The Schrdinger equation for the fractional Laplacian in negative - - PowerPoint PPT Presentation

Introduction Homogeneous trees Hyperbolic spaces

Conference Probability and Analysis (Będlewo, May 15 – 17, 2017)

The Schrödinger equation for the fractional Laplacian in negative curvature

Jean–Philippe Anker (Université d’Orléans) Joint work in progress with Yannick Sire (Johns Hopkins University, Baltimore)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Equations

Schrödinger equation

• i ∂tu(t, x) − ∆xu(t, x) = F(t, x)

u(0, x)=f(x) Half wave equation

• i ∂tu(t, x) +
• −∆x u(t, x) = F(t, x)

u(0, x)=f(x) Schrödinger equation for the fractional Laplacian

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) = F(t, x)

u(0, x)=f(x) (1) where 1<κ <2 (possibly also 0<κ <1)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Equations

Schrödinger equation

• i ∂tu(t, x) − ∆xu(t, x) = F(t, x)

u(0, x)=f(x) Half wave equation

• i ∂tu(t, x) +
• −∆x u(t, x) = F(t, x)

u(0, x)=f(x) Schrödinger equation for the fractional Laplacian

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) = F(t, x)

u(0, x)=f(x) (1) where 1<κ <2 (possibly also 0<κ <1)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Equations

Schrödinger equation

• i ∂tu(t, x) − ∆xu(t, x) = F(t, x)

u(0, x)=f(x) Half wave equation

• i ∂tu(t, x) +
• −∆x u(t, x) = F(t, x)

u(0, x)=f(x) Schrödinger equation for the fractional Laplacian

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) = F(t, x)

u(0, x)=f(x) (1) where 1<κ <2 (possibly also 0<κ <1)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy

Linear homogeneous equation : F = 0

Solution : u(t, x) = e it(−∆)κ/2f(x) =

• kκ

t (x, y) f(y) dy

Kernel estimate Dispersive estimate :

• e it(−∆)κ/2
• Lq′→Lq

∀ t∈R∗, ∀ 2≤q ≤∞

Linear inhomogeneous equation : F = 0

Duhamel formula : u(t, x) = e it(−∆)κ/2f(x) + t e i(t−s)(−∆x)κ/2F(s, x) ds Strichartz inequality : u(t, x)Lp

tLq x fL2 + u(t, x)L ˜ p′ t L˜ q′ x

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy

Linear homogeneous equation : F = 0

Solution : u(t, x) = e it(−∆)κ/2f(x) =

• kκ

t (x, y) f(y) dy

Kernel estimate Dispersive estimate :

• e it(−∆)κ/2
• Lq′→Lq

∀ t∈R∗, ∀ 2≤q ≤∞

Linear inhomogeneous equation : F = 0

Duhamel formula : u(t, x) = e it(−∆)κ/2f(x) + t e i(t−s)(−∆x)κ/2F(s, x) ds Strichartz inequality : u(t, x)Lp

tLq x fL2 + u(t, x)L ˜ p′ t L˜ q′ x

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy

Linear homogeneous equation : F = 0

Solution : u(t, x) = e it(−∆)κ/2f(x) =

• kκ

t (x, y) f(y) dy

Kernel estimate Dispersive estimate :

• e it(−∆)κ/2
• Lq′→Lq

∀ t∈R∗, ∀ 2≤q ≤∞

Linear inhomogeneous equation : F = 0

Duhamel formula : u(t, x) = e it(−∆)κ/2f(x) + t e i(t−s)(−∆x)κ/2F(s, x) ds Strichartz inequality : u(t, x)Lp

tLq x fL2 + u(t, x)L ˜ p′ t L˜ q′ x

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy

Linear homogeneous equation : F = 0

Solution : u(t, x) = e it(−∆)κ/2f(x) =

• kκ

t (x, y) f(y) dy

Kernel estimate Dispersive estimate :

