Nonlinear Schrödinger equation under a magnetic field kept ... · Nonlinear Schrödinger equation...

Nonlinear Schrödinger equation under amagnetic field kept in the foreground

Jean VAN SCHAFTINGEN

Institut de Recherche enMathématique et Physique

Workshop in Nonlinear PDEsBrussels, September 7–11, 2015

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 1 / 23

Nonlinear Schrödinger equation when themagnetic field is kept in the foreground

1 The magnetic nonlinear Schrödinger equation

2 Semiclassical results

3 Constant magnetic field problem

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 2 / 23

Nonlinear Schrödinger equation when themagnetic field is kept in the foreground

1 The magnetic nonlinear Schrödinger equation

2 Semiclassical results

3 Constant magnetic field problem

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 3 / 23

Magnetic nonlinear Schödinger functional

∫

Ω

ϵ2|DA/ϵu|2

2+

V|u|2

2−|u|p

p

where

u : Ω→ C ' R2 is a wave-function,−|u|p−2 is an interaction by δ-functions,V is an (adimensionalized) electric potential,ϵ is a characteristic length (adimensionalized Planckconstant),DA/ϵ is a covariant derivative.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 4 / 23

Magnetic nonlinear Schödinger functional

∫

Ω

ϵ2|DA/ϵu|2

2+

V|u|2

2−|u|p

p

where

u : Ω→ C ' R2 is a wave-function,−|u|p−2 is an interaction by δ-functions,V is an (adimensionalized) electric potential,ϵ is a characteristic length (adimensionalized Planckconstant),DA/ϵ is a covariant derivative.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 4 / 23

Covariant derivative and connectionDifferential geometry meets physics

DAu4= Du− iuA,

where A : Ω→∧1

R3 ' R3.1 differential calculus:

DA(ηu) = ηDAu + uDη, D|u|2 = 2(u|DAu),

2 the curvature is the magnetic field B = dA ' ∇ × B,

KA[v,w]u4= DA(DAu[w])[v]− DA(DAu[v])[w]

= −idA[v,w]u ' −i(∇ × A|v × w)u,

3 gauge invariance: if A′4= A + dΛ ' A + ∇Λ and u′

4= eiΛu,

thenDA′u′ = eiΛDAu.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 5 / 23

Covariant derivative and connectionDifferential geometry meets physics

DAu4= Du− iuA,

where A : Ω→∧1

R3 ' R3.1 differential calculus:

DA(ηu) = ηDAu + uDη, D|u|2 = 2(u|DAu),

2 the curvature is the magnetic field B = dA ' ∇ × B,

KA[v,w]u4= DA(DAu[w])[v]− DA(DAu[v])[w]

= −idA[v,w]u ' −i(∇ × A|v × w)u,

3 gauge invariance: if A′4= A + dΛ ' A + ∇Λ and u′

4= eiΛu,

thenDA′u′ = eiΛDAu.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 5 / 23

Covariant derivative and connectionDifferential geometry meets physics

DAu4= Du− iuA,

where A : Ω→∧1

R3 ' R3.1 differential calculus:

DA(ηu) = ηDAu + uDη, D|u|2 = 2(u|DAu),

2 the curvature is the magnetic field B = dA ' ∇ × B,

KA[v,w]u4= DA(DAu[w])[v]− DA(DAu[v])[w]

= −idA[v,w]u ' −i(∇ × A|v × w)u,

3 gauge invariance: if A′4= A + dΛ ' A + ∇Λ and u′

4= eiΛu,

thenDA′u′ = eiΛDAu.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 5 / 23

Magnetic nonlinear Schrödinger equation

The Euler–Lagrange equation associated to

∫

Ω

ϵ2|DA/ϵu|2

2+

V|u|2

2−|u|p

p

is−ϵ2∆A/ϵu + Vu = |u|p−2u,

where the magnetic Laplacian is defined as

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 6 / 23

Magnetic nonlinear Schrödinger equation

The Euler–Lagrange equation associated to

∫

Ω

ϵ2|DA/ϵu|2

2+

V|u|2

2−|u|p

p

is−ϵ2∆A/ϵu + Vu = |u|p−2u,

where the magnetic Laplacian is defined as

−ϵ2∆A/ϵu = −ϵ2∆u + 2iϵ⟨A,Du⟩+ (|A|2 − id∗A)u

' −ϵ2∆u + 2iϵA · ∇u + (|A|2 − i∇ · A)u.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 6 / 23

