counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the...
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![Page 1: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/1.jpg)
Counting stationary modes: a discrete view of geometry anddynamics
Stephane Nonnenmacher
Institut de Physique Theorique, CEA Saclay
September 20th, 2012Weyl Law at 100, Fields Institute
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Outline
• A (sketchy) history of Weyl’s law, from 19th century physics to V.Ivrii’sproof of the 2-term asymptotics
• resonances of open wave systems : counting them all, vs. selecting onlythe long-living ones
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An urgent mathematical challenge from theoretical physics
In October 1910, Hendrik Lorentz delivered lectures inGottingen, Old and new problems of physics. He mentionedan “urgent question” related with the black body radiationproblem :
Prove that the density of standing electromagneticwaves inside a bounded cavity Ω ⊂ R3 is, at highfrequency, independent of the shape of Ω.
A similar conjecture had been expressed a month earlier byArnold Sommerfeld, for scalar waves. In this case, theproblem boils down to counting the solutions of the Helmholtzequation inside a bounded domain Ω ⊂ Rd :
(∆ + λ2n)un = 0, un∂Ω = 0 (Dirichlet b.c.)
Our central object : the counting function N(λ)def= #0 ≤ λn ≤ λ.
In 1910, N(λ) could be computed only for simple, separable domains(rectangle, disk, ball..) : one gets in the high-frequeny limit (λ→∞)
N(λ) ∼ |Ω|4π
λ2 (2− dim.), N(λ) ∼ |Ω|6π2 λ
3 (3− dim.)
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An efficient postdoc
Hermann Weyl (a fresh PhD) was attending Lorentz’slectures. A few months later he had proved the 2-dimensionalscalar case,
N(λ) =|Ω|4π
λ2 + o(λ2), λ→∞,
which was presented by D.Hilbert in front of the RoyalAcademy of Sciences.
Within a couple of years, Weyl had generalized his result in various ways : 3dimensions, electromagnetic waves, elasticity waves.
In 1913 he conjectured (based on the case of the rectangle) a 2-termasymptotics, depending on the boundary conditions :
N(d)D/N(λ) =
ωd |Ω|(2π)d λ
d ∓ ωd−1|∂Ω|4(2π)d−1 λ
d−1 + o(λd−1)
Then he switched to other topics (general relativity, gauge theory etc.)
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Why were (prominent) physicists so interested in this question ?The black body radiation problem had puzzled physicistsfor several decades [KIRCHHOFF’1859].At thermal equilibrium, a black body emits EM waves with aspectral distribution ρ(λ,T ), which depends on the densityD(λ) = dN(λ)
dλ of stationary waves inside Ω.
Around 1900, all physicists took for granted that theasymptotics for D(λ) was independent of the shape.Theywere confronting a more annoying puzzle : Ultravioletcatastrophe
Equipartition of energy =⇒ ρ(λ,T ) ∝ D(λ)T ∝ |Ω|λ2T=⇒ the full emitted power P(T ) =
R∞0 ρ(λ,T ) dλ is infinite as
soon as T > 0 !
Several heuristic laws were proposed (Wien, Rayleigh,Jeans. . . ). Finally, Planck’s ’1900 guess ρ(λ,T ) ∝ λ3
ehλ/T−1gave a good experimental fit, and initiated quantummechanics.
Once the “catastrophe” was put aside, it was time to put theassumption D(λ) ∝ |Ω|λ2 on rigorous grounds.
![Page 6: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/6.jpg)
Why were (prominent) physicists so interested in this question ?The black body radiation problem had puzzled physicistsfor several decades [KIRCHHOFF’1859].At thermal equilibrium, a black body emits EM waves with aspectral distribution ρ(λ,T ), which depends on the densityD(λ) = dN(λ)
dλ of stationary waves inside Ω.
Around 1900, all physicists took for granted that theasymptotics for D(λ) was independent of the shape.Theywere confronting a more annoying puzzle : Ultravioletcatastrophe
Equipartition of energy =⇒ ρ(λ,T ) ∝ D(λ)T ∝ |Ω|λ2T=⇒ the full emitted power P(T ) =
R∞0 ρ(λ,T ) dλ is infinite as
soon as T > 0 !
Several heuristic laws were proposed (Wien, Rayleigh,Jeans. . . ). Finally, Planck’s ’1900 guess ρ(λ,T ) ∝ λ3
ehλ/T−1gave a good experimental fit, and initiated quantummechanics.
Once the “catastrophe” was put aside, it was time to put theassumption D(λ) ∝ |Ω|λ2 on rigorous grounds.
![Page 7: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/7.jpg)
Why were (prominent) physicists so interested in this question ?The black body radiation problem had puzzled physicistsfor several decades [KIRCHHOFF’1859].At thermal equilibrium, a black body emits EM waves with aspectral distribution ρ(λ,T ), which depends on the densityD(λ) = dN(λ)
dλ of stationary waves inside Ω.
