.3. Cauchy data of Dirichlet eigenfunctions for manifolds
with boundary 259
12.4. Restriction bounds for Neumann eigenfunctions 260
12.5. Periods and Fourier coefficients of eigenfunctions on
a closed geodesic 260
12.6. Kuznecov sum formula: Proofs of Theorems 12.8 and 12.10 262
12.7. Restricted Weyl laws 263
12.8. Relating matrix elements of restrictions to global matrix elements 265
12.9. Geodesic geometry of hypersurfaces 266
CONTENTS ix
12.10. Tangential cutoffs 268
12.11. Canonical relation of 7// 268
12.12. The canonical relation of 7^ Op//(a)7// 269
12.13. The canonical relation T* o C o T 271
12.14. The pullback Yn := A*r* o o F 272
12.15. The pushforward 7r£*A*r* o Ch 0 r 272
12.16. The symbol of U{t{)*{rr*H OvH{aY(H) £U{t2) 274
12.17. Proof of the restricted local Weyl law: Proposition 12.14 275
12.18. Asymptotic completeness and orthogonality of Cauchy data 276
12.19. Expansions in Cauchy data of eigenfunctions 278
12.20. Bochner-Riesz means for Cauchy data 280
12.21. Quantum ergodic restriction theorems 281
12.22. Rellich approach to QER: Proof of Theorem 12.33 283
12.23. Proof of Theorem 12.36 and Corollary 12.37 286
12.24. Quantum ergodic restriction (QER) theorems for Dirichlet data 287
12.25. Time averaging 289
12.26. Completion of the proofs of Theorems 12.39 and 12.40 292
Bibliography 295
Chapter 13. Nodal sets: Real domain 299
13.1. Fundamental existence theorem for nodal sets 300
13.2. Curvature of nodal lines and level lines 301
13.3. Sub-level sets of eigenfunctions 302
13.4. Nodal sets of real homogeneous polynomials 304
13.5. Rectifiability of the nodal set 305
13.6. Doubling estimates 307
13.7. Lower bounds for 7tm~l(J f ) for C°° metrics 309
13.8. Counting nodal domains 315
Bibliography 327
Chapter 14. Eigenfunctions in the complex domain 333
14.1. Grauert tubes and complex geodesic flow 334
14.2. Analytic continuation of the exponential map 335
14.3. Maximal Grauert tubes 335
14.4. Model examples 336
14.5. Analytic continuation of eigenfunctions 337
14.6. Maximal holomorphic extension 338
14.7. Husimi functions 339
14.8. Poisson wave operator and Szego projector on Grauert tubes 339
14.9. Poisson operator and analytic continuation of eigenfunctions 339
14.10. Analytic continuation of the Poisson wave group 340
14.11. Complexified spectral projections 340
14.12. Poisson operator as a complex Fourier integral operator 341
14.13. Complexified Poisson kernel as a complex Fourier integral operator 342
14.14. Analytic continuation of the Poisson wave kernel 343
14.15. Hormander parametrix for the Poisson wave kernel 343
14.16. Subordination to the heat kernel 343
X
CONTENTS
14.17. Fourier integral distributions
with complex phase 344
14.18. Analytic continuation of the Hadamard parametrix 344
14.19. Analytic continuation of the Hormander parametrix 345
14.20. Ag, and characteristics 345
14.21. Characteristic variety and characteristic conoid 346
14.22. Hadamard parametrix for the Poisson wave kernel 346
14.23. Hadamard parametrix as an oscillatory integral
with complex phase 347
14.24. Tempered spectral projector and Poisson semi-group as complex
Fourier integral operators 350
14.25. Complexified wave group and Szego kernels 351
14.26. Growth of complexified eigenfunctions 352
14.27. Siciak extremal functions: Proof of Theorem 14.14 (1) 354
14.28. Pointwise phase space Weyl laws on Grauert tubes 356
14.29. Proof of Corollary 14.16 358
14.30. Complex nodal sets and sequences of logarithms 359
14.31. Real zeros and complex analysis 361
14.32. Background on hypersurfaces and geodesics 362
14.33. Proof of the Donnelly-Fefferman lower bound (A. Brudnyi) 367
14.34. Properties of eigenfunctions in good balls 369
14.35. Background on good-ness 369
14.36. A. Brudnyi’s proof of Proposition 14.38 370
14.37. Equidistribution of complex nodal sets of real ergodic eigenfunctions372
14.38. Sketch of the proof 373
14.39. Growth properties of complexified eigenfunctions 374
14.40. Proof of Lemma 14.48 377
14.41. Proof of Lemma 14.47 377
14.42. Intersections of nodal sets and analytic curves on real
analytic surfaces 378
14.43. Counting nodal lines which touch the boundary in analytic
plane domains 379
14.44. Application to Pleijel’s conjecture 384
14.45. Equidistribution of intersections of nodal lines and geodesics
on surfaces 384
Bibliography 389
Index 393
|
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| id | DE-604.BV044892234 |
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| indexdate | 2024-07-10T08:04:00Z |
| institution | BVB |
| isbn | 9781470410377 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030286215 |
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| owner | DE-355 DE-BY-UBR DE-188 |
| owner_facet | DE-355 DE-BY-UBR DE-188 |
| physical | xiv, 394 Seiten |
| publishDate | 2017 |
| publishDateSearch | 2017 |
| publishDateSort | 2017 |
| publisher | American Mathematical Society |
| record_format | marc |
| series | Regional conference series in mathematics |
| series2 | Regional conference series in mathematics |
| spelling | Zelditch, Steven 1953- Verfasser (DE-588)1145810098 aut Eigenfunctions of the Laplacian on a Riemannian manifold Steve Zelditch Providence, Rhode Island American Mathematical Society [2017] © 2017 xiv, 394 Seiten txt rdacontent n rdamedia nc rdacarrier Regional conference series in mathematics number 125 Laplace-Operator (DE-588)4166772-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 gnd rswk-swf Riemannscher Raum (DE-588)4128295-4 s Laplace-Operator (DE-588)4166772-4 s DE-604 Regional conference series in mathematics number 125 (DE-604)BV000004346 125 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030286215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zelditch, Steven 1953- Eigenfunctions of the Laplacian on a Riemannian manifold Regional conference series in mathematics Laplace-Operator (DE-588)4166772-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| subject_GND | (DE-588)4166772-4 (DE-588)4128295-4 |
| title | Eigenfunctions of the Laplacian on a Riemannian manifold |
| title_auth | Eigenfunctions of the Laplacian on a Riemannian manifold |
| title_exact_search | Eigenfunctions of the Laplacian on a Riemannian manifold |
| title_full | Eigenfunctions of the Laplacian on a Riemannian manifold Steve Zelditch |
| title_fullStr | Eigenfunctions of the Laplacian on a Riemannian manifold Steve Zelditch |
| title_full_unstemmed | Eigenfunctions of the Laplacian on a Riemannian manifold Steve Zelditch |
| title_short | Eigenfunctions of the Laplacian on a Riemannian manifold |
| title_sort | eigenfunctions of the laplacian on a riemannian manifold |
| topic | Laplace-Operator (DE-588)4166772-4 gnd Riemannscher Raum (DE-588)4128295-4 gnd |
| topic_facet | Laplace-Operator Riemannscher Raum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030286215&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004346 |
| work_keys_str_mv | AT zelditchsteven eigenfunctionsofthelaplacianonariemannianmanifold |