spectral theory, geometry and dynamical systems · applications eigenvalue questions counting...
TRANSCRIPT
![Page 1: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/1.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
Spectral theory, geometry and dynamicalsystems
Dmitry Jakobson
8th January 2010
![Page 2: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/2.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• M is n-dimensional compact connected manifold,n ≥ 2. g is a Riemannian metric on M: for anyU, V ∈ TxM, their inner product is g(U, V ).g(∂/∂xi , ∂/∂xj) := gij .
• g defines analogs of div and grad. The Laplacian ∆ of afunction f is given by
∆f = div(gradf ).
An eigenfunction φ with eigenvalue λ ≥ 0 satisfies
∆f + λf = 0.
![Page 3: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/3.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• M is n-dimensional compact connected manifold,n ≥ 2. g is a Riemannian metric on M: for anyU, V ∈ TxM, their inner product is g(U, V ).g(∂/∂xi , ∂/∂xj) := gij .
• g defines analogs of div and grad. The Laplacian ∆ of afunction f is given by
∆f = div(gradf ).
An eigenfunction φ with eigenvalue λ ≥ 0 satisfies
∆f + λf = 0.
![Page 4: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/4.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example 1: R2.
∆f =∂2f∂x2 +
∂2f∂y2 .
Periodic eigenfunctions on the 2-torus T2:f (x ± 2π, y ± 2π) = f (x , y). They are
sin(m · x + n · y), cos(m · x + n · y), λ = m2 + n2.
• Fact: any square-integrable function F (x , y) on T2 (s.t.∫T2 |F (x , y)|2dxdy < ∞), can be expanded into Fourier
series,
F =+∞∑
m,n=−∞am,n sin(mx + ny) + bm,n cos(mx + ny).
![Page 5: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/5.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example 1: R2.
∆f =∂2f∂x2 +
∂2f∂y2 .
Periodic eigenfunctions on the 2-torus T2:f (x ± 2π, y ± 2π) = f (x , y). They are
sin(m · x + n · y), cos(m · x + n · y), λ = m2 + n2.
• Fact: any square-integrable function F (x , y) on T2 (s.t.∫T2 |F (x , y)|2dxdy < ∞), can be expanded into Fourier
series,
F =+∞∑
m,n=−∞am,n sin(mx + ny) + bm,n cos(mx + ny).
![Page 6: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/6.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example 2: sphere S2 = {(x , y , z) : x2 + y2 + z2 = 1}.Spherical coordinates: (φ, θ) ∈ [0, π]× [0, 2π], wherex = sin φ cos θ, y = sin φ sin θ, z = cos φ.
∆f =1
sin2 φ· ∂2f∂θ2 +
cos φ
sin φ· ∂f∂φ
+∂2f∂φ2
• Eigenfunctions are called spherical harmonics:
Y ml (φ, θ) = Pm
l (cos φ)(a cos(mθ) + b sin(mθ)).
Here λ = l(l + 1); Pml , |m| ≤ l is associated Legendre
function,
Pml (x) =
(−1)m
2l · l! (1− x2)m2
d l+m
dx l+m
((x2 − 1)l
).
Any square-integrable function F on S2 can beexpanded in a series of spherical harmonics.
![Page 7: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/7.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example 2: sphere S2 = {(x , y , z) : x2 + y2 + z2 = 1}.Spherical coordinates: (φ, θ) ∈ [0, π]× [0, 2π], wherex = sin φ cos θ, y = sin φ sin θ, z = cos φ.
∆f =1
sin2 φ· ∂2f∂θ2 +
cos φ
sin φ· ∂f∂φ
+∂2f∂φ2
• Eigenfunctions are called spherical harmonics:
Y ml (φ, θ) = Pm
l (cos φ)(a cos(mθ) + b sin(mθ)).
Here λ = l(l + 1); Pml , |m| ≤ l is associated Legendre
function,
Pml (x) =
(−1)m
2l · l! (1− x2)m2
d l+m
dx l+m
((x2 − 1)l
).
Any square-integrable function F on S2 can beexpanded in a series of spherical harmonics.
![Page 8: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/8.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• The same is true on any compact M. Eigenvalues of ∆:
0 = λ0 < λ1 ≤ λ2 ≤ . . .
Any square-integrable function F on M can beexpanded in a series of eigenfunctions φi of ∆.
• Similar results hold for domains with boundary (youneed to specify boundary conditions).
• Standard boundary conditions: Dirichlet (φ vanishes onthe boundary); Neumann (normal derivative of φvanishes on the boundary).
• Example: M = [0, π]. sin x satisfies Dirichlet b.c; cos xsatisfies Neumann b.c.