Around 1900, all physicists took for granted that theasymptotics for D(λ) was independent of the shape.Theywere confronting a more annoying puzzle : Ultravioletcatastrophe
Equipartition of energy =⇒ ρ(λ,T ) ∝ D(λ)T ∝ |Ω|λ2T=⇒ the full emitted power P(T ) =
R∞0 ρ(λ,T ) dλ is infinite as
soon as T > 0 !
Several heuristic laws were proposed (Wien, Rayleigh,Jeans. . . ). Finally, Planck’s ’1900 guess ρ(λ,T ) ∝ λ3
ehλ/T−1gave a good experimental fit, and initiated quantummechanics.
Once the “catastrophe” was put aside, it was time to put theassumption D(λ) ∝ |Ω|λ2 on rigorous grounds.
![Page 8: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/8.jpg)
Why were (prominent) physicists so interested in this question ?The black body radiation problem had puzzled physicistsfor several decades [KIRCHHOFF’1859].At thermal equilibrium, a black body emits EM waves with aspectral distribution ρ(λ,T ), which depends on the densityD(λ) = dN(λ)
dλ of stationary waves inside Ω.
Around 1900, all physicists took for granted that theasymptotics for D(λ) was independent of the shape.Theywere confronting a more annoying puzzle : Ultravioletcatastrophe
Equipartition of energy =⇒ ρ(λ,T ) ∝ D(λ)T ∝ |Ω|λ2T=⇒ the full emitted power P(T ) =
R∞0 ρ(λ,T ) dλ is infinite as
soon as T > 0 !
Several heuristic laws were proposed (Wien, Rayleigh,Jeans. . . ). Finally, Planck’s ’1900 guess ρ(λ,T ) ∝ λ3
ehλ/T−1gave a good experimental fit, and initiated quantummechanics.
Once the “catastrophe” was put aside, it was time to put theassumption D(λ) ∝ |Ω|λ2 on rigorous grounds.
![Page 9: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/9.jpg)
Why were (prominent) physicists so interested in this question ?The black body radiation problem had puzzled physicistsfor several decades [KIRCHHOFF’1859].At thermal equilibrium, a black body emits EM waves with aspectral distribution ρ(λ,T ), which depends on the densityD(λ) = dN(λ)
dλ of stationary waves inside Ω.
Around 1900, all physicists took for granted that theasymptotics for D(λ) was independent of the shape.Theywere confronting a more annoying puzzle : Ultravioletcatastrophe
Equipartition of energy =⇒ ρ(λ,T ) ∝ D(λ)T ∝ |Ω|λ2T=⇒ the full emitted power P(T ) =
R∞0 ρ(λ,T ) dλ is infinite as
soon as T > 0 !
Several heuristic laws were proposed (Wien, Rayleigh,Jeans. . . ). Finally, Planck’s ’1900 guess ρ(λ,T ) ∝ λ3
ehλ/T−1gave a good experimental fit, and initiated quantummechanics.
Once the “catastrophe” was put aside, it was time to put theassumption D(λ) ∝ |Ω|λ2 on rigorous grounds.
![Page 10: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/10.jpg)
Rectangular cavities
2
1 !/"
L !/"
L
The (Dirichlet) stationary modes of strings or rectangles are explicit :
string : un(x) = sin(πxn/L), λ =πnL, n ≥ 1 =⇒ ND(λ) = [λL/π]
rectangle : un1,n2 (x , y) = sin(πxn1/L1) sin(πyn2/L2), λ = π
r“n1
L1
”2+“n2
L2
”2,
Gauss’s lattice point problem in a 1/4-ellipse. Leads to the 2-termasymptotics, in any dimension d ≥ 2 :
ND/N(λ) =L1L2
4πλ2∓2(L1 + L2)
4πλ+ o(λ) (2− dim) .
NB : even in this case, estimating the remainder is a difficult task.
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What if Ω is not separable ?
If the domain Ω doesn’t allow separation of variables, the eigenmodes/valuesare not known explicitly (true PDE problem).Weyl’s achievement : nevertheless obtain global information on the spectrum,like the asymptotics of N(λ).
Weyl used the result for rectangles + a variationalmethod, consequence of the minimax principle :Dirichlet-Neumann bracketing.Pave Ω with (small) rectangles. Then,X
N,D(λ) ≤ NΩ(λ) ≤X+
N,N(λ)
Refine the paving when λ→∞
ND(λ) =|Ω|4π
λ2 + o(λ2)
[COURANT’1924] : this variational method can beimproved to give a remainder O(λ logλ), but no better.
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What if Ω is not separable ?
If the domain Ω doesn’t allow separation of variables, the eigenmodes/valuesare not known explicitly (true PDE problem).Weyl’s achievement : nevertheless obtain global information on the spectrum,like the asymptotics of N(λ).
Weyl used the result for rectangles + a variationalmethod, consequence of the minimax principle :Dirichlet-Neumann bracketing.Pave Ω with (small) rectangles. Then,X
N,D(λ) ≤ NΩ(λ) ≤X+
N,N(λ)
Refine the paving when λ→∞
ND(λ) =|Ω|4π
λ2 + o(λ2)
[COURANT’1924] : this variational method can beimproved to give a remainder O(λ logλ), but no better.