Magnetic nonlinear Schrödinger equation

The Euler–Lagrange equation associated to

∫

Ω

ϵ2|DA/ϵu|2

2+

V|u|2

2−|u|p

p

is−ϵ2∆A/ϵu + Vu = |u|p−2u,

where the magnetic Laplacian is defined as

−ϵ2∆A/ϵu4= −ϵ2∆u + 2iϵ⟨A,Du⟩+ |A|2u

' −ϵ2∆u + 2iϵA · ∇u + |A|2u,

in the Coulomb/Lorenz gauge (d∗A ' ∇ · A = 0).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 6 / 23

Existence of groundstatesSeminal work of Esteban and Lions

Groundstate solutions are minimizers of∫

Ω

ϵ2|DA/ϵu|2 + V|u|2

∫

Ω

|u|p

2p

.

Theorem (M. Esteban and P.-L. Lions, 1989)Groundstates exist when either

1 Ω is bounded,2 Ω = R3, and V and dA are constant,3 a strict inequality is satisfied.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 7 / 23

Existence of groundstatesSeminal work of Esteban and Lions

Groundstate solutions are minimizers of∫

Ω

ϵ2|DA/ϵu|2 + V|u|2

∫

Ω

|u|p

2p

.

Theorem (M. Esteban and P.-L. Lions, 1989)Groundstates exist when either

1 Ω is bounded,2 Ω = R3, and V and dA are constant,3 a strict inequality is satisfied.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 7 / 23

Nonlinear Schrödinger equation when themagnetic field is kept in the foreground

1 The magnetic nonlinear Schrödinger equation

2 Semiclassical results

3 Constant magnetic field problem

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 8 / 23

Magnetic field going into the backgroundMagnetic field in the background

Theorem (Kurata (2000))

If 12 −

13 <

1p <

12 and V0

4= infV < lim inf∞V, then there exist

(xϵ)ϵ>0 in R3 and (ωϵ)ϵ>0 such that V(xϵ)→ V0 and

uϵ(x) ' αϵUx− xϵ

ϵ

ei

A(xϵ)[x−xϵ ]

ϵ +ωϵ

,

where U is the unique positive radial groundstate of thelimiting problem

−∆U + V0U = |U|p−2U.

See also Cingolani (2003), Cingolani & Secchi (2002), Secchi &Squassina (2005), Cao & Tang (2006), Barile (2008), Cingolani,Jeanjean & Secchi (2009), Cingolani & Clapp (2009, 2010), Barile,Cingolani & Secchi (2011), Ding & Liu (2013). . .

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 9 / 23

Magnetic field going into the backgroundMagnetic field in the background

Theorem (Kurata (2000))

If 12 −

13 <

1p <

12 and V0

4= infV < lim inf∞V, then there exist

(xϵ)ϵ>0 in R3 and (ωϵ)ϵ>0 such that V(xϵ)→ V0 and

uϵ(x) ' αϵUx− xϵ

ϵ

ei

A(xϵ)[x−xϵ ]

ϵ +ωϵ

,

where U is the unique positive radial groundstate of thelimiting problem

−∆U + V0U = |U|p−2U.

The behavior of the concentration points and of the densitiesare independent of the magnetic field:

|uϵ(x)|2 '

Ux− xϵ

ϵ

2.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 9 / 23

Magnetic field going into the backgroundMagnetic field in the background

Theorem (Kurata (2000))

If 12 −

13 <

1p <

12 and V0

4= infV < lim inf∞V, then there exist

(xϵ)ϵ>0 in R3 and (ωϵ)ϵ>0 such that V(xϵ)→ V0 and

uϵ(x) ' αϵUx− xϵ

ϵ

ei

A(xϵ)[x−xϵ ]

ϵ +ωϵ

,

where U is the unique positive radial groundstate of thelimiting problem

−∆U + V0U = |U|p−2U.