![Page 9: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/9.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• The same is true on any compact M. Eigenvalues of ∆:
0 = λ0 < λ1 ≤ λ2 ≤ . . .
Any square-integrable function F on M can beexpanded in a series of eigenfunctions φi of ∆.
• Similar results hold for domains with boundary (youneed to specify boundary conditions).
• Standard boundary conditions: Dirichlet (φ vanishes onthe boundary); Neumann (normal derivative of φvanishes on the boundary).
• Example: M = [0, π]. sin x satisfies Dirichlet b.c; cos xsatisfies Neumann b.c.
![Page 10: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/10.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• The same is true on any compact M. Eigenvalues of ∆:
0 = λ0 < λ1 ≤ λ2 ≤ . . .
Any square-integrable function F on M can beexpanded in a series of eigenfunctions φi of ∆.
• Similar results hold for domains with boundary (youneed to specify boundary conditions).
• Standard boundary conditions: Dirichlet (φ vanishes onthe boundary); Neumann (normal derivative of φvanishes on the boundary).
• Example: M = [0, π]. sin x satisfies Dirichlet b.c; cos xsatisfies Neumann b.c.
![Page 11: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/11.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• The same is true on any compact M. Eigenvalues of ∆:
0 = λ0 < λ1 ≤ λ2 ≤ . . .
Any square-integrable function F on M can beexpanded in a series of eigenfunctions φi of ∆.
• Similar results hold for domains with boundary (youneed to specify boundary conditions).
• Standard boundary conditions: Dirichlet (φ vanishes onthe boundary); Neumann (normal derivative of φvanishes on the boundary).
• Example: M = [0, π]. sin x satisfies Dirichlet b.c; cos xsatisfies Neumann b.c.
![Page 12: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/12.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Solving partial differential equations like heat equation∂u(x , t)/∂t = c ·∆xu(x , t) and wave equation∂2u(x , t)/∂t2 = c ·∆xu(x , t).
• Stationary solutions of Schrodinger equation or “purequantum states.”
• Inverse problems: suppose you know someeigenvalues and eigenfunctions; describe the domain S(related problems appear in radar/remote sensing,x-ray/MRI, oil/gas/metal exploration etc).
![Page 13: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/13.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Solving partial differential equations like heat equation∂u(x , t)/∂t = c ·∆xu(x , t) and wave equation∂2u(x , t)/∂t2 = c ·∆xu(x , t).
• Stationary solutions of Schrodinger equation or “purequantum states.”
• Inverse problems: suppose you know someeigenvalues and eigenfunctions; describe the domain S(related problems appear in radar/remote sensing,x-ray/MRI, oil/gas/metal exploration etc).
![Page 14: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/14.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Solving partial differential equations like heat equation∂u(x , t)/∂t = c ·∆xu(x , t) and wave equation∂2u(x , t)/∂t2 = c ·∆xu(x , t).
• Stationary solutions of Schrodinger equation or “purequantum states.”
• Inverse problems: suppose you know someeigenvalues and eigenfunctions; describe the domain S(related problems appear in radar/remote sensing,x-ray/MRI, oil/gas/metal exploration etc).
![Page 15: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/15.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Determine the smallest λ > 0 for a given surface S (its“bass note”), and other small eigenvalues.
• Mark Kac: “Can you hear the shape of a drum?” Canyou determine the domain if you know its spectrum (allthe λi -s)?Answer: No! Two different domains S can have thesame spectrum (sound the same). Example below isdue to Gordon, Webb and Wolpert.
• Count the eigenvalues: N(λ) = #{λj < λ}. Study N(λ)as λ →∞.
• How do eigenvalue differences λi+1 − λi behave whenλ is large?
![Page 16: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/16.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Determine the smallest λ > 0 for a given surface S (its“bass note”), and other small eigenvalues.
• Mark Kac: “Can you hear the shape of a drum?” Canyou determine the domain if you know its spectrum (allthe λi -s)?Answer: No! Two different domains S can have thesame spectrum (sound the same). Example below isdue to Gordon, Webb and Wolpert.
• Count the eigenvalues: N(λ) = #{λj < λ}. Study N(λ)as λ →∞.
• How do eigenvalue differences λi+1 − λi behave whenλ is large?
![Page 17: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/17.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Determine the smallest λ > 0 for a given surface S (its“bass note”), and other small eigenvalues.
• Mark Kac: “Can you hear the shape of a drum?” Canyou determine the domain if you know its spectrum (allthe λi -s)?Answer: No! Two different domains S can have thesame spectrum (sound the same). Example below isdue to Gordon, Webb and Wolpert.
