Schr¨ odinger equation in presence of a magnetic field
Jean Dolbeault
http://www.ceremade.dauphine.fr/ ∼ dolbeaul Ceremade, Universit´e Paris-Dauphine
September 18, 2019 Nonlinear days in Alghero
September 16-20, 2019
Outline
Without magnetic fields: symmetry and symmetry breaking in interpolation inequalities
B Gagliardo-Nirenberg-Sobolev inequalities on the sphere B Keller-Lieb-Thirring inequalities on the sphere
B Caffarelli-Kohn-Nirenberg inequalities With magnetic fields in dimensions 2 and 3 B Interpolation inequalities and spectral estimates
B Estimates, numerics; an open question on constant magnetic fields Magnetic rings: the case of S 1
B A one-dimensional magnetic interpolation inequality
B Consequences: Keller-Lieb-Thirring estimates, Aharonov-Bohm magnetic fields and a new Hardy inequality in R 2
Aharonov-Bohm magnetic fields in R 2 B Aharonov-Bohm effect
B Interpolation and Keller-Lieb-Thirring inequalities in R 2 B Aharonov-Symmetry and symmetry breaking
J. Dolbeault Magnetic ground state & symmetry
A joint research program (mostly) with...
M.J. Esteban, Ceremade, Universit´e Paris-Dauphine B symmetry, interpolation, Keller-Lieb-Thirring, magnetic fields
M. Loss, Georgia Institute of Technology (Atlanta) B symmetry, interpolation, Keller-Lieb-Thirring, magnetic fields
A. Laptev, Imperial College London B Keller-Lieb-Thirring, magnetic fields
D. Bonheure, Universit´e Libre de Bruxelles B Aharonov-Bohm magnetic fields
J. Dolbeault Magnetic ground state & symmetry
Definitions in presence of a magnetic field
The magnetic covariant derivative / magnetic gradient
∇ A := ∇ + i A The magnetic Dirichlet energy: if ψ = u e iS , then
ˆ
R
d|∇ A ψ | 2 dx = ˆ
R
d|∇ ψ | 2 dx + ˆ
R
d|∇ S + A | 2 | ψ | 2 dx The magnetic Sobolev space
H 1 A ( R d ) :=
ψ ∈ L 2 ( R d ) : ∇ A ψ ∈ L 2 ( R d ) The magnetic Laplacian
− ∆ A ψ = − ∆ ψ − 2 i A · ∇ ψ + | A | 2 ψ − i (div A) ψ
J. Dolbeault Magnetic ground state & symmetry
Magnetic interpolation, ground state of the nonlinear Schr¨ odinger equation and symmetry
For any p ∈ (2, 2 ∗ ), the magnetic interpolation inequality k∇ A ψ k 2 L
2( R
d) + α k ψ k 2 L
2( R
d) ≥ µ(α) k ψ k 2 L
p( R
d) ∀ ψ ∈ H 1 A (R d ) is a consequence of the Gagliardo-Nirenberg inequality
k∇ ψ k 2 L
2( R
d) + α k ψ k 2 L
2( R
d) ≥ µ GN (α) k ψ k 2 L
p( R
d) ∀ ψ ∈ H 1 (R d ) and of the diamagnetic inequality
k∇ A ψ k 2 L
2( R
d) ≥ k∇ u k 2 L
2( R
d) , u = | ψ | An optimal function solves
− ∆ A ψ + α ψ = | ψ | p−2 ψ
If A is invariant under rotation, is there a (complex valued) ground state (minimum of the energy) which is depending only on | x | ?
