DispersiveQuantization - University of Minnesotaolver/t_/dq.pdf · 2011-02-10 · Dispersion...
Transcript of DispersiveQuantization - University of Minnesotaolver/t_/dq.pdf · 2011-02-10 · Dispersion...
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Dispersive Quantization
Peter J. Olver
University of Minnesota
http://www.math.umn.edu/∼ olver
Arizona, February, 2011
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Peter J. OlverIntroduction to Partial Differential Equations
Pearson Publ., to appear (2011?)
Amer. Math. Monthly 117 (2010) 599–610
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Dispersion
Definition. A linear partial differential equation is calleddispersive if the different Fourier modes travel unalteredbut at different speeds.
Substitutingu(t, x) = e i (kx−ωt)
produces the dispersion relation
ω = ω(k)
relating frequency ω and wave number k.
Phase velocity: cp =ω(k)
k
Group velocity: cg =dω
dk(stationary phase)
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The simplest linear dispersive wave equation:
∂u
∂t=
∂3u
∂x3
Dispersion relation: ω = −k3
Phase velocity: cp =ω
k= −k2
Group velocity: cg =dω
dk= −3k2
Thus, wave packets (and energy) move faster (to the left) thanthe individual waves.
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Linear Dispersion on the Line
∂u
∂t=
∂3u
∂x3u(0, x) = f(x)
Fourier transform solution:
u(t, x) =1√2π
∫ ∞
−∞f(k) e i (kx+k3 t) dk
Fundamental solution u(0, x) = δ(x)
u(t, x) =1
2π
∫ ∞
−∞e i (kx+k3 t) dk =
13√
3 tAi
(x
3√
3 t
)
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Linear Dispersion on the Line
∂u
∂t=
∂3u
∂x3u(0, x) = f(x)
Fourier transform solution:
u(t, x) =1√2π
∫ ∞
−∞f(k) e i (kx+k3 t) dk
Fundamental solution u(0, x) = δ(x)
u(t, x) =1
2π
∫ ∞
−∞e i (kx+k3 t) dk =
13√
3 tAi
(x
3√
3 t
)
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Linear Dispersion on the Line
∂u
∂t=
∂3u
∂x3u(0, x) = f(x)
Fourier transform solution:
u(t, x) =1√2π
∫ ∞
−∞f(k) e i (kx+k3 t) dk
Fundamental solution u(0, x) = δ(x)
u(t, x) =1
2π
∫ ∞
−∞e i (kx+k3 t) dk =
13√
3 tAi
(x
3√
3 t
)
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t = .03 t = .1 t = 1/3
t = 1 t = 5 t = 20
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Linear Dispersion on the Line
∂u
∂t=
∂3u
∂x3u(0, x) = f(x)
u(t, x) =1
3√
3 t
∫ ∞
−∞f(ξ) Ai
(x − ξ
3√
3 t
)
dξ
Step function initial data: u(0, x) = σ(x) =
{0, x < 0,
1, x > 0.
u(t, x) =1
3− H
(
− x3√
3 t
)
H(z) =z Γ
(23
)1F2
(13 ; 2
3 , 43 ; 1
9 z3)
35/3 Γ(
23
)Γ(
43
) −z2 Γ
(23
)1F2
(23 ; 4
3 , 53 ; 1
9 z3)
37/3 Γ(
43
)Γ(
53
)
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Linear Dispersion on the Line
∂u
∂t=
∂3u
∂x3u(0, x) = f(x)
u(t, x) =1
3√
3 t
∫ ∞
−∞f(ξ) Ai
(x − ξ
3√
3 t
)
dξ
Step function initial data: u(0, x) = σ(x) =
{0, x < 0,
1, x > 0.