• Count the eigenvalues: N(λ) = #{λj < λ}. Study N(λ)as λ →∞.
• How do eigenvalue differences λi+1 − λi behave whenλ is large?
![Page 18: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/18.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Determine the smallest λ > 0 for a given surface S (its“bass note”), and other small eigenvalues.
• Mark Kac: “Can you hear the shape of a drum?” Canyou determine the domain if you know its spectrum (allthe λi -s)?Answer: No! Two different domains S can have thesame spectrum (sound the same). Example below isdue to Gordon, Webb and Wolpert.
• Count the eigenvalues: N(λ) = #{λj < λ}. Study N(λ)as λ →∞.
• How do eigenvalue differences λi+1 − λi behave whenλ is large?
![Page 19: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/19.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example: 2-torus T2: λm,n = m2 + n2. Let λ = t2. Then
N(t2) = #{(m, n) : m2 + n2 < t2} =
#{(m, n) :√
m2 + n2 < t}.How many lattice points are inside the circle of radiust? Leading term is given by the area:
N(t2) = πt2 + R(t), (1)
where R(t) is the remainder.• Question: How big is R(t)? Conjecture (Hardy): for
any δ > 0,
R(t) < C(δ) · t1/2+δ, as t →∞.
Best known estimate (Huxley, 2003):
R(t) < C · t131/208(log t)2.26.
Note: 131/208 = 0.629807....
![Page 20: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/20.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example: 2-torus T2: λm,n = m2 + n2. Let λ = t2. Then
N(t2) = #{(m, n) : m2 + n2 < t2} =
#{(m, n) :√
m2 + n2 < t}.How many lattice points are inside the circle of radiust? Leading term is given by the area:
N(t2) = πt2 + R(t), (1)
where R(t) is the remainder.• Question: How big is R(t)? Conjecture (Hardy): for
any δ > 0,
R(t) < C(δ) · t1/2+δ, as t →∞.
Best known estimate (Huxley, 2003):
R(t) < C · t131/208(log t)2.26.
Note: 131/208 = 0.629807....
![Page 21: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/21.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• An analogue of (1) holds for very general domains; it iscalled Weyl’s law (Weyl, 1911).M is n-dimensional:
N(λ) = cn · vol(M)λn/2 + R(λ), R is a remainder .
• It is known (Avakumovic, Levitan, Hormander) that
|R(λ)| < Cλ(n−1)/2.
• More detailed study of R(λ) is difficult and interesting; itis related to the properties of the geodesic flow on M.
![Page 22: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/22.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• An analogue of (1) holds for very general domains; it iscalled Weyl’s law (Weyl, 1911).M is n-dimensional:
N(λ) = cn · vol(M)λn/2 + R(λ), R is a remainder .
• It is known (Avakumovic, Levitan, Hormander) that
|R(λ)| < Cλ(n−1)/2.
• More detailed study of R(λ) is difficult and interesting; itis related to the properties of the geodesic flow on M.
![Page 23: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/23.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• An analogue of (1) holds for very general domains; it iscalled Weyl’s law (Weyl, 1911).M is n-dimensional:
N(λ) = cn · vol(M)λn/2 + R(λ), R is a remainder .
• It is known (Avakumovic, Levitan, Hormander) that
|R(λ)| < Cλ(n−1)/2.
• More detailed study of R(λ) is difficult and interesting; itis related to the properties of the geodesic flow on M.
![Page 24: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/24.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Nodal set N (φλ) = {x ∈ M : φλ(x) = 0}, codimension1 is M. On a surface, it’s a union of curves.First pictures: Chladni plates. E. Chladni, 18th century.He put sand on a plate and played with a violin bow tomake it vibrate.
![Page 25: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/25.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Chladni patterns are still used to tune violins.
![Page 26: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/26.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• How large are nodal sets, i.e. how large isvoln−1(N (φλ))?Dimension n = 1: eigenfunction sin(nx) has ∼ n =
√λ
zeros in [0, 2π].• For real-analytic metrics, Donnelly and Fefferman
showed that there exists C1, C2 (independent of λ > 0)s.t.
C1√
λ ≤ voln−1(N (φλ)) ≤ C2√
λ
Similar bounds are conjectured for arbitrary smoothmetrics in all dimensions (Cheng and Yau).
• In general, the best known result in dimension 2 forarbitrary metrics is
C1√
λ ≤ voln−1(N (φλ)) ≤ C2λ3/4.
The lower bound is due to Bruning, the upper bound toDonnelly and Fefferman.