![Page 13: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/13.jpg)
What if Ω is not separable ?
If the domain Ω doesn’t allow separation of variables, the eigenmodes/valuesare not known explicitly (true PDE problem).Weyl’s achievement : nevertheless obtain global information on the spectrum,like the asymptotics of N(λ).
Weyl used the result for rectangles + a variationalmethod, consequence of the minimax principle :Dirichlet-Neumann bracketing.Pave Ω with (small) rectangles. Then,X
N,D(λ) ≤ NΩ(λ) ≤X+
N,N(λ)
Refine the paving when λ→∞
ND(λ) =|Ω|4π
λ2 + o(λ2)
[COURANT’1924] : this variational method can beimproved to give a remainder O(λ logλ), but no better.
![Page 14: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/14.jpg)
What to do next ?
• Estimate the remainder R(λ) = N(λ)− ωd |Ω|(2π)d λ
d . Possibly, prove Weyl’s2-term asymptotics
• generalize the counting problem to other settings• Laplace-Beltrami op. (or more general elliptic op.) on a compact Riemannian
manifold (with or without boundary).• Quantum mechanics : Schrodinger op. H~ = −~2∆
2m + V (x), with Planck’sconstant ~ ≈ 10−34J.s.
Semiclassical analysis : in the semiclassical limit ~→ 0, deduce properties
of H~ from those of the classical Hamiltonian H(x , ξ) = |ξ|22m + V (x) and the
flow Φt it generates on the phase space T∗Rd .
Weyl’s law : count the eigenvalues of H~ in a fixed interval [E1,E2], as~→ 0 :
N~([E1,E2]) =1
(2π~)dVol
n(x , ξ) ∈ T∗Rd , H(x , ξ) ∈ [E1,E2]
o+ o(~−d )
Each quantum state occupies a volume ≈ (2π~)d in phase space.
H~ = −~2∆2 : back to the geometric setting, Φt = geodesic flow, λ ∼ ~−1.
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What to do next ?
• Estimate the remainder R(λ) = N(λ)− ωd |Ω|(2π)d λ
d . Possibly, prove Weyl’s2-term asymptotics
• generalize the counting problem to other settings
• Laplace-Beltrami op. (or more general elliptic op.) on a compact Riemannianmanifold (with or without boundary).
• Quantum mechanics : Schrodinger op. H~ = −~2∆2m + V (x), with Planck’s
constant ~ ≈ 10−34J.s.
Semiclassical analysis : in the semiclassical limit ~→ 0, deduce properties
of H~ from those of the classical Hamiltonian H(x , ξ) = |ξ|22m + V (x) and the
flow Φt it generates on the phase space T∗Rd .
Weyl’s law : count the eigenvalues of H~ in a fixed interval [E1,E2], as~→ 0 :
N~([E1,E2]) =1
(2π~)dVol
n(x , ξ) ∈ T∗Rd , H(x , ξ) ∈ [E1,E2]
o+ o(~−d )
Each quantum state occupies a volume ≈ (2π~)d in phase space.
H~ = −~2∆2 : back to the geometric setting, Φt = geodesic flow, λ ∼ ~−1.
![Page 16: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/16.jpg)
What to do next ?
• Estimate the remainder R(λ) = N(λ)− ωd |Ω|(2π)d λ
d . Possibly, prove Weyl’s2-term asymptotics
• generalize the counting problem to other settings• Laplace-Beltrami op. (or more general elliptic op.) on a compact Riemannian
manifold (with or without boundary).
• Quantum mechanics : Schrodinger op. H~ = −~2∆2m + V (x), with Planck’s
constant ~ ≈ 10−34J.s.
Semiclassical analysis : in the semiclassical limit ~→ 0, deduce properties
of H~ from those of the classical Hamiltonian H(x , ξ) = |ξ|22m + V (x) and the
flow Φt it generates on the phase space T∗Rd .
Weyl’s law : count the eigenvalues of H~ in a fixed interval [E1,E2], as~→ 0 :
N~([E1,E2]) =1
(2π~)dVol
n(x , ξ) ∈ T∗Rd , H(x , ξ) ∈ [E1,E2]
o+ o(~−d )
Each quantum state occupies a volume ≈ (2π~)d in phase space.
H~ = −~2∆2 : back to the geometric setting, Φt = geodesic flow, λ ∼ ~−1.
![Page 17: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/17.jpg)
What to do next ?
• Estimate the remainder R(λ) = N(λ)− ωd |Ω|(2π)d λ
d . Possibly, prove Weyl’s2-term asymptotics
• generalize the counting problem to other settings• Laplace-Beltrami op. (or more general elliptic op.) on a compact Riemannian
manifold (with or without boundary).• Quantum mechanics : Schrodinger op. H~ = −~2∆
2m + V (x), with Planck’sconstant ~ ≈ 10−34J.s.
Semiclassical analysis : in the semiclassical limit ~→ 0, deduce properties
of H~ from those of the classical Hamiltonian H(x , ξ) = |ξ|22m + V (x) and the
flow Φt it generates on the phase space T∗Rd .