Asymptoticaly consistent with the Lorentz force when v = 0:

0 = F = −q(dV + vùdA) ' q(−∇V + v × (∇ × A)).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 9 / 23

Magnetic field going into the backgroundMagnetic field in the background

Theorem (Kurata (2000))

If 12 −

13 <

1p <

12 and V0

4= infV < lim inf∞V, then there exist

(xϵ)ϵ>0 in R3 and (ωϵ)ϵ>0 such that V(xϵ)→ V0 and

uϵ(x) ' αϵUx− xϵ

ϵ

ei

A(xϵ)[x−xϵ ]

ϵ +ωϵ

,

where U is the unique positive radial groundstate of thelimiting problem

−∆U + V0U = |U|p−2U.

Question: How to have interaction with the magnetic field inthe semiclassical limit?

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 9 / 23

Magnetic semiclassical soliton dynamicsThe Lorentz force is validated in the dynamics

Theorem (Selvitella (2008) & Squassina (2009))

If 2 < p < 2 + 43 , then there exists solutions to

iϵ∂tψ = −ϵ2∆A/ϵψ+ Vψ − |ψ|p−2ψ,

of the form

ψϵ(x, t) ' Ux− x(t)

ϵ

ei

A(x(t))[x−x(t)]+x(t)·(x−x(t))

ϵ +ωϵ(t)

.

with

x(t) = −dV(x(t))− x(t)ùdA(x(t))

' −∇V(x(t)) + x(t)× (∇ × A)(x(t)).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 10 / 23

Why is the magnetic field fading out?

By the Heisenberg uncertainty principle, a groundstate mighthave a magnetic momentum μ ∈

∧2R3 ' R3. It is thus subject

to a force of

−qdV + d⟨dA, μ⟩ ' −q∇V + ∇(∇ × A · μ).

There is no asymptotic magnetic effect:

q =

∫

R3|∇uϵ|2 ' ϵ3, μ =

∫

R3((x− xϵ)∧ (−iϵ∇A/ϵuϵ(x))|uϵ(x)) dx ' ϵ4.

Idea: Interesting phenomena should happen in the strongmagnetic field régime A ' ϵ−1.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 11 / 23

Why is the magnetic field fading out?

By the Heisenberg uncertainty principle, a groundstate mighthave a magnetic momentum μ ∈

∧2R3 ' R3. It is thus subject

to a force of

−qdV + d⟨dA, μ⟩ ' −q∇V + ∇(∇ × A · μ).

There is no asymptotic magnetic effect:

q =

∫

R3|∇uϵ|2 ' ϵ3, μ =

∫

R3((x− xϵ)∧ (−iϵ∇A/ϵuϵ(x))|uϵ(x)) dx ' ϵ4.

Idea: Interesting phenomena should happen in the strongmagnetic field régime A ' ϵ−1.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 11 / 23

Why is the magnetic field fading out?

By the Heisenberg uncertainty principle, a groundstate mighthave a magnetic momentum μ ∈

∧2R3 ' R3. It is thus subject

to a force of

−qdV + d⟨dA, μ⟩ ' −q∇V + ∇(∇ × A · μ).

There is no asymptotic magnetic effect:

q =

∫

R3|∇uϵ|2 ' ϵ3, μ =

∫

R3((x− xϵ)∧ (−iϵ∇A/ϵuϵ(x))|uϵ(x)) dx ' ϵ4.