![Page 27: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/27.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• How large are nodal sets, i.e. how large isvoln−1(N (φλ))?Dimension n = 1: eigenfunction sin(nx) has ∼ n =
√λ
zeros in [0, 2π].• For real-analytic metrics, Donnelly and Fefferman
showed that there exists C1, C2 (independent of λ > 0)s.t.
C1√
λ ≤ voln−1(N (φλ)) ≤ C2√
λ
Similar bounds are conjectured for arbitrary smoothmetrics in all dimensions (Cheng and Yau).
• In general, the best known result in dimension 2 forarbitrary metrics is
C1√
λ ≤ voln−1(N (φλ)) ≤ C2λ3/4.
The lower bound is due to Bruning, the upper bound toDonnelly and Fefferman.
![Page 28: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/28.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• How large are nodal sets, i.e. how large isvoln−1(N (φλ))?Dimension n = 1: eigenfunction sin(nx) has ∼ n =
√λ
zeros in [0, 2π].• For real-analytic metrics, Donnelly and Fefferman
showed that there exists C1, C2 (independent of λ > 0)s.t.
C1√
λ ≤ voln−1(N (φλ)) ≤ C2√
λ
Similar bounds are conjectured for arbitrary smoothmetrics in all dimensions (Cheng and Yau).
• In general, the best known result in dimension 2 forarbitrary metrics is
C1√
λ ≤ voln−1(N (φλ)) ≤ C2λ3/4.
The lower bound is due to Bruning, the upper bound toDonnelly and Fefferman.
![Page 29: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/29.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Nodal domain of φ is a connected component ofM \ N (φ). Courant nodal domain theorem. φk has≤ k + 1 nodal domains. The constant in the estimatewas improved by Pleijel.
• Nazarov and Sodin showed that a random sphericalharmonic φn has ∼ cn2 nodal domains.
• Eigenfunctions with large λ can have few nodaldomains: φn(x , y) = sin(nx + y) on T2 has two nodaldomains for all n; λn = n2 + 1.
• John Toth (McGill) and Zelditch recently studiednumber of open nodal lines (intersecting the boundary)in certain planar domains, and got a bound ≤ C · √n.
• Any collection of disjoint closed curves on S2, invariantunder (x , y , z) → (−x ,−y ,−z) is equivalent to a nodalset of a spherical harmonic (Eremenko, J, Nadirashvili)
![Page 30: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/30.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Nodal domain of φ is a connected component ofM \ N (φ). Courant nodal domain theorem. φk has≤ k + 1 nodal domains. The constant in the estimatewas improved by Pleijel.
• Nazarov and Sodin showed that a random sphericalharmonic φn has ∼ cn2 nodal domains.
• Eigenfunctions with large λ can have few nodaldomains: φn(x , y) = sin(nx + y) on T2 has two nodaldomains for all n; λn = n2 + 1.
• John Toth (McGill) and Zelditch recently studiednumber of open nodal lines (intersecting the boundary)in certain planar domains, and got a bound ≤ C · √n.
• Any collection of disjoint closed curves on S2, invariantunder (x , y , z) → (−x ,−y ,−z) is equivalent to a nodalset of a spherical harmonic (Eremenko, J, Nadirashvili)
![Page 31: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/31.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Nodal domain of φ is a connected component ofM \ N (φ). Courant nodal domain theorem. φk has≤ k + 1 nodal domains. The constant in the estimatewas improved by Pleijel.
• Nazarov and Sodin showed that a random sphericalharmonic φn has ∼ cn2 nodal domains.
• Eigenfunctions with large λ can have few nodaldomains: φn(x , y) = sin(nx + y) on T2 has two nodaldomains for all n; λn = n2 + 1.
• John Toth (McGill) and Zelditch recently studiednumber of open nodal lines (intersecting the boundary)in certain planar domains, and got a bound ≤ C · √n.
• Any collection of disjoint closed curves on S2, invariantunder (x , y , z) → (−x ,−y ,−z) is equivalent to a nodalset of a spherical harmonic (Eremenko, J, Nadirashvili)
![Page 32: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/32.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Nodal domain of φ is a connected component ofM \ N (φ). Courant nodal domain theorem. φk has≤ k + 1 nodal domains. The constant in the estimatewas improved by Pleijel.
• Nazarov and Sodin showed that a random sphericalharmonic φn has ∼ cn2 nodal domains.
• Eigenfunctions with large λ can have few nodaldomains: φn(x , y) = sin(nx + y) on T2 has two nodaldomains for all n; λn = n2 + 1.
• John Toth (McGill) and Zelditch recently studiednumber of open nodal lines (intersecting the boundary)in certain planar domains, and got a bound ≤ C · √n.