Weyl’s law : count the eigenvalues of H~ in a fixed interval [E1,E2], as~→ 0 :
N~([E1,E2]) =1
(2π~)dVol
n(x , ξ) ∈ T∗Rd , H(x , ξ) ∈ [E1,E2]
o+ o(~−d )
Each quantum state occupies a volume ≈ (2π~)d in phase space.
H~ = −~2∆2 : back to the geometric setting, Φt = geodesic flow, λ ∼ ~−1.
![Page 18: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/18.jpg)
What to do next ?
• Estimate the remainder R(λ) = N(λ)− ωd |Ω|(2π)d λ
d . Possibly, prove Weyl’s2-term asymptotics
• generalize the counting problem to other settings• Laplace-Beltrami op. (or more general elliptic op.) on a compact Riemannian
manifold (with or without boundary).• Quantum mechanics : Schrodinger op. H~ = −~2∆
2m + V (x), with Planck’sconstant ~ ≈ 10−34J.s.
Semiclassical analysis : in the semiclassical limit ~→ 0, deduce properties
of H~ from those of the classical Hamiltonian H(x , ξ) = |ξ|22m + V (x) and the
flow Φt it generates on the phase space T∗Rd .
Weyl’s law : count the eigenvalues of H~ in a fixed interval [E1,E2], as~→ 0 :
N~([E1,E2]) =1
(2π~)dVol
n(x , ξ) ∈ T∗Rd , H(x , ξ) ∈ [E1,E2]
o+ o(~−d )
Each quantum state occupies a volume ≈ (2π~)d in phase space.
H~ = −~2∆2 : back to the geometric setting, Φt = geodesic flow, λ ∼ ~−1.
![Page 19: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/19.jpg)
What to do next ?
• Estimate the remainder R(λ) = N(λ)− ωd |Ω|(2π)d λ
d . Possibly, prove Weyl’s2-term asymptotics
• generalize the counting problem to other settings• Laplace-Beltrami op. (or more general elliptic op.) on a compact Riemannian
manifold (with or without boundary).• Quantum mechanics : Schrodinger op. H~ = −~2∆
2m + V (x), with Planck’sconstant ~ ≈ 10−34J.s.
Semiclassical analysis : in the semiclassical limit ~→ 0, deduce properties
of H~ from those of the classical Hamiltonian H(x , ξ) = |ξ|22m + V (x) and the
flow Φt it generates on the phase space T∗Rd .
Weyl’s law : count the eigenvalues of H~ in a fixed interval [E1,E2], as~→ 0 :
N~([E1,E2]) =1
(2π~)dVol
n(x , ξ) ∈ T∗Rd , H(x , ξ) ∈ [E1,E2]
o+ o(~−d )
Each quantum state occupies a volume ≈ (2π~)d in phase space.
H~ = −~2∆2 : back to the geometric setting, Φt = geodesic flow, λ ∼ ~−1.
![Page 20: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/20.jpg)
Alternative to variational method : mollifying N(λ)The spectral density can be expressed as a trace :
N(λ) = Tr Θ(λ−√−∆) .
Easier to analyze operators given by smooth functions of ∆ or√−∆
• resolvent (z + ∆)−1 defined for z ∈ C \ R [CARLEMAN’34]
• heat semigroup et∆. Heat kernel et∆(x , y) = diffusion of a Brownianparticle. Tr et∆ =
Pn e−tλ2
n is a smoothing of N(λ), with λ ∼ t−1/2
[MINAKSHISUNDARAM-PLEIJEL’52]
• Wave group e−it√−∆, solves the wave equation. Propagates at unit
speed.
Once one has a good control on the trace of either of these operators, getestimates on N(λ) through some Tauberian theorem.
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Using the wave equation - X without boundaryThe wave group provides the most precise estimates for R(λ) (Fouriertransform is easily inverted)Uncertainty principle : control e−it
√−∆ on a time scale |t | ≤ T ⇐⇒ control
D(λ) smoothed on a scale δλ ∼ 1T .
• [LEVITAN’52, AVAKUMOVIC’56,HORMANDER’68] : Parametrix of e−it√−∆ for
|t | ≤ T0 R(λ) = O(λd−1).Optimal for X = Sd due to large degeneracies.
• [CHAZARAIN’73,DUISTERMAAT-GUILLEMIN’75] Tr e−it√−∆ has singularities at
t = Tγ the lengths of closed geodesics R(λ) = o(λd−1), provided theset of periodic points has measure zero.
![Page 22: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/22.jpg)
Using the wave equation - X without boundaryThe wave group provides the most precise estimates for R(λ) (Fouriertransform is easily inverted)Uncertainty principle : control e−it
√−∆ on a time scale |t | ≤ T ⇐⇒ control
D(λ) smoothed on a scale δλ ∼ 1T .
• [LEVITAN’52, AVAKUMOVIC’56,HORMANDER’68] : Parametrix of e−it√−∆ for
|t | ≤ T0 R(λ) = O(λd−1).Optimal for X = Sd due to large degeneracies.