Idea: Interesting phenomena should happen in the strongmagnetic field régime A ' ϵ−1.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 11 / 23

The new problem

We study solutions of the Euler–Lagrange equation associatedto

∫

Ω

ϵ2|DA/ϵ2u|2

2+

V|u|2

2−|u|p

p

which is−ϵ2∆A/ϵ2u + Vu = |u|p−2u,

where

−ϵ2∆A/ϵu = −ϵ2∆u− 2i⟨A,Du⟩+|A|2 − id∗A

ϵ2u

' −ϵ2∆u− 2iA · ∇u +|A|2 − i∇ · A

ϵ2u.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 12 / 23

The new problem

We study solutions of the Euler–Lagrange equation associatedto

∫

Ω

ϵ2|DA/ϵ2u|2

2+

V|u|2

2−|u|p

p

which is−ϵ2∆A/ϵ2u + Vu = |u|p−2u,

where

−ϵ2∆A/ϵu4= −ϵ2∆u + 2i⟨A,Du⟩+

|A|2

ϵ2u

' −ϵ2∆u + 2iA · ∇u +|A|2

ϵ2u,

in the Coulomb/Lorenz gauge (d∗A ' ∇ · A = 0).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 12 / 23

The concentration function

For V ∈ R and B ∈∧2

R3 ' R3, define

A(x)[v] =B(x,v)

2'

(B× x) · v

2,

so that dA = B

and set

E(V,B) =1

2−

1

p

infu∈C∞c (R3;C)

∫

R3|DAu|2 + V|u|2

p

p−2

∫

R3|u|p

2

p−2

.

1 E is continuous,2 diamagnetic inequality:

E(V,B) ≥ E(V,0).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 13 / 23

The concentration function

For V ∈ R and B ∈∧2

R3 ' R3, define

A(x)[v] =B(x,v)

2'

(B× x) · v

2,

so that dA = B and set

E(V,B) =1

2−

1

p

infu∈C∞c (R3;C)

∫

R3|DAu|2 + V|u|2

p

p−2

∫

R3|u|p

2

p−2

.

1 E is continuous,

2 diamagnetic inequality:

E(V,B) ≥ E(V,0).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 13 / 23

The concentration function

For V ∈ R and B ∈∧2

R3 ' R3, define

A(x)[v] =B(x,v)

2'

(B× x) · v

2,

so that dA = B and set

E(V,B) =1

2−

1

p

infu∈C∞c (R3;C)

∫

R3|DAu|2 + V|u|2

p

p−2

∫

R3|u|p

2

p−2

.

1 E is continuous,2 diamagnetic inequality:

E(V,B) ≥ E(V,0).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 13 / 23

Asymptotic result

Theorem (Di Cosmo and Van Schaftingen (2015))

If 12 −

13 <

1p <

12 and Ω is bounded, then there exist (xϵ)ϵ>0 in

R3 such thatlim

R→∞ϵ→0

‖uϵ‖L∞(Ω\BϵR(xϵ)) = 0,

limϵ→0

ϵ−NFϵ(uϵ) = limϵ→0

E(V(xϵ),dA(xϵ)) = infΩ

E(V,dA).

If V(xϵn)→ V∗, dA(xn)→ B∗ and A∗(x)[v] =B∗(x,v)

2 , up to asubsequence

uϵn ' UV∗,A∗

x− xn

ϵn

ei

A(xϵn )[x−xϵn ]

ϵ2n+ωn

,

with UV∗,A∗ a groundstate of

−∆A∗UV∗,A∗ + V∗UV∗,A∗ = |UV∗,A∗ |p−2UV∗,A∗ .

See also Fournais and Raymond (2015).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 14 / 23

Asymptotic result

Theorem (Di Cosmo and Van Schaftingen (2015))

If 12 −

13 <

1p <

12 and Ω is bounded, then there exist (xϵ)ϵ>0 in

R3 such thatlim

R→∞ϵ→0

‖uϵ‖L∞(Ω\BϵR(xϵ)) = 0,

limϵ→0

ϵ−NFϵ(uϵ) = limϵ→0

E(V(xϵ),dA(xϵ)) = infΩ

E(V,dA).