• Any collection of disjoint closed curves on S2, invariantunder (x , y , z) → (−x ,−y ,−z) is equivalent to a nodalset of a spherical harmonic (Eremenko, J, Nadirashvili)
![Page 33: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/33.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Nodal domain of φ is a connected component ofM \ N (φ). Courant nodal domain theorem. φk has≤ k + 1 nodal domains. The constant in the estimatewas improved by Pleijel.
• Nazarov and Sodin showed that a random sphericalharmonic φn has ∼ cn2 nodal domains.
• Eigenfunctions with large λ can have few nodaldomains: φn(x , y) = sin(nx + y) on T2 has two nodaldomains for all n; λn = n2 + 1.
• John Toth (McGill) and Zelditch recently studiednumber of open nodal lines (intersecting the boundary)in certain planar domains, and got a bound ≤ C · √n.
• Any collection of disjoint closed curves on S2, invariantunder (x , y , z) → (−x ,−y ,−z) is equivalent to a nodalset of a spherical harmonic (Eremenko, J, Nadirashvili)
![Page 34: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/34.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question (Yau): Does the number of critical points(maxima, minima and saddle points) of φλ always growas λ →∞?
• Answer (Jakobson, Nadirashvili): Not always! Forexample, on T2 with a “metric of revolution”
(100 + cos(4x))(dx2 + dy2)
there exists a sequence φi such that λi →∞, and eachφi has exactly 16 critical points.
• It is not known if the number of critical points growsgenerically.
![Page 35: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/35.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question (Yau): Does the number of critical points(maxima, minima and saddle points) of φλ always growas λ →∞?
• Answer (Jakobson, Nadirashvili): Not always! Forexample, on T2 with a “metric of revolution”
(100 + cos(4x))(dx2 + dy2)
there exists a sequence φi such that λi →∞, and eachφi has exactly 16 critical points.
• It is not known if the number of critical points growsgenerically.
![Page 36: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/36.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question (Yau): Does the number of critical points(maxima, minima and saddle points) of φλ always growas λ →∞?
• Answer (Jakobson, Nadirashvili): Not always! Forexample, on T2 with a “metric of revolution”
(100 + cos(4x))(dx2 + dy2)
there exists a sequence φi such that λi →∞, and eachφi has exactly 16 critical points.
• It is not known if the number of critical points growsgenerically.
![Page 37: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/37.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Let∆φ + λφ = 0, λ-large (high energy). Correspondenceprinciple (Niels Bohr) predicts that at high energies,certain properties of eigenvalues and eigenfunctions of∆ on M (quantum system) would depend on thedynamics of the geodesic flow on M (classical system).
• Geodesic is a curve that locally minimizes distancebetween points lying on it. Examples: straight lines inRn, great circles on Sn (that’s how planes fly on S2).
• Geodesic flow Gt is defined as follows: letx ∈ M, v ∈ TxM, g(v , v) = 1. Consider a uniquegeodesic γv (t) s.t. γ(0) = x , γ′(0) = v . Then
Gt(v) := γ′v (t)
Light travels along geodesics.
![Page 38: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/38.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Let∆φ + λφ = 0, λ-large (high energy). Correspondenceprinciple (Niels Bohr) predicts that at high energies,certain properties of eigenvalues and eigenfunctions of∆ on M (quantum system) would depend on thedynamics of the geodesic flow on M (classical system).
• Geodesic is a curve that locally minimizes distancebetween points lying on it. Examples: straight lines inRn, great circles on Sn (that’s how planes fly on S2).
• Geodesic flow Gt is defined as follows: letx ∈ M, v ∈ TxM, g(v , v) = 1. Consider a uniquegeodesic γv (t) s.t. γ(0) = x , γ′(0) = v . Then
Gt(v) := γ′v (t)
Light travels along geodesics.
![Page 39: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/39.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Let∆φ + λφ = 0, λ-large (high energy). Correspondenceprinciple (Niels Bohr) predicts that at high energies,certain properties of eigenvalues and eigenfunctions of∆ on M (quantum system) would depend on thedynamics of the geodesic flow on M (classical system).
• Geodesic is a curve that locally minimizes distancebetween points lying on it. Examples: straight lines inRn, great circles on Sn (that’s how planes fly on S2).
• Geodesic flow Gt is defined as follows: letx ∈ M, v ∈ TxM, g(v , v) = 1. Consider a uniquegeodesic γv (t) s.t. γ(0) = x , γ′(0) = v . Then
Gt(v) := γ′v (t)
Light travels along geodesics.