• [CHAZARAIN’73,DUISTERMAAT-GUILLEMIN’75] Tr e−it√−∆ has singularities at
t = Tγ the lengths of closed geodesics R(λ) = o(λd−1), provided theset of periodic points has measure zero.
![Page 23: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/23.jpg)
Using the wave equation - X without boundaryThe wave group provides the most precise estimates for R(λ) (Fouriertransform is easily inverted)Uncertainty principle : control e−it
√−∆ on a time scale |t | ≤ T ⇐⇒ control
D(λ) smoothed on a scale δλ ∼ 1T .
• [LEVITAN’52, AVAKUMOVIC’56,HORMANDER’68] : Parametrix of e−it√−∆ for
|t | ≤ T0 R(λ) = O(λd−1).Optimal for X = Sd due to large degeneracies.
• [CHAZARAIN’73,DUISTERMAAT-GUILLEMIN’75] Tr e−it√−∆ has singularities at
t = Tγ the lengths of closed geodesics R(λ) = o(λd−1), provided theset of periodic points has measure zero.
![Page 24: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/24.jpg)
Periodic orbits as oscillations of D(λ)
Around 1970, (some) physicists want to understand the oscillations of D(λ).Motivations : nuclear physics, semiconductors. . . [GUTZWILLER’70,BALIAN-BLOCH’72]
In the case of a classically chaotic system, the Gutzwiller trace formularelates quantum and classical informations :
D(λ) = D(λ) + Dfl (λ)λ→∞∼
Xj≥0
A0,jλd−j + Re
Xγ per. geod.
eiλTγXj≥0
Aγ,jλ−j
D(λ) contains the information on the periodic geodesics (and vice-versa)
”Generalizes” Selberg’s trace formula (1956) for X = Γ\H2 compact.
• 1 application : (X , g) negatively curved, lower bound for R(λ), in terms ofthe full set of per. orbits [JAKOBSON-POLTEROVICH-TOTH’07]
![Page 25: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/25.jpg)
Periodic orbits as oscillations of D(λ)
Around 1970, (some) physicists want to understand the oscillations of D(λ).Motivations : nuclear physics, semiconductors. . . [GUTZWILLER’70,BALIAN-BLOCH’72]
In the case of a classically chaotic system, the Gutzwiller trace formularelates quantum and classical informations :
D(λ) = D(λ) + Dfl (λ)λ→∞∼
Xj≥0
A0,jλd−j + Re
Xγ per. geod.
eiλTγXj≥0
Aγ,jλ−j
D(λ) contains the information on the periodic geodesics (and vice-versa)
”Generalizes” Selberg’s trace formula (1956) for X = Γ\H2 compact.
• 1 application : (X , g) negatively curved, lower bound for R(λ), in terms ofthe full set of per. orbits [JAKOBSON-POLTEROVICH-TOTH’07]
![Page 26: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/26.jpg)
Periodic orbits as oscillations of D(λ)
Around 1970, (some) physicists want to understand the oscillations of D(λ).Motivations : nuclear physics, semiconductors. . . [GUTZWILLER’70,BALIAN-BLOCH’72]
In the case of a classically chaotic system, the Gutzwiller trace formularelates quantum and classical informations :
D(λ) = D(λ) + Dfl (λ)λ→∞∼
Xj≥0
A0,jλd−j + Re
Xγ per. geod.
eiλTγXj≥0
Aγ,jλ−j
D(λ) contains the information on the periodic geodesics (and vice-versa)
”Generalizes” Selberg’s trace formula (1956) for X = Γ\H2 compact.
• 1 application : (X , g) negatively curved, lower bound for R(λ), in terms ofthe full set of per. orbits [JAKOBSON-POLTEROVICH-TOTH’07]
![Page 27: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/27.jpg)
Domains with (smooth) boundaries
Even if ∂Ω is smooth, describing the wave propagation inpresence of boundaries is difficult, mostly due to gliding orgrazing rays. Lots of progresses in the 1970s [MELROSE,SJOSTRAND, TAYLOR]
[SEELEY’78] : R(λ) = O(λd−1).
[IVRII’80, MELROSE’80] : FINALLY, Weyl’s 2-term asymptotics
RD/N(λ) = ∓ωd−1|∂Ω|4(2π)d−1 λ
d−1 + o(λd−1),
provided the set of periodic (broken) geodesics has measure zero.
![Page 28: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/28.jpg)
Domains with (smooth) boundaries
Even if ∂Ω is smooth, describing the wave propagation inpresence of boundaries is difficult, mostly due to gliding orgrazing rays. Lots of progresses in the 1970s [MELROSE,SJOSTRAND, TAYLOR]
[SEELEY’78] : R(λ) = O(λd−1).
[IVRII’80, MELROSE’80] : FINALLY, Weyl’s 2-term asymptotics
RD/N(λ) = ∓ωd−1|∂Ω|4(2π)d−1 λ
d−1 + o(λd−1),
provided the set of periodic (broken) geodesics has measure zero.