J. Dolbeault Magnetic ground state & symmetry
Symmetry and symmetry breaking in interpolation inequalities
without magnetic field
Gagliardo-Nirenberg-Sobolev inequalities on the sphere Keller-Lieb-Thirring inequalities on the sphere
Caffarelli-Kohn-Nirenberg inequalities on R 2
J. Dolbeault Magnetic ground state & symmetry
A result of uniqueness on a classical example
On the sphere S d , let us consider the positive solutions of
− ∆u + λ u = u p−1 p ∈ [1, 2) ∪ (2, 2 ∗ ] if d ≥ 3, 2 ∗ = d−2 2 d
p ∈ [1, 2) ∪ (2, + ∞ ) if d = 1, 2 Theorem
If λ ≤ d , u ≡ λ 1/(p−2) is the unique solution
[Gidas & Spruck, 1981], [Bidaut-V´eron & V´eron, 1991]
J. Dolbeault Magnetic ground state & symmetry
Bifurcation point of view and symmetry breaking
2 4 6 8 10
2 4 6 8
Figure: (p − 2) λ 7→ (p − 2) µ(λ) with d = 3
k∇ u k 2 L
2( S
d) + λ k u k 2 L
2( S
d) ≥ µ(λ) k u k 2 L
p( S
d)
Taylor expansion of u = 1 + ε ϕ 1 as ε → 0 with − ∆ϕ 1 = d ϕ 1
µ(λ) < λ if and only if λ > p−2 d
B The inequality holds with µ(λ) = λ = p−2 d [Bakry & Emery, 1985]
[Beckner, 1993], [Bidaut-V´eron & V´eron, 1991, Corollary 6.1]
J. Dolbeault Magnetic ground state & symmetry
The Bakry-Emery method on the sphere
Entropy functional E p [ρ] := p−2 1 h ´
S
dρ
2pdµ − ´
S
dρ dµ
2pi
if p 6 = 2 E 2 [ρ] := ´
S
dρ log
ρ kρk
L1 (Sd)dµ Fisher information functional
I p [ρ] := ´
S
d|∇ ρ
1p| 2 d µ
[Bakry & Emery, 1985] carr´ e du champ method: use the heat flow
∂ρ
∂t = ∆ρ and observe that dt d E p [ρ] = − I p [ρ]
d dt
I p [ρ] − d E p [ρ]
≤ 0 = ⇒ I p [ρ] ≥ d E p [ρ]
with ρ = | u | p , if p ≤ 2 # := (d−1) 2d
2+1
2 J. Dolbeault Magnetic ground state & symmetryThe evolution under the fast diffusion flow
To overcome the limitation p ≤ 2 # , one can consider a nonlinear diffusion of fast diffusion / porous medium type
∂ρ
∂t = ∆ρ m
[Demange], [JD, Esteban, Kowalczyk, Loss]: for any p ∈ [1, 2 ∗ ] K p [ρ] := d
dt
I p [ρ] − d E p [ρ]
≤ 0
1.0 1.5 2.5 3.0
0.0 0.5 1.5 2.0
(p, m) admissible region, d = 5
J. Dolbeault Magnetic ground state & symmetry
References
JD, M. J. Esteban, M. Kowalczyk, and M. Loss. Improved interpolation inequalities on the sphere. Discrete and Continuous Dynamical Systems Series S, 7 (4): 695-724, 2014.
JD, M.J. Esteban, and M. Loss. Interpolation inequalities on the sphere: linear vs. nonlinear flows. Annales de la facult´e des sciences de Toulouse S´er. 6, 26 (2): 351-379, 2017
JD, M.J. Esteban. Improved interpolation inequalities and stability. Preprint, 2019 arXiv:1908.08235
J. Dolbeault Magnetic ground state & symmetry
Optimal inequalities
With µ(λ) = λ = p−2 d : [Bakry & Emery, 1985]
[Beckner, 1993], [Bidaut-V´eron & V´eron, 1991, Corollary 6.1]
k∇ u k 2 L
2( S
d) ≥ d p − 2
k u k 2 L
p( S
d) − k u k 2 L
2( S
d)
∀ u ∈ H 1 (S d )
d ≥ 3, p ∈ [1, 2) or p ∈ (2, d−2 2 d ) d = 1 or d = 2, p ∈ [1, 2) or p ∈ (2, ∞ )
p = − 2 = 2 d/(d − 2) = − 2 with d = 1 [Exner, Harrell, Loss, 1998]
k∇ u k 2 L
2( S
d) + 1 4
ˆ
S
d1 u 2 dµ
−1
≥ 1
4 k u k 2 L
2( S
d) ∀ u ∈ H 1 + (S d )
J. Dolbeault Magnetic ground state & symmetry
Keller-Lieb-Thirring inequalities on the sphere
The Keller-Lieb-Tirring inequality is equivalent to an interpolation inequality of Gagliardo-Nirenberg-Sobolev type
We measure a quantitative deviation with respect to the semi-classical regime due to finite size effects
Joint work with M.J. Esteban and A. Laptev
J. Dolbeault Magnetic ground state & symmetry
An introduction to (Keller)-Lieb-Thirring inequalities in R d
(λ k ) k≥1 : eigenvalues of the Schr¨odinger operator H = − ∆ − V on R d Euclidean case [Keller, 1961]
| λ 1 | γ ≤ L 1 γ,d ˆ
R
dV + γ+
d2[Lieb-Thirring, 1976]
P
k≥1 | λ k | γ ≤ L γ,d
´
R
dV γ+
d
+
2γ ≥ 1/2 if d = 1, γ > 0 if d = 2 and γ ≥ 0 if d ≥ 3 [Weidl], [Cwikel], [Rosenbljum], [Aizenman], [Laptev-Weidl], [Helffer], [Robert], [JD-Felmer-Loss-Paturel], [JD-Laptev-Loss]...[Frank, Hundertmark, Jex, Nam]
Compact manifolds: log Sobolev case: [Federbusch], [Rothaus];
case γ = 0 (Rozenbljum-Lieb-Cwikel inequality): [Levin-Solomyak];
[Lieb], [Levin], [Ouabaz-Poupaud]... [Ilyin]
B How does one take into account the finite size effects on S d ?