![Page 40: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/40.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: how Gt behaves for t →∞: do closetrajectories converge (focusing), or diverge(de-focusing)?
• Focusing: Sn. All geodesics (meridians) starting at thenorth pole at t = 0 focus at the south pole at t = π (Nand S are called conjugate points).
• De-focusing: Hn, the n-dimensional hyperbolic space.Geodesics diverge exponentially fast; no conjugatepoints.
![Page 41: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/41.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: how Gt behaves for t →∞: do closetrajectories converge (focusing), or diverge(de-focusing)?
• Focusing: Sn. All geodesics (meridians) starting at thenorth pole at t = 0 focus at the south pole at t = π (Nand S are called conjugate points).
• De-focusing: Hn, the n-dimensional hyperbolic space.Geodesics diverge exponentially fast; no conjugatepoints.
![Page 42: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/42.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: how Gt behaves for t →∞: do closetrajectories converge (focusing), or diverge(de-focusing)?
• Focusing: Sn. All geodesics (meridians) starting at thenorth pole at t = 0 focus at the south pole at t = π (Nand S are called conjugate points).
• De-focusing: Hn, the n-dimensional hyperbolic space.Geodesics diverge exponentially fast; no conjugatepoints.
![Page 43: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/43.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Curvature: S surface in R3, given by z = f (x , y). Letgradf (p) = 0, then K = det(∂2f/∂x∂y). If K > 0, then Sis convex or concave at p; if K < 0, then S looks like asaddle at p. Also,vol(BS(x0, r)) = vol(BR2(r))
[1− K (x0)r2
12 + O(r4)].
• Positive curvature ⇒ focusing; K = +1 on S2.• Examples of regular geodesic flows: flat torus (move
along straight lines); and surfaces of revolution. Theflow on a 2-dimensional surface has 2 first integrals.Such flows are called integrable.
• Negative curvature ⇒ de-focusing; K = −1 on H2. If allsectional curvatures on M are negative, then geodesicflow on M is ergodic: all invariant sets have measure 0or 1, almost every geodesic γ(t) becomes uniformlydistributed (on the unit sphere bundle in TM) as t →∞.
![Page 44: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/44.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Curvature: S surface in R3, given by z = f (x , y). Letgradf (p) = 0, then K = det(∂2f/∂x∂y). If K > 0, then Sis convex or concave at p; if K < 0, then S looks like asaddle at p. Also,vol(BS(x0, r)) = vol(BR2(r))
[1− K (x0)r2
12 + O(r4)].
• Positive curvature ⇒ focusing; K = +1 on S2.• Examples of regular geodesic flows: flat torus (move
along straight lines); and surfaces of revolution. Theflow on a 2-dimensional surface has 2 first integrals.Such flows are called integrable.
• Negative curvature ⇒ de-focusing; K = −1 on H2. If allsectional curvatures on M are negative, then geodesicflow on M is ergodic: all invariant sets have measure 0or 1, almost every geodesic γ(t) becomes uniformlydistributed (on the unit sphere bundle in TM) as t →∞.
![Page 45: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/45.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Curvature: S surface in R3, given by z = f (x , y). Letgradf (p) = 0, then K = det(∂2f/∂x∂y). If K > 0, then Sis convex or concave at p; if K < 0, then S looks like asaddle at p. Also,vol(BS(x0, r)) = vol(BR2(r))
[1− K (x0)r2
12 + O(r4)].
• Positive curvature ⇒ focusing; K = +1 on S2.• Examples of regular geodesic flows: flat torus (move
along straight lines); and surfaces of revolution. Theflow on a 2-dimensional surface has 2 first integrals.Such flows are called integrable.
• Negative curvature ⇒ de-focusing; K = −1 on H2. If allsectional curvatures on M are negative, then geodesicflow on M is ergodic: all invariant sets have measure 0or 1, almost every geodesic γ(t) becomes uniformlydistributed (on the unit sphere bundle in TM) as t →∞.
![Page 46: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/46.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Curvature: S surface in R3, given by z = f (x , y). Letgradf (p) = 0, then K = det(∂2f/∂x∂y). If K > 0, then Sis convex or concave at p; if K < 0, then S looks like asaddle at p. Also,vol(BS(x0, r)) = vol(BR2(r))
[1− K (x0)r2
12 + O(r4)].
• Positive curvature ⇒ focusing; K = +1 on S2.• Examples of regular geodesic flows: flat torus (move
along straight lines); and surfaces of revolution. Theflow on a 2-dimensional surface has 2 first integrals.Such flows are called integrable.
