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E-grāmata: Eigenfunctions of the Laplacian on a Riemannian Manifold

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Eigenfunctions of the Laplacian on a Riemannian ManifoldPalielināt
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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.
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(28)
1.1 What are eigenfunctions and why are they useful
1(2)
1.2 Notation for eigenvalues
3(1)
1.3 Weyl's law for (--Delta;)-eigenvalues
3(1)
1.4 Quantum mechanics
4(2)
1.5 Dynamics of the geodesic or billiard flow
6(1)
1.6 Intensity plots and excursion sets
7(2)
1.7 Nodal sets and critical point sets
9(1)
1.8 Local versus global analysis of eigenfunctions
10(1)
1.9 High frequency limits, oscillation and concentration
10(1)
1.10 Spectral projections
11(1)
1.11 V norms
12(1)
1.12 Matrix elements and Wigner distributions
13(1)
1.13 Egorov's theorem
14(1)
1.14 Eherenfest time
14(1)
1.15 Weak* limit problem
14(3)
1.16 Ergodic versus completely integrable geodesic flow
17(1)
1.17 Ergodic eigenfunctions
18(1)
1.18 Quantum unique ergodicity (QUE)
18(1)
1.19 Completely integrable eigenfunctions
18(1)
1.20 Heisenberg uncertainty principle
19(1)
1.21 Sequences of eigenfunctions and length scales
20(1)
1.22 Localization of eigenfunctions on closed geodesies
21(1)
1.23 Some remarks on the contents and on other texts
22(1)
1.24 References
23(6)
Bibliography
25(4)
Chapter 2 Geometric preliminaries
29(12)
2.1 Symplectic linear algebra and geometry
29(2)
2.2 Symplectic manifolds and cotangent bundles
31(1)
2.3 Lagrangian submanifolds
32(1)
2.4 Jacobi fields and Poincare map
33(1)
2.5 Pseudo-differential operators
34(1)
2.6 Symbols
35(1)
2.7 Quantization of symbols
36(1)
2.8 Action of a pseudo-differential operator on a rapidly oscillating exponential
37(4)
Bibliography
39(2)
Chapter 3 Main results
41(20)
3.1 Universal LP bounds
41(1)
3.2 Self-focal points and extremal LP bounds for high p
42(1)
3.3 Low LP norms and concentration of eigenfunctions around geodesies
43(1)
3.4 Kakeya-Nikodym maximal function and extremal LP bounds for small p
44(1)
3.5 Concentration of joint eigenfunctions of quantum integrable Δ around closed geodesies
45(3)
3.6 Quantum ergodic restriction theorems for Cauchy data
48(2)
3.7 Quantum ergodic restriction theorems for Dirichlet data
50(2)
3.8 Counting nodal domains and nodal intersections with curves
52(4)
3.9 Intersections of nodal lines and general curves on negatively curved surfaces
56(1)
3.10 Complex zeros of eigenfunctions
56(5)
Bibliography
59(2)
Chapter 4 Model spaces of constant curvature
61(34)
4.1 Euclidean space
61(4)
4.2 Euclidean wave kernels
65(8)
4.3 Flat torus TP
73(1)
4.4 Spheres Sn
74(6)
4.5 Hyperbolic space and non-Euclidean plane waves
80(1)
4.6 Dynamics and group theory of G = PSL(2, R)
81(1)
4.7 The Hyperbolic Laplacian
82(1)
4.8 Wave kernel and Poisson kernel on Hyperbolic space Hn
83(3)
4.9 Poisson kernel
86(1)
4.10 Spherical functions on H2
87(1)
4.11 The non-Euclidean Fourier transform
87(1)
4.12 Hyperbolic cylinders
87(1)
4.13 Irreducible representations of G
88(1)
4.14 Compact hyperbolic quotients XΓ = Γ\H2
88(1)
4.15 Representation theory of G and spectral theory of Δ on compact quotients
89(1)
4.16 Appendix on the Fourier transform
89(6)
Bibliography
93(2)
Chapter 5 Local structure of eigenfunctions
95(24)
5.1 Local versus global eigenfunctions
95(1)
5.2 Small balls and local dilation
96(2)
5.3 Local elliptic estimates of eigenfunctions
98(4)
5.4 A-Poisson operators
102(2)
5.5 Bernstein estimates
104(1)
5.6 Frequency function and doubling index
105(2)
5.7 Carleman estimates
107(2)
5.8 Norm square of the Cauchy data
109(4)
5.9 Hyperbolic aspects
113(6)
Bibliography
115(4)
Chapter 6 Hadamard parametrices on Riemannian manifolds
119(16)
6.1 Hadamard parametrix
119(2)