If V(xϵn)→ V∗, dA(xn)→ B∗ and A∗(x)[v] =B∗(x,v)

2 , up to asubsequence

uϵn ' UV∗,A∗

x− xn

ϵn

ei

A(xϵn )[x−xϵn ]

ϵ2n+ωn

,

See also Fournais and Raymond (2015).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 14 / 23

Forces in the semiclassical limit

If x∗ ∈ Ω and

E(V(x∗),dA(x∗)) = infΩ

E(V,dA),

and UV∗,A∗ is the corresponding groundstate, then

qdV(x∗) = d⟨dA, μ⟩(x∗),

with

q =

∫

R3|UV∗,A∗ |

2

and

μ =

∫

R3(y∧ (−i∇AUV∗,A∗(y))|UV∗,A∗(y)) dy.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 15 / 23

About the proof

Challenges:1 uniqueness and nondegeneracy of solutions of the

limiting problem are unknown,

2 the coefficients of the rescaled problem are not locallyuniformly bounded coefficients,

3 natural function space of the limiting problem at x∗depends on x∗.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 16 / 23

About the proof

Challenges:1 uniqueness and nondegeneracy of solutions of the

limiting problem are unknown,2 the coefficients of the rescaled problem are not locally

uniformly bounded coefficients,

3 natural function space of the limiting problem at x∗depends on x∗.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 16 / 23

About the proof

Challenges:1 uniqueness and nondegeneracy of solutions of the

limiting problem are unknown,2 the coefficients of the rescaled problem are not locally

uniformly bounded coefficients,3 natural function space of the limiting problem at x∗

depends on x∗.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 16 / 23

About the proof

Challenges:1 uniqueness and nondegeneracy of solutions of the

limiting problem are unknown,2 the coefficients of the rescaled problem are not locally

uniformly bounded coefficients,3 natural function space of the limiting problem at x∗

depends on x∗.

Also covers local minimizers (del Pino & Felmer penalization),with slow decay at infinity

lim|x|→∞

V(x)|x|2 > 0.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 16 / 23

Nonlinear Schrödinger equation when themagnetic field is kept in the foreground

1 The magnetic nonlinear Schrödinger equation

2 Semiclassical results

3 Constant magnetic field problem

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 17 / 23

The constant electromagnetic potential problem

What happens to the groundstate of

−∆A + Vu = |u|p−2u

when V = 1 and A(x)[v] = 12B[x,v] ' 1

2(B× x) · v for someB ∈

∧2R3?

How does the solution depend on B?

Is the solution symmetric?How does the solution decay?

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 18 / 23

The constant electromagnetic potential problem

What happens to the groundstate of

−∆A + Vu = |u|p−2u

when V = 1 and A(x)[v] = 12B[x,v] ' 1

2(B× x) · v for someB ∈

∧2R3?

How does the solution depend on B?Is the solution symmetric?

How does the solution decay?

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 18 / 23

The constant electromagnetic potential problem

What happens to the groundstate of

−∆A + Vu = |u|p−2u

when V = 1 and A(x)[v] = 12B[x,v] ' 1

2(B× x) · v for someB ∈

∧2R3?

How does the solution depend on B?Is the solution symmetric?How does the solution decay?

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 18 / 23

Free electron in a magnetic fieldThe first Landau level

Groundstate of the linear Schrödinger equation satisfy

−∆Aψ = Eψ.

If A(x)[v] = 12B(x,v) ' 1

2(B× x) · v, then a groundstate is givenby

ψ(x) = e−|B×x|2

4|B| ,

withE = |B|.

PropertiesE increases with |B|,

ψ is axially symmetric with respect to B,ψ has Gaussian decay perpendiculary to B.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 19 / 23

Free electron in a magnetic fieldThe first Landau level

Groundstate of the linear Schrödinger equation satisfy

−∆Aψ = Eψ.

If A(x)[v] = 12B(x,v) ' 1

2(B× x) · v, then a groundstate is givenby

ψ(x) = e−|B×x|2

4|B| ,

withE = |B|.

PropertiesE increases with |B|,ψ is axially symmetric with respect to B,

ψ has Gaussian decay perpendiculary to B.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 19 / 23

Free electron in a magnetic fieldThe first Landau level

Groundstate of the linear Schrödinger equation satisfy

−∆Aψ = Eψ.