• Negative curvature ⇒ de-focusing; K = −1 on H2. If allsectional curvatures on M are negative, then geodesicflow on M is ergodic: all invariant sets have measure 0or 1, almost every geodesic γ(t) becomes uniformlydistributed (on the unit sphere bundle in TM) as t →∞.
![Page 47: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/47.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Planar billiards, billiard map: angle of incidenceequals angle of reflection.
• Regular billiard map: billiard in an ellipse. Orbits stayforever tangent to confocal ellipses (or confocalhyperbolas); such curves are called caustics.
• If the bounding curve is smooth enough and strictlyconvex (curvature never vanishes), then causticsalways exist (Lazutkin).
![Page 48: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/48.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Planar billiards, billiard map: angle of incidenceequals angle of reflection.
• Regular billiard map: billiard in an ellipse. Orbits stayforever tangent to confocal ellipses (or confocalhyperbolas); such curves are called caustics.
• If the bounding curve is smooth enough and strictlyconvex (curvature never vanishes), then causticsalways exist (Lazutkin).
![Page 49: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/49.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Planar billiards, billiard map: angle of incidenceequals angle of reflection.
• Regular billiard map: billiard in an ellipse. Orbits stayforever tangent to confocal ellipses (or confocalhyperbolas); such curves are called caustics.
• If the bounding curve is smooth enough and strictlyconvex (curvature never vanishes), then causticsalways exist (Lazutkin).
![Page 50: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/50.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Ergodic planar billiards: Sinai billiard and Bunimovichstadium
![Page 51: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/51.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Other examples of ergodic and integrable billiards:cardioid and circular billiards
![Page 52: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/52.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: Where do eigenfunctions concentrate?• Answer: “Quantum ergodicity” theorem (Shnirelman,
Zelditch, Colin de Verdiere): If the geodesic flow iserogdic (“almost all” trajectories become uniformlydistributed), then “almost all” eigenfunctions becomeuniformly distributed.
• Billiard version: Gerard-Leichtnam, Zelditch-Zworski.• What is the precise meaning? Eigenfunction φλ
describes a quantum particle; |φ|2 - probability densityof that particle. Let A ⊂ M; then
∫A |φ|2 - probability of
finding the particle in A. For almost all φλ,
limλ→∞
∫A |φλ|2∫M |φλ|2
=vol(A)
vol(M)
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: Where do eigenfunctions concentrate?• Answer: “Quantum ergodicity” theorem (Shnirelman,
Zelditch, Colin de Verdiere): If the geodesic flow iserogdic (“almost all” trajectories become uniformlydistributed), then “almost all” eigenfunctions becomeuniformly distributed.
• Billiard version: Gerard-Leichtnam, Zelditch-Zworski.• What is the precise meaning? Eigenfunction φλ
describes a quantum particle; |φ|2 - probability densityof that particle. Let A ⊂ M; then
∫A |φ|2 - probability of
finding the particle in A. For almost all φλ,
limλ→∞
∫A |φλ|2∫M |φλ|2
=vol(A)
vol(M)
![Page 54: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/54.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: Where do eigenfunctions concentrate?• Answer: “Quantum ergodicity” theorem (Shnirelman,
Zelditch, Colin de Verdiere): If the geodesic flow iserogdic (“almost all” trajectories become uniformlydistributed), then “almost all” eigenfunctions becomeuniformly distributed.
• Billiard version: Gerard-Leichtnam, Zelditch-Zworski.• What is the precise meaning? Eigenfunction φλ
describes a quantum particle; |φ|2 - probability densityof that particle. Let A ⊂ M; then
∫A |φ|2 - probability of
finding the particle in A. For almost all φλ,
limλ→∞
∫A |φλ|2∫M |φλ|2
=vol(A)
vol(M)
![Page 55: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/55.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Question: Where do eigenfunctions concentrate?• Answer: “Quantum ergodicity” theorem (Shnirelman,
Zelditch, Colin de Verdiere): If the geodesic flow iserogdic (“almost all” trajectories become uniformlydistributed), then “almost all” eigenfunctions becomeuniformly distributed.