![Page 29: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/29.jpg)
Domains with (smooth) boundaries
Even if ∂Ω is smooth, describing the wave propagation inpresence of boundaries is difficult, mostly due to gliding orgrazing rays. Lots of progresses in the 1970s [MELROSE,SJOSTRAND, TAYLOR]
[SEELEY’78] : R(λ) = O(λd−1).
[IVRII’80, MELROSE’80] : FINALLY, Weyl’s 2-term asymptotics
RD/N(λ) = ∓ωd−1|∂Ω|4(2π)d−1 λ
d−1 + o(λd−1),
provided the set of periodic (broken) geodesics has measure zero.
![Page 30: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/30.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages 750 pages
so far, 2282 pages...
![Page 31: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/31.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages 750 pages
so far, 2282 pages...
![Page 32: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/32.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages
750 pages
so far, 2282 pages...
![Page 33: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/33.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages
750 pages
so far, 2282 pages...
![Page 34: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/34.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages 750 pages
so far, 2282 pages...
![Page 35: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/35.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages 750 pages
so far, 2282 pages...
![Page 36: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/36.jpg)
If you really want to understand Victor’s tricks, you may try
238 pages 750 pages
so far, 2282 pages...
![Page 37: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/37.jpg)
How did Victor manage to find these tricks ?
Working hard... ...in an inspiring environment
![Page 38: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/38.jpg)
How did Victor manage to find these tricks ?
Working hard...
...in an inspiring environment
![Page 39: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/39.jpg)
How did Victor manage to find these tricks ?
Working hard... ...in an inspiring environment
![Page 40: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/40.jpg)
From eigenvalues to resonancesIn quantum or wave physics, stationary modes are most often a mathematicalidealization.A system always interacts with its environment (measuring device,absorption, spontaneous emission. . . ) each excited state has a finitelifetime τn.
=⇒ the spectrum is not a sum of δ peaks, but rather of Lorentzian peakscentered at En ∈ R, of widths Γn = 1
τn(decay rates).
0
0.2
0.4
0.6
0.8
1
1.2
-2 0 2 4 6 8 10
1/((x-1)**2+1)+1/((x-4)**2+2)1/((x-1)**2+1)1/((x-4)**2+2)
0
0.2
0.4
0.6
0.8
1
1.2
-2 0 2 4 6 8 10
1/((x-1)**2+1)+1/((x-4)**2+2)1/((x-1)**2+1)1/((x-4)**2+2)
0
0.2
0.4
0.6
0.8
1
1.2
-2 0 2 4 6 8 10
1/((x-1)**2+1)+1/((x-4)**2+2)1/((x-1)**2+1)1/((x-4)**2+2)
0
0.2
0.4
0.6
0.8
1
1.2
-2 0 2 4 6 8 10
1/((x-1)**2+1)+1/((x-4)**2+2)1/((x-1)**2+1)1/((x-4)**2+2)
0
0.2
0.4
0.6
0.8
1
1.2
-2 0 2 4 6 8 10
1/((x-1)**2+1)+1/((x-4)**2+2)1/((x-1)**2+1)1/((x-4)**2+2)
Each Lorentzian↔ a complex resonance zn = λn − iΓn.
![Page 41: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/41.jpg)
Clean mathematical setting : geometric scatteringOpen cavity with waveguide.Obstacle / potential V ∈ Cc(Rd ) / (X , g) Euclidean near infinity
−∆Ω (or −∆ + V ) has abs. cont. spectrum on R+. Yet, the cutoff resolventRχ(z)
def= χ(∆ + z2)−1χ admits (in odd dimension) a meromorphic
continuation from Im z > 0 to C, with isolated poles of finite multiplicities zn,the resonances (or scattering poles) of ∆X .
z!
jz
0plane
![Page 42: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/42.jpg)
Counting (all) resonances
Can we estimate N(r)def= #j ; |zj | ≤ r?
Cannot use selfadjoint methods (minimax)=⇒ Upper bounds are much easier to obtainthan lower bounds.
0!
N(r)r
z!plane
Main tool : complex analysis.Construct an entire function d(z) which vanishes at the resonances.d(z) = det(I − K (z)) with K (z) holomorphic family of compact ops.
• Control the growth of d(z) when |z| → ∞ (count singular values of K (z),use self-adjoint Weyl’s law)
Jensen=⇒ N(r) ≤ C r d , r →∞ [MELROSE,ZWORSKI,VODEV,SJOSTRAND-ZWORSKI. . . ]
Connection with a volume : C = cd ad if Supp V ⊂ B(0, a)[ZWORSKI’87,STEFANOV’06]
• [CHRISTIANSEN’05. . . DINH-VU’12] For generic obstacle / metric perturbation /potential supported in B(0, a), the upper bound is sharp.
• [CHRISTIANSEN’10] Distribution in angular sectors, higher density near R.[SJOSTRAND’12] Semiclassical setting, potential with a small randomperturbation : Weyl law for resonances in a thin strip below R.
![Page 43: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/43.jpg)
Counting (all) resonances
Can we estimate N(r)def= #j ; |zj | ≤ r?
Cannot use selfadjoint methods (minimax)=⇒ Upper bounds are much easier to obtainthan lower bounds.