u(t, x) =1
3− H
(
− x3√
3 t
)
H(z) =z Γ
(23
)1F2
(13 ; 2
3 , 43 ; 1
9 z3)
35/3 Γ(
23
)Γ(
43
) −z2 Γ
(23
)1F2
(23 ; 4
3 , 53 ; 1
9 z3)
37/3 Γ(
43
)Γ(
53
)
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Linear Dispersion on the Line
∂u
∂t=
∂3u
∂x3u(0, x) = f(x)
u(t, x) =1
3√
3 t
∫ ∞
−∞f(ξ) Ai
(x − ξ
3√
3 t
)
dξ
Step function initial data: u(0, x) = σ(x) =
{0, x < 0,
1, x > 0.
u(t, x) =1
3− H
(
− x3√
3 t
)
H(z) =z Γ
(23
)1F2
(13 ; 2
3 , 43 ; 1
9 z3)
35/3 Γ(
23
)Γ(
43
) −z2 Γ
(23
)1F2
(23 ; 4
3 , 53 ; 1
9 z3)
37/3 Γ(
43
)Γ(
53
)
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t = .005 t = .01 t = .05
t = .1 t = .5 t = 1.
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Periodic Linear Dispersion
∂u
∂t=
∂3u
∂x3
u(t, 0) = u(t, 2π)∂u
∂x(t, 0) =
∂u
∂x(t, 2π)
∂2u
∂x2(t, 0) =
∂2u
∂x2(t, 2π)
Step function initial data:
u(0, x) = σ(x) =
{0, x < 0,
1, x > 0.
u⋆(t, x) ∼ 1
2+
2
π
∞∑
j =0
sin( (2j + 1)x − (2j + 1)3 t )
2j + 1.
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Periodic Linear Dispersion
∂u
∂t=
∂3u
∂x3
u(t, 0) = u(t, 2π)∂u
∂x(t, 0) =
∂u
∂x(t, 2π)
∂2u
∂x2(t, 0) =
∂2u
∂x2(t, 2π)
Step function initial data:
u(0, x) = σ(x) =
{0, x < 0,
1, x > 0.
u⋆(t, x) ∼ 1
2+
2
π
∞∑
j =0
sin( (2j + 1)x − (2j + 1)3 t )
2j + 1.
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Periodic Linear Dispersion
∂u
∂t=
∂3u
∂x3
u(t, 0) = u(t, 2π)∂u
∂x(t, 0) =
∂u
∂x(t, 2π)
∂2u
∂x2(t, 0) =
∂2u
∂x2(t, 2π)
Step function initial data:
u(0, x) = σ(x) =
{0, x < 0,
1, x > 0.
u⋆(t, x) ∼ 1
2+
2
π
∞∑
j =0
sin( (2j + 1)x − (2j + 1)3 t )
2j + 1.
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t = 0. t = .1 t = .2
t = .3 t = .4 t = .5
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t = 130 π t = 1
15 π t = 110 π
t = 215 π t = 1
6 π t = 15 π
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t = π t = 12 π t = 1
3 π
t = 14 π t = 1
5 π t = 16 π
t = 17 π t = 1
8 π t = 19 π
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Theorem. At rational time t = πp/q, the solution u⋆(t, x)is constant on every subinterval πj/q < x < π(j + 1)/q.At irrational time u⋆(t, x) is a non-differentiable continuousfunction.
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Lemma.
f(x) ∼∞∑
k=−∞
ck e i kx
is piecewise constant on intervals pπ/q < x < (p + 1)π/q ifand only if
ck = cl, k ≡ l 6≡ 0 mod 2q, ck = 0, 0 6= k ≡ 0 mod 2q.
where
ck =πk ck
i q (e− i πk/q − 1)k 6≡ 0 mod 2q.
=⇒ DFT
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The Fourier coefficients of the solution u⋆(t, x) at rational timet = πp/q are
ck = bk
(π p
q
)= bk(0) e i (kx−k3πp/q),
where
bk(0) =
− i /(πk), k odd,
1/2, k = 0,
0, 0 6= k even.
Crucial observation:
if k ≡ l mod 2q, then k3 ≡ l3 mod 2q
and soe i (kx−k3πp/q) = e i (lx−l3πp/q)
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Theorem. At rational time t = πp/q, the fundamentalsolution to the initial-boundary value problem is a linearcombination of finitely many delta functions.