• Billiard version: Gerard-Leichtnam, Zelditch-Zworski.• What is the precise meaning? Eigenfunction φλ
describes a quantum particle; |φ|2 - probability densityof that particle. Let A ⊂ M; then
∫A |φ|2 - probability of
finding the particle in A. For almost all φλ,
limλ→∞
∫A |φλ|2∫M |φλ|2
=vol(A)
vol(M)
![Page 56: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/56.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• QE for Dirac operator, Laplacian on forms: Jakobson,Strohmaier, Zelditch
• QE for restrictions of eigenfunctions to submanifolds (orto the boundary in billiards): Toth, Zelditch
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• QE for Dirac operator, Laplacian on forms: Jakobson,Strohmaier, Zelditch
• QE for restrictions of eigenfunctions to submanifolds (orto the boundary in billiards): Toth, Zelditch
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example: M = S1 ∼= [0, 2π], φn(x) = 1√π
sin(nx). Letf (x) be an “observable.” Then
∫ 2π
0f (x)(φn(x))2 dx =
12π
∫ 2π
0f (x)(1− cos(2nx)) dx
which by Riemann-Lebesgue lemma converges to
12π
∫ 2π
0f (x)dx
as n →∞.• True for all eigenfunctions (quantum unique ergodicity
or QUE). Does QUE hold on any other manifolds?• Conjecture: QUE should hold on compact
negatively-curved manifolds (Rudnick-Sarnak). Provedfor arithmetic hyperbolic manifolds (Lindenstrauss).
![Page 59: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/59.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example: M = S1 ∼= [0, 2π], φn(x) = 1√π
sin(nx). Letf (x) be an “observable.” Then
∫ 2π
0f (x)(φn(x))2 dx =
12π
∫ 2π
0f (x)(1− cos(2nx)) dx
which by Riemann-Lebesgue lemma converges to
12π
∫ 2π
0f (x)dx
as n →∞.• True for all eigenfunctions (quantum unique ergodicity
or QUE). Does QUE hold on any other manifolds?• Conjecture: QUE should hold on compact
negatively-curved manifolds (Rudnick-Sarnak). Provedfor arithmetic hyperbolic manifolds (Lindenstrauss).
![Page 60: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/60.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Example: M = S1 ∼= [0, 2π], φn(x) = 1√π
sin(nx). Letf (x) be an “observable.” Then
∫ 2π
0f (x)(φn(x))2 dx =
12π
∫ 2π
0f (x)(1− cos(2nx)) dx
which by Riemann-Lebesgue lemma converges to
12π
∫ 2π
0f (x)dx
as n →∞.• True for all eigenfunctions (quantum unique ergodicity
or QUE). Does QUE hold on any other manifolds?• Conjecture: QUE should hold on compact
negatively-curved manifolds (Rudnick-Sarnak). Provedfor arithmetic hyperbolic manifolds (Lindenstrauss).
![Page 61: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/61.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
Eigenfunctions of the hyperbolic Laplacian on H2/PSL(2, Z),Hejhal:
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Eigenfunctions on S2 = {x2 + y2 + z2 = 1}. Letφn(x , y , z) = (x + iy)n. Then |φn|2 = (1− z2)n. Thatexpression is = 1 on the equator {z = 0}, and decaysexponentially fast for z > 0 as n →∞. Therefore,φ2
n → δequator as n →∞.• In fact, for any even nonnegative function f on S2, there
exists a sequence of spherical harmonics φn s.t. φ2n → f
as n →∞ (Jakobson, Zelditch).
![Page 63: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/63.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Eigenfunctions on S2 = {x2 + y2 + z2 = 1}. Letφn(x , y , z) = (x + iy)n. Then |φn|2 = (1− z2)n. Thatexpression is = 1 on the equator {z = 0}, and decaysexponentially fast for z > 0 as n →∞. Therefore,φ2
n → δequator as n →∞.• In fact, for any even nonnegative function f on S2, there
exists a sequence of spherical harmonics φn s.t. φ2n → f
as n →∞ (Jakobson, Zelditch).
![Page 64: Spectral theory, geometry and dynamical systems · Applications Eigenvalue questions Counting eigenvalues Nodal sets and domains Critical points ... j2dxdy < 1), can be expanded into](https://reader034.vdocuments.net/reader034/viewer/2022050118/5f4e7a5eb6f9633f2c3bd1ab/html5/thumbnails/64.jpg)
SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Billiards: QUE conjectures does not hold for theBunimovich stadium (Hassell). Ergodic eigenfunction:
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Other stadium eigenfunctions, including “bouncing ball”eigenfunctions, for which QUE fails (they have density 0among all eigenfunctions, so QE still holds).
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Ergodic eigenfunction on a cardioid billiard.
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
• Billiards with caustics: there exist eigenfunctions thatconcentrate in the region between the caustic and theboundary (“whispering gallery”) eigenfunctions.Example: circular billiard.
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SpectralTheory ofLaplacian
Applications
Eigenvaluequestions
Countingeigenvalues
Nodal setsand domains
Critical points
Geodesic flowand curvature
Billiards
Limits ofeigenfunctions
Billiardeigenfunctions
Another eigenfunction for a circular billiard.