• e it(−∆)κ/2
• Lq′→Lq

∀ t∈R∗, ∀ 2≤q ≤∞

Linear inhomogeneous equation : F = 0

Duhamel formula : u(t, x) = e it(−∆)κ/2f(x) + t e i(t−s)(−∆x)κ/2F(s, x) ds Strichartz inequality : u(t, x)Lp

tLq x fL2 + u(t, x)L ˜ p′ t L˜ q′ x

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy

Linear homogeneous equation : F = 0

Solution : u(t, x) = e it(−∆)κ/2f(x) =

• kκ

t (x, y) f(y) dy

Kernel estimate Dispersive estimate :

• e it(−∆)κ/2
• Lq′→Lq

∀ t∈R∗, ∀ 2≤q ≤∞

Linear inhomogeneous equation : F = 0

Duhamel formula : u(t, x) = e it(−∆)κ/2f(x) + t e i(t−s)(−∆x)κ/2F(s, x) ds Strichartz inequality : u(t, x)Lp

tLq x fL2 + u(t, x)L ˜ p′ t L˜ q′ x

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy (continued)

Nonlinear equation : F(t, x) = F(u(t, x)) with F(u) power–like e.g.

• F(u) = const.

     uγ (γ integer ≥ 2) |u|γ (γ >1) u |u|γ−1 (γ >1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools :

Strichartz inequality Fixed point theorem Conservation law

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy (continued)

Nonlinear equation : F(t, x) = F(u(t, x)) with F(u) power–like e.g.

• F(u) = const.

     uγ (γ integer ≥ 2) |u|γ (γ >1) u |u|γ−1 (γ >1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools :

Strichartz inequality Fixed point theorem Conservation law

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy (continued)

Nonlinear equation : F(t, x) = F(u(t, x)) with F(u) power–like e.g.

• F(u) = const.

     uγ (γ integer ≥ 2) |u|γ (γ >1) u |u|γ−1 (γ >1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools :

Strichartz inequality Fixed point theorem Conservation law

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Standard strategy (continued)

Nonlinear equation : F(t, x) = F(u(t, x)) with F(u) power–like e.g.

• F(u) = const.

     uγ (γ integer ≥ 2) |u|γ (γ >1) u |u|γ−1 (γ >1) Main problem = local/global well–posedness ∼ existence and uniqueness of solutions Tools :

Strichartz inequality Fixed point theorem Conservation law

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Homogeneous trees

T= TQ homogeneous tree with Q+1≥3 edges Example : Q=5 Discrete analogs of hyperbolic spaces Volume of balls of radius r∈N : V (r) = 1+ Q+1

Q−1 (Qr−1) ≍ Qr

Combinatorial Laplacian on (the vertices of) T : ∆f(x) =

1 Q+1

• d(y,x)=1

f(y) − f(x)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Homogeneous trees

T= TQ homogeneous tree with Q+1≥3 edges Example : Q=5 Discrete analogs of hyperbolic spaces Volume of balls of radius r∈N : V (r) = 1+ Q+1

Q−1 (Qr−1) ≍ Qr

Combinatorial Laplacian on (the vertices of) T : ∆f(x) =

1 Q+1

• d(y,x)=1

f(y) − f(x)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Homogeneous trees

T= TQ homogeneous tree with Q+1≥3 edges Example : Q=5 Discrete analogs of hyperbolic spaces Volume of balls of radius r∈N : V (r) = 1+ Q+1

Q−1 (Qr−1) ≍ Qr

Combinatorial Laplacian on (the vertices of) T : ∆f(x) =

1 Q+1

• d(y,x)=1

f(y) − f(x)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Homogeneous trees

T= TQ homogeneous tree with Q+1≥3 edges Example : Q=5 Discrete analogs of hyperbolic spaces Volume of balls of radius r∈N : V (r) = 1+ Q+1

Q−1 (Qr−1) ≍ Qr

Combinatorial Laplacian on (the vertices of) T : ∆f(x) =

1 Q+1

• d(y,x)=1

f(y) − f(x)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Schrödinger equation on T

On T, the Schrödinger equation (with continuous time)

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) = F(t, x)

u(0, x)=f(x) (1) can be solved by using the Fourier transform : u(t, x) = e it(−∆)κ/2f(x)

• homogeneous

+ t e i(t−s)(−∆x)κ/2F(s, x) ds

• inhomogeneous

where e it(−∆)κ/2f(x) =

• y∈T f(y) kκ

t (d(x, y))

• f ∗ kκ

t (x) J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Schrödinger kernel on T

Inverse spherical Fourier transform kκ

t (r) = const.