J. Dolbeault Magnetic ground state & symmetry
H¨ older duality and link with interpolation inequalities
Let p = q−2 q . Consider the Schr¨ odinger energy ˆ
S
d|∇ u | 2 − ˆ
S
dV | u | 2 ≥ ˆ
S
d|∇ u | 2 − µ k u k 2 L
q( S
d)
≥ − λ(µ) k u k 2 L
2( S
d) if µ = k V + k L
p( S
d)
We deduce from k∇ u k 2 L
2( S
d) + λ k u k 2 L
2( S
d) ≥ µ(λ) k u k 2 L
q( S
d) that k∇ u k 2 L
2( S
d) − µ(λ) k u k 2 L
q( S
d) ≥ − λ k u k 2 L
2( S
d)
2 4 6 8 10 12
2 4 6 8 10 12
2 4 6 8 10 12
2 4 6 8 10
µ(λ)
12λ(µ)
q = d = 3
J. Dolbeault Magnetic ground state & symmetry
A Keller-Lieb-Thirring inequality on the sphere
Let d ≥ 1, p ∈
max { 1, d/2 } , + ∞
and µ ∗ := d 2 (p − 1) Theorem (JD-Esteban-Laptev)
There exists a convex increasing function λ s.t. λ(µ) = µ if µ ∈ 0, µ ∗ and λ(µ) > µ if µ ∈ µ ∗ , + ∞
and, for any p < d/2,
| λ 1 ( − ∆ − V ) | ≤ λ k V k L
p( S
d)
∀ V ∈ L p (S d ) This estimate is optimal
For large values of µ, we have λ(µ) p−
d2= L 1 p−
d2
,d (κ q,d µ) p (1 + o(1))
If p = d /2 and d ≥ 3, the inequality holds with λ(µ) = µ iff µ ∈ [0, µ ∗ ]
J. Dolbeault Magnetic ground state & symmetry
A Keller-Lieb-Thirring inequality: second formulation
Let d ≥ 1, γ = p − d/2 Corollary (JD-Esteban-Laptev)
| λ 1 ( − ∆ − V ) | γ . L 1 γ,d ˆ
S
dV γ+
d2as µ = k V k L
γ+d2
( S
d) → ∞ if either γ > max { 0, 1 − d /2 } or γ = 1/2 and d = 1 However, if µ = k V k L
γ+d2( S
d) ≤ µ ∗ , then we have
| λ 1 ( − ∆ − V ) | γ+
d2≤ ˆ
S
dV γ+
d2for any γ ≥ max { 0, 1 − d/2 } and this estimate is optimal
L 1 γ,d is the optimal constant in the Euclidean one bound state ineq.
| λ 1 ( − ∆ − φ) | γ ≤ L 1 γ,d ˆ
R
dφ γ+ +
d2dx
J. Dolbeault Magnetic ground state & symmetry
References
JD, M. J. Esteban, and A. Laptev. Spectral estimates on the sphere. Analysis & PDE, 7 (2): 435-460, 2014
JD, M.J. Esteban, A. Laptev, M. Loss. Spectral properties of Schr¨ odinger operators on compact manifolds: Rigidity, flows,
interpolation and spectral estimates. Comptes Rendus Math´ematique, 351 (11-12): 437-440, 2013
J. Dolbeault Magnetic ground state & symmetry
Caffarelli-Kohn-Nirenberg, symmetry and symmetry breaking results, and weighted nonlinear flows
Joint work with M.J. Esteban and M. Loss
J. Dolbeault Magnetic ground state & symmetry
Critical Caffarelli-Kohn-Nirenberg inequality
Let D a,b := n
v ∈ L p R d , | x | −b dx
: | x | −a |∇ v | ∈ L 2