6.2 Hadamard-Riesz parametrix
121(1)
6.3 The Hadamard-Feynman fundamental solution and Hadamard's parametrix
122(1)
6.4 Sketch of proof of Hadamard's construction
123(3)
6.5 Convergence in the real analytic case
126(1)
6.6 Away from CR
126(1)
6.7 Hadamard parametrix on a manifold without conjugate points
127(1)
6.8 Dimension 3
127(4)
6.9 Appendix on homogeneous distributions
131(4)
Bibliography
133(2)
Chapter 7 Lagrangian distributions and Fourier integral operators
135(26)
7.1 Introduction
135(2)
7.2 Homogeneous Fourier integral operators
137(9)
7.3 Semi-classical Fourier integral operators
146(4)
7.4 Principal symbol, testing and matrix elements
150(7)
7.5 Composition of half-densities on canonical relations in cotangent bundles
157(4)
Bibliography
159(2)
Chapter 8 Small time wave group and Weyl asymptotics
161(14)
8.1 Hormander parametrix
161(1)
8.2 Wave group and spectral projections
162(1)
8.3 Small-time asymptotics for microlocal wave operators
163(2)
8.4 Weyl law and local Weyl law
165(2)
8.5 Fourier Tauberian approach
167(4)
8.6 Tauberian lemmas
171(4)
Bibliography
173(2)
Chapter 9 Matrix elements
175(22)
9.1 Invariance properties
176(1)
9.2 Proof of Egorov's theorem
176(2)
9.3 Weak* limit problem
178(1)
9.4 Matrix elements of spherical harmonics
179(1)
9.5 Quantum ergodicity and mixing of eigenfunctions
180(8)
9.6 Hassell's scarring result for stadia
188(4)
9.7 Appendix on Duhamel's formula
192(5)
Bibliography
195(2)
Chapter 10 LP norms
197(42)
10.1 Discrete Restriction theorems
199(1)
10.2 Random spherical harmonics and extremal spherical harmonics
200(1)
10.3 Sketch of proof of the Sogge Lp estimates
201(2)
10.4 Maximal eigenfunction growth
203(7)
10.5 Geometry of loops and return maps
210(6)
10.6 Proof of Theorem 10.21. Step 1: Safarov's pre-trace formula
216(6)
10.7 Proof of Theorem 10.29. Step 2: Estimates of remainders at L-points
222(1)
10.8 Completion of the proof of Proposition 10.30 and Theorem 10.29: study of Rj1
223(4)
10.9 Infinitely many twisted self-focal points
227(1)
10.10 Dynamics of the first return map at a self-focal point
228(1)
10.11 Proof of Proposition 10.20
229(2)
10.12 Uniformly bounded orthonormal basis
231(1)
10.13 Appendix: Integrated Weyl laws in the real domain
232(7)
Bibliography
235(4)
Chapter 11 Quantum Integrable systems
239(16)
11.1 Classical integrable systems
239(3)
11.2 Normal forms of integrable Hamiltonians near non-degenerate singular orbits
242(1)
11.3 Joint eigenfunctions
243(1)
11.4 Quantum toral integrable systems
243(3)
11.5 Lagrangian torus fibration and classical moment map
246(1)
11.6 LP norms of Quantum integrable eigenfunctions
246(1)
11.7 Sketch of proof of Theorem 11.8
247(2)
11.8 Mass concentration of special eigenfunctions on hyperbolic orbits in the quantum integrable case
249(1)
11.9 Details on Mh
250(1)
11.10 Concentration of quantum integrable eigenfunctions on submanifolds
251(4)
Bibliography
253(2)
Chapter 12 Restriction theorems
255(44)
12.1 Null restrictions, degenerate restrictions and `goodness'
256(2)
12.2 L2 upper bounds on Dirichlet or Neumann data of eigenfunctions
258(1)
12.3 Cauchy data of Dirichlet eigenfunctions for manifolds with boundary
259(1)
12.4 Restriction bounds for Neumann eigenfunctions
260(1)
12.5 Periods and Fourier coefficients of eigenfunctions on a closed geodesic
260(2)
12.6 Kuznecov sum formula: Proofs of Theorems 12.8 and 12.10
262(1)
12.7 Restricted Weyl laws
263(2)
12.8 Relating matrix elements of restrictions to global matrix elements
265(1)
12.9 Geodesic geometry of hypersurfaces
266(2)
12.10 Tangential cutoffs
268(1)
12.11 Canonical relation of γH
268(1)
12.12 The canonical relation of γ*H OpH(a)γH
269(2)
12.13 The canonical relation Γ o CH o Γ
271(1)
12.14 The pullback ΓH:= Δ*tΓ* o CH o Γ
272(1)
12.15 The pushforward πt*ΔΓ* o CH o Γ
272(2)
12.16 The symbol of U(t1)*(γ*H OpH(a)γH)≥εU(t2)
274(1)