• π

log Q

e it[1−γ(λ)]κ/2 ϕλ(r) |c(λ)|−2 dλ where γ(λ) =

Q iλ+ Q−iλ Q1/2 + Q−1/2

c(λ) =

1 Q1/2 + Q−1/2 Q1/2+iλ − Q−1/2−iλ Q iλ − Q−iλ

ϕλ(r) = c(λ)Q(−1/2+iλ)r + c(−λ)Q(−1/2−iλ)r

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Kernel estimate on T

Hence kκ

t (r) = const. Q−r/2

• R/2πZ

e it[1−γ(0)cos λ]κ/2 − irλ ×

sin λ Q1/2 eiλ − Q−1/2 e−iλ dλ

By stationary phase analysis, one gets Global upper bound Assume that 0<κ ≤2. Then |kκ

t (r)| Q− r

2

∀ t∈R∗, ∀ r∈N Moreover there exists a constant C >0 such that |kκ

t (r)| |t|− 3

2 (1+r)Q− r 2

∀ t∈R∗, ∀ r∈N if 1+r ≤C |t|

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Kernel estimate on T

Hence kκ

t (r) = const. Q−r/2

• R/2πZ

e it[1−γ(0)cos λ]κ/2 − irλ ×

sin λ Q1/2 eiλ − Q−1/2 e−iλ dλ

By stationary phase analysis, one gets Global upper bound Assume that 0<κ ≤2. Then |kκ

t (r)| Q− r

2

∀ t∈R∗, ∀ r∈N Moreover there exists a constant C >0 such that |kκ

t (r)| |t|− 3

2 (1+r)Q− r 2

∀ t∈R∗, ∀ r∈N if 1+r ≤C |t|

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Linear estimates on T

Dispersive estimate Let 0<κ ≤2 and 2<q ≤∞. Then e it(−∆)κ/2ℓq′→ℓq (1+|t|)− 3

2

∀ t∈R∗ Main tool = following version of the Kunze–Stein phenomenon Lemma Let 2≤q <∞ and q

2 ≤ p<q. Then

Lq ′(T) ∗ L p

rad(T) ⊂ Lq(T)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Linear estimates on T

Dispersive estimate Let 0<κ ≤2 and 2<q ≤∞. Then e it(−∆)κ/2ℓq′→ℓq (1+|t|)− 3

2

∀ t∈R∗ Main tool = following version of the Kunze–Stein phenomenon Lemma Let 2≤q <∞ and q

2 ≤ p<q. Then

Lq ′(T) ∗ L p

rad(T) ⊂ Lq(T)

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Linear estimates on T

Strichartz inequality u(t, x)Lp

t ℓq x fℓ2 + F(t, x)L˜ p′ t ℓ ˜ q′ x

for all admissible pairs (1

p, 1 q) and (1 ˜ p, 1 ˜ q) in the following square

1 1

1 2 1 2 1 p 1 q

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

NLS on T

Consider the nonlinear Schrödinger equation

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) =

F(u(t, x)) u(0, x)=f(x) (2) where

• |

F(u)| |u|γ | F(u)− F (v)| {|u|γ−1+|v|γ−1}|u−v| for some exponent γ >1 Theorem (2) is locally well–posed for arbitrary initial data in ℓ2 (2) is globally well–posed for small initial data in ℓ2 (2) is globally well–posed for arbitrary initial data in ℓ2 under the additional condition Im{ F(u)u}= 0