0!
N(r)r
z!plane
Main tool : complex analysis.Construct an entire function d(z) which vanishes at the resonances.d(z) = det(I − K (z)) with K (z) holomorphic family of compact ops.
• Control the growth of d(z) when |z| → ∞ (count singular values of K (z),use self-adjoint Weyl’s law)
Jensen=⇒ N(r) ≤ C r d , r →∞ [MELROSE,ZWORSKI,VODEV,SJOSTRAND-ZWORSKI. . . ]
Connection with a volume : C = cd ad if Supp V ⊂ B(0, a)[ZWORSKI’87,STEFANOV’06]
• [CHRISTIANSEN’05. . . DINH-VU’12] For generic obstacle / metric perturbation /potential supported in B(0, a), the upper bound is sharp.
• [CHRISTIANSEN’10] Distribution in angular sectors, higher density near R.[SJOSTRAND’12] Semiclassical setting, potential with a small randomperturbation : Weyl law for resonances in a thin strip below R.
![Page 44: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/44.jpg)
Counting (all) resonances
Can we estimate N(r)def= #j ; |zj | ≤ r?
Cannot use selfadjoint methods (minimax)=⇒ Upper bounds are much easier to obtainthan lower bounds.
0!
N(r)r
z!plane
Main tool : complex analysis.Construct an entire function d(z) which vanishes at the resonances.d(z) = det(I − K (z)) with K (z) holomorphic family of compact ops.
• Control the growth of d(z) when |z| → ∞ (count singular values of K (z),use self-adjoint Weyl’s law)
Jensen=⇒ N(r) ≤ C r d , r →∞ [MELROSE,ZWORSKI,VODEV,SJOSTRAND-ZWORSKI. . . ]
Connection with a volume : C = cd ad if Supp V ⊂ B(0, a)[ZWORSKI’87,STEFANOV’06]
• [CHRISTIANSEN’05. . . DINH-VU’12] For generic obstacle / metric perturbation /potential supported in B(0, a), the upper bound is sharp.
• [CHRISTIANSEN’10] Distribution in angular sectors, higher density near R.[SJOSTRAND’12] Semiclassical setting, potential with a small randomperturbation : Weyl law for resonances in a thin strip below R.
![Page 45: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/45.jpg)
Counting (all) resonances
Can we estimate N(r)def= #j ; |zj | ≤ r?
Cannot use selfadjoint methods (minimax)=⇒ Upper bounds are much easier to obtainthan lower bounds.
0!
N(r)r
z!plane
Main tool : complex analysis.Construct an entire function d(z) which vanishes at the resonances.d(z) = det(I − K (z)) with K (z) holomorphic family of compact ops.
• Control the growth of d(z) when |z| → ∞ (count singular values of K (z),use self-adjoint Weyl’s law)
Jensen=⇒ N(r) ≤ C r d , r →∞ [MELROSE,ZWORSKI,VODEV,SJOSTRAND-ZWORSKI. . . ]
Connection with a volume : C = cd ad if Supp V ⊂ B(0, a)[ZWORSKI’87,STEFANOV’06]
• [CHRISTIANSEN’05. . . DINH-VU’12] For generic obstacle / metric perturbation /potential supported in B(0, a), the upper bound is sharp.
• [CHRISTIANSEN’10] Distribution in angular sectors, higher density near R.[SJOSTRAND’12] Semiclassical setting, potential with a small randomperturbation : Weyl law for resonances in a thin strip below R.
![Page 46: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/46.jpg)
Counting ”long living” resonances
From a physics point of view, the resonances with| Im z| 1 are not very significant (very small lifetime) rather count resonances of bounded decay rates :N(λ, γ)
def= #j ; |zj − λ| ≤ γ), γ > 0 fixed, λ→∞.
///jz
!"0
This counting gives informations on the classical dynamics on the trapped set
K def=˘
(x , ξ) ∈ S∗X ; Φt (x , ξ) uniformly bounded for all t ∈ R¯
(compact subset of S∗X , invariant through Φt ).
• K = ∅ =⇒ no long-living resonance• K = a single hyperbolic orbit. Resonances form a (projected) deformed
lattice, encoding the length and Lyapunov exponents of the orbit[IKAWA’85,GERARD’87]
N( , )=O(1)
!"0
" !
![Page 47: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/47.jpg)
Counting ”long living” resonances
From a physics point of view, the resonances with| Im z| 1 are not very significant (very small lifetime) rather count resonances of bounded decay rates :N(λ, γ)
def= #j ; |zj − λ| ≤ γ), γ > 0 fixed, λ→∞.
///jz
!"0
This counting gives informations on the classical dynamics on the trapped set
K def=˘
(x , ξ) ∈ S∗X ; Φt (x , ξ) uniformly bounded for all t ∈ R¯
(compact subset of S∗X , invariant through Φt ).• K = ∅ =⇒ no long-living resonance
• K = a single hyperbolic orbit. Resonances form a (projected) deformedlattice, encoding the length and Lyapunov exponents of the orbit[IKAWA’85,GERARD’87]
N( , )=O(1)
!"0
" !