If A(x)[v] = 12B(x,v) ' 1

2(B× x) · v, then a groundstate is givenby

ψ(x) = e−|B×x|2

4|B| ,

withE = |B|.

PropertiesE increases with |B|,ψ is axially symmetric with respect to B,ψ has Gaussian decay perpendiculary to B.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 19 / 23

Groundstates under a small constant magneticfield

Theorem (Bonheure, Nys and Van Schaftingen, 2015)

If V ' 1 and |A|2 ® 1 then1 UA,V is real and radially symmetric,

2 UA,V(x) ® e−|xùB|24|B| ' e−

|B×x|24|B| ,

3 E(V,B) ' E(1,0) +V − 1

2

∫

R3|U|2 +

|B|2

4 · 3

∫

R3|y|2|U(y)|2 dy

.

“Essentially” means up to translations in R3 and rotation in C.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 20 / 23

Groundstates under a small constant magneticfield

Theorem (Bonheure, Nys and Van Schaftingen, 2015)

If V ' 1 and |A|2 ® 1 then1 UA,V is real and radially symmetric,

2 UA,V(x) ® e−|xùB|24|B| ' e−

|B×x|24|B| ,

3 E(V,B) ' E(1,0) +V − 1

2

∫

R3|U|2 +

|B|2

4 · 3

∫

R3|y|2|U(y)|2 dy

.

By scaling, we cover in fact |A|2/V ® 1.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 20 / 23

Groundstates under a small constant magneticfield

Theorem (Bonheure, Nys and Van Schaftingen, 2015)

If V ' 1 and |A|2 ® 1 then1 UA,V is real and radially symmetric,

2 UA,V(x) ® e−|xùB|24|B| ' e−

|B×x|24|B| ,

3 E(V,B) ' E(1,0) +V − 1

2

∫

R3|U|2 +

|B|2

4 · 3

∫

R3|y|2|U(y)|2 dy

.

In R2, the Gaussian decay follows from the linear theory (L.Erdos, 1996).

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 20 / 23

Proving the properties by essential uniqueness

Idea: Prove that when |A|2 ® 1, then the groundstate isessentially unique.

Good news: When A = 0, U is essentially nondegenerate(Weinstein, 1985).

Problem: UA lives naturally in the space

H1A(R3) = u ∈ L2(R3) : DAu ∈ L2(R3);

the natural functional space depends on the parameter A.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 21 / 23

Proving the properties by essential uniqueness

Idea: Prove that when |A|2 ® 1, then the groundstate isessentially unique.

Good news: When A = 0, U is essentially nondegenerate(Weinstein, 1985).

Problem: UA lives naturally in the space

H1A(R3) = u ∈ L2(R3) : DAu ∈ L2(R3);

the natural functional space depends on the parameter A.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 21 / 23

Proving the properties by essential uniqueness

Idea: Prove that when |A|2 ® 1, then the groundstate isessentially unique.

Good news: When A = 0, U is essentially nondegenerate(Weinstein, 1985).

Problem: UA lives naturally in the space

H1A(R3) = u ∈ L2(R3) : DAu ∈ L2(R3);

the natural functional space depends on the parameter A.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 21 / 23

Working across functional spaces

the gap between U1 ∈ H1A1

(R3) and U2 ∈ H1A2

(R3) can bemeasured by

∫

R3|DA1U1 − DA2U2|2 + |U1 − U2|2,

the uniqueness part of the implicit function theorem justuses nondegeneracy and continuity of the linearizedoperator.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 22 / 23

Working across functional spaces

the gap between U1 ∈ H1A1

(R3) and U2 ∈ H1A2

(R3) can bemeasured by

∫

R3|DA1U1 − DA2U2|2 + |U1 − U2|2,

the uniqueness part of the implicit function theorem justuses nondegeneracy and continuity of the linearizedoperator.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 22 / 23

Thank you for your attention.

J. Van Schaftingen (UCLouvain) Magnetic nonlinear Schrödinger Brussels, 2015-09-11 23 / 23