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

NLS on T

Consider the nonlinear Schrödinger equation

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) =

F(u(t, x)) u(0, x)=f(x) (2) where

• |

F(u)| |u|γ | F(u)− F (v)| {|u|γ−1+|v|γ−1}|u−v| for some exponent γ >1 Theorem (2) is locally well–posed for arbitrary initial data in ℓ2 (2) is globally well–posed for small initial data in ℓ2 (2) is globally well–posed for arbitrary initial data in ℓ2 under the additional condition Im{ F(u)u}= 0

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Hyperbolic spaces Hn

Some data Ball model : Hn = {x∈Rn | x<1} Riemannian metric : ds2 = 1−x2

2

−2 dx2 Hyperbolic distance : d(x, 0) = log 1+x

1−x

Riemannian volume : dV(x) = 1−x2

2

• −n dx1 . . . dxn

Laplacian : ∆ = 1−x2

2

2 n

j=1

∂

∂xj

2+(n−2) 1−x2

2

n

j=1xj ∂ ∂xj

Remarks Continuous analogs of homogeneous trees Similar large scale analysis + local analysis

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Hyperbolic spaces Hn

Some data Ball model : Hn = {x∈Rn | x<1} Riemannian metric : ds2 = 1−x2

2

−2 dx2 Hyperbolic distance : d(x, 0) = log 1+x

1−x

Riemannian volume : dV(x) = 1−x2

2

• −n dx1 . . . dxn

Laplacian : ∆ = 1−x2

2

2 n

j=1

∂

∂xj

2+(n−2) 1−x2

2

n

j=1xj ∂ ∂xj

Remarks Continuous analogs of homogeneous trees Similar large scale analysis + local analysis

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Schrödinger equation Hn

On Hn, the Schrödinger equation

• i ∂tu(t, x) + (−∆x)

κ 2 u(t, x) = F(t, x)

u(0, x)=f(x) (1) can be solved again by using the Fourier transform : u(t, x) = e it(−∆)κ/2f(x)

• homogeneous

+ t e i(t−s)(−∆x)κ/2F(s, x) ds

• inhomogeneous

where e it(−∆)κ/2f(x) =

• Hnf(y) kκ

t (d(x, y)) dy

• f ∗ kκ

t (x)

Inverse spherical Fourier transform kκ

t (r) = const.

∞ e it[λ2+( n−1

2 )2] κ/2

ϕλ(r) |c(λ)|−2 dλ

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Large time estimates on Hn

Large time kernel estimate Assume that 1<κ <2. Then |kκ

t (r)|

• |t|− 3

2 (1+r) e− n−1 2

r

if |t|≥1+r (1+r)N e− n−1

2

r

if 1≤|t|≤1+r Large time dispersive estimate Let 1<κ ≤2 and 2<q ≤∞. Then e it(−∆)κ/2Lq′→Lq |t|− 3

2

for large |t| Tool = Kunze–Stein phenomenon

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Large time estimates on Hn

Large time kernel estimate Assume that 1<κ <2. Then |kκ

t (r)|

• |t|− 3

2 (1+r) e− n−1 2

r

if |t|≥1+r (1+r)N e− n−1

2

r

if 1≤|t|≤1+r Large time dispersive estimate Let 1<κ ≤2 and 2<q ≤∞. Then e it(−∆)κ/2Lq′→Lq |t|− 3

2

for large |t| Tool = Kunze–Stein phenomenon

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Strichartz inequality on Hn

Strichartz inequality Assume that 1<κ ≤2. Then u(t, x)Lp

t H−σ,q x

fL2 + F(t, x)L˜

p′ t H ˜ σ,˜ q′ x

Here (1

p, 1 q), (1 ˜ p, 1 ˜ q) are admissible pairs

and σ ≥ max

• 0, n(1

2 − 1 q)− κ p

• , ˜

σ ≥ max

• 0, n(1

2 − 1 ˜ q)− κ ˜ p

• 1

1

1 2 1 2 1 p 1 q 1 2 − κ 2n σ n

J.–Ph. Anker P&A Będlewo 2017

Introduction Homogeneous trees Hyperbolic spaces

Thank you for your attention

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J.–Ph. Anker P&A Będlewo 2017