![Page 48: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/48.jpg)
Counting ”long living” resonances
From a physics point of view, the resonances with| Im z| 1 are not very significant (very small lifetime) rather count resonances of bounded decay rates :N(λ, γ)
def= #j ; |zj − λ| ≤ γ), γ > 0 fixed, λ→∞.
///jz
!"0
This counting gives informations on the classical dynamics on the trapped set
K def=˘
(x , ξ) ∈ S∗X ; Φt (x , ξ) uniformly bounded for all t ∈ R¯
(compact subset of S∗X , invariant through Φt ).• K = ∅ =⇒ no long-living resonance• K = a single hyperbolic orbit. Resonances form a (projected) deformed
lattice, encoding the length and Lyapunov exponents of the orbit[IKAWA’85,GERARD’87]
N( , )=O(1)
!"0
" !
![Page 49: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/49.jpg)
Counting ”long living” resonances (2)• K contains an elliptic periodic orbit⇒ many resonances with
Im z = O(λ−∞) =⇒ N(λ, γ) λd−1 [POPOV,VODEV,STEFANOV]
N( )~C
!"
0
!," d!1!
• K a fractal subset carrying a chaotic (hyperbolic) flow. Quantum chaos
!
"!0
///
N( , ) < C ! " #
Fractal Weyl upper bound[SJOSTRAND,SJOSTRAND-ZWORSKI,N-SJOSTRAND-ZWORSKI]
∀γ > 0,∃Cγ > 0, N(λ, γ) ≤ Cγ λν , λ→∞,
where dimMink (K ) = 2ν + 1 (so that 0 < ν < d − 1).
![Page 50: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/50.jpg)
Counting ”long living” resonances (2)• K contains an elliptic periodic orbit⇒ many resonances with
Im z = O(λ−∞) =⇒ N(λ, γ) λd−1 [POPOV,VODEV,STEFANOV]
N( )~C
!"
0
!," d!1!
• K a fractal subset carrying a chaotic (hyperbolic) flow. Quantum chaos
!
"!0
///
N( , ) < C ! " #
Fractal Weyl upper bound[SJOSTRAND,SJOSTRAND-ZWORSKI,N-SJOSTRAND-ZWORSKI]
∀γ > 0,∃Cγ > 0, N(λ, γ) ≤ Cγ λν , λ→∞,
where dimMink (K ) = 2ν + 1 (so that 0 < ν < d − 1).
![Page 51: Counting stationary modes: a discrete view of geometry and … · 2012. 9. 20. · proof of the 2-term asymptotics resonances of open wave systems : counting them all, vs. selecting](https://reader036.vdocument.in/reader036/viewer/2022063018/5fdb97bb9513a44fcd1582a6/html5/thumbnails/51.jpg)
Fractal Weyl law ?
N(λ, γ) ≤ Cγ λν , λ→∞,
This bound also results from a volume estimate : count the number ofquantum states ”living” in the λ−1/2-neighbourhood of K .
Fractal Weyl Law conjecture : this upper bound is sharp, at least at the levelof the power ν.
Several numerical studies confirm the conjecture[LU-SRIDHAR-ZWORSKI,GUILLOPE-LIN-ZWORSKI,SCHOMERUS-TWORZYDŁO].
Only proved for a discrete-time toy model(quantum baker’s map) [N-ZWORSKI]A chaotic open map B : T2 → T2 is quantizedinto a family (BN)N∈N of subunitary N × Nmatrices, where N ≡ ~−1.
Fractal Weyl law in this context :# Spec(BN) ∩ e−γ ≤ |z| ≤ 1 ∼ Cγ Nν asN →∞, where ν = dim(trapped set of B)
2 < 1.
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
be81
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
be243
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
be729
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
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Fractal Weyl law ?
N(λ, γ) ≤ Cγ λν , λ→∞,
This bound also results from a volume estimate : count the number ofquantum states ”living” in the λ−1/2-neighbourhood of K .
Fractal Weyl Law conjecture : this upper bound is sharp, at least at the levelof the power ν.
Several numerical studies confirm the conjecture[LU-SRIDHAR-ZWORSKI,GUILLOPE-LIN-ZWORSKI,SCHOMERUS-TWORZYDŁO].
Only proved for a discrete-time toy model(quantum baker’s map) [N-ZWORSKI]A chaotic open map B : T2 → T2 is quantizedinto a family (BN)N∈N of subunitary N × Nmatrices, where N ≡ ~−1.
Fractal Weyl law in this context :# Spec(BN) ∩ e−γ ≤ |z| ≤ 1 ∼ Cγ Nν asN →∞, where ν = dim(trapped set of B)
2 < 1.
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Fractal Weyl law galore
FWL for quantum maps search for FWL in certain families (MN)N→∞ oflarge matrices
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Fractal Weyl law galore
FWL for quantum maps search for FWL in certain families (MN)N→∞ oflarge matrices
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Experimental studies on microwave billiards. [KUHL et al.’12]Major difficulty : extract the ”true” resonances from the signal.