12.17 Proof of the restricted local Weyl law: Proposition 12.14
275(1)
12.18 Asymptotic completeness and orthogonality of Cauchy data
276(2)
12.19 Expansions in Cauchy data of eigenfunctions
278(2)
12.20 Bochner-Riesz means for Cauchy data
280(1)
12.21 Quantum ergodic restriction theorems
281(2)
12.22 Rellich approach to QER: Proof of Theorem 12.33
283(3)
12.23 Proof of Theorem 12.36 and Corollary 12.37
286(1)
12.24 Quantum ergodic restriction (QER) theorems for Dirichlet data
287(2)
12.25 Time averaging
289(3)
12.26 Completion of the proofs of Theorems 12.39 and 12.40
292(7)
Bibliography
295(4)
Chapter 13 Nodal sets: Real domain
299(34)
13.1 Fundamental existence theorem for nodal sets
300(1)
13.2 Curvature of nodal lines and level lines
301(1)
13.3 Sub-level sets of eigenfunctions
302(2)
13.4 Nodal sets of real homogeneous polynomials
304(1)
13.5 Rectifiability of the nodal set
305(2)
13.6 Doubling estimates
307(2)
13.7 Lower bounds for Hm--1(Nλ) for C∞ metrics
309(6)
13.8 Counting nodal domains
315(18)
Bibliography
327(6)
Chapter 14 Eigenfunctions in the complex domain
333(60)
14.1 Grauert tubes and complex geodesic flow
334(1)
14.2 Analytic continuation of the exponential map
335(1)
14.3 Maximal Grauert tubes
335(1)
14.4 Model examples
336(1)
14.5 Analytic continuation of eigenfunctions
337(1)
14.6 Maximal holomorphic extension
338(1)
14.7 Husimi functions
339(1)
14.8 Poisson wave operator and Szego projector on Grauert tubes
339(1)
14.9 Poisson operator and analytic continuation of eigenfunctions
339(1)
14.10 Analytic continuation of the Poisson wave group
340(1)
14.11 Complexified spectral projections
340(1)
14.12 Poisson operator as a complex Fourier integral operator
341(1)
14.13 Complexified Poisson kernel as a complex Fourier integral operator
342(1)
14.14 Analytic continuation of the Poisson wave kernel
343(1)
14.15 Hormander parametrix for the Poisson wave kernel
343(1)
14.16 Subordination to the heat kernel
343(1)
14.17 Fourier integral distributions with complex phase
344(1)
14.18 Analytic continuation of the Hadamard parametrix
344(1)
14.19 Analytic continuation of the Hormander parametrix
345(1)
14.20 Ag, Dg and characteristics
345(1)
14.21 Characteristic variety and characteristic conoid
346(1)
14.22 Hadamard parametrix for the Poisson wave kernel
346(1)
14.23 Hadamard parametrix as an oscillatory integral with complex phase
347(3)
14.24 Tempered spectral projector and Poisson semi-group as complex Fourier integral operators
350(1)
14.25 Complexified wave group and Szego kernels
351(1)
14.26 Growth of complexified eigenfunctions
352(2)
14.27 Siciak extremal functions: Proof of Theorem 14.14 (1)
354(2)
14.28 Pointwise phase space Weyl laws on Grauert tubes
356(2)
14.29 Proof of Corollary 14.16
358(1)
14.30 Complex nodal sets and sequences of logarithms
359(2)
14.31 Real zeros and complex analysis
361(1)
14.32 Background on hypersurfaces and geodesies
362(5)
14.33 Proof of the Donnelly-Fefferman lower bound (A. Brudnyi)
367(2)
14.34 Properties of eigenfunctions in good balls
369(1)
14.35 Background on good-ness
369(1)
14.36 A. Brudnyi's proof of Proposition 14.38
370(2)
14.37 Equidistribution of complex nodal sets of real ergodic eigenfunctions
372(1)
14.38 Sketch of the proof
373(1)
14.39 Growth properties of complexified eigenfunctions
374(3)
14.40 Proof of Lemma 14.48
377(1)
14.41 Proof of Lemma 14.47
377(1)
14.42 Intersections of nodal sets and analytic curves on real analytic surfaces
378(1)
14.43 Counting nodal lines which touch the boundary in analytic plane domains
379(5)
14.44 Application to Pleijel's conjecture
384(1)
14.45 Equidistribution of intersections of nodal lines and geodesies on surfaces
384(9)
Bibliography
389(4)
Index 393
Steve Zelditch, Northwestern University, Evanston, IL.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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