Corollary. At rational time, any solution profile u(πp/q, x)to the periodic initial-boundary value problem dependson only finitely many values of the initial data, namelyu(0, xj) = f(xj) where xj = πj/q for j = 0, . . . , 2q − 1when p is odd, or xj = 2πj/q for j = 0, . . . , q − 1 when p iseven.
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Theorem. At rational time t = πp/q, the fundamentalsolution to the initial-boundary value problem is a linearcombination of finitely many delta functions.
Corollary. At rational time, any solution profile u(πp/q, x)to the periodic initial-boundary value problem dependson only finitely many values of the initial data, namelyu(0, xj) = f(xj) where xj = πj/q for j = 0, . . . , 2q − 1when p is odd, or xj = 2πj/q for j = 0, . . . , q − 1 when p iseven.
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⋆ ⋆ The same phenomenon appears in any linearly dispersiveequation with “integral” dispersion relation:
ω(k) =n∑
m=0
cmkm
wherecm/cn ∈ Q
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Linear Schrodinger Equation
i∂u
∂t=
∂2u
∂x2
Dispersion relation: ω = k2
Phase velocity: cp =ω
k= −k
Group velocity: cg =dω
dk= −2k
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Periodic Linear Schrodinger Equation
i∂u
∂t=
∂2u
∂x2
u(t, 0) = u(t, 2π)∂u
∂x(t, 0) =
∂u
∂x(t, 2π)
∂2u
∂x2(t, 0) =
∂2u
∂x2(t, 2π)
• Michael Berry, et. al.
• Oskolkov
• Michael Taylor
• Fulling, Gunturk
• Kapitanski, Rodnianski
“Does a quantum particle know the time?”
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William Henry Fox Talbot (1800–1877)
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⋆ Talbot’s 1835 image of a latticed window in Lacock Abbey
=⇒ oldest photographic negative in existence.
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The Talbot Effect
Fresnel diffraction by periodic gratings (1836)
“It was very curious to observe that though the grating was
greatly out of the focus of the lens . . . the appearance of
the bands was perfectly distinct and well defined . . . the
experiments are communicated in the hope that they may
prove interesting to the cultivators of optical science.”
— Fox Talbot
=⇒ Lord Rayleigh calculates the Talbot distance (1881)
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The Quantized/Fractal Talbot Effect
• Optical experiments — Berry & Klein
• Diffraction of matter waves (helium attoms) — Nowak et. al.
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Quantum Revival
• Electrons in potassium ions — Yeazell & Stroud
• Vibrations of bromine molecules —Vrakking, Villeneuve, Stolow
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Periodic Schrodinger Equation
i∂u
∂t=
∂2u
∂x2
u(t, 0) = u(t, 2π)∂u
∂x(t, 0) =
∂u
∂x(t, 2π)
∂2u
∂x2(t, 0) =
∂2u
∂x2(t, 2π)
Integrated fundamental solution:
u(t, x) =1
2π
∞∑
06=k=−∞
e i (kx+k2t)
k.
⋆ For x/t ∈ Q, this is known as a Gauss (or, more generally,Weyl) sum, of importance in number theory
=⇒ Hardy, Littlewood, Weil, I. Vinogradov, etc.
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Integrated fundamental solution:
u(t, x) =1
2π
∞∑
06=k=−∞
e i (kx+k2t)
k.
Theorem.
• The fundamental solution ∂u/∂x is a Jacobi theta function.At rational times t = pπ/q, it linear combination of deltafunctions concentrated at rational nodes xj = πj/q.
• At irrational times t, the integrated fundamental solution is acontinuous but nowhere differentiable function.(The fractal dimension of its graph is 3
2 .)
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Dispersive Carpet
Schrodinger Carpet
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Future Directions
• Other boundary conditions (Fokas/Bona)
• Higher space dimensions and other domains (e.g., tori, spheres)
• Numerical solution techniques?
• Dispersive nonlinear partial differential equations:periodic Korteweg–deVries — Zabusky & Kruskal
• Experimental verification in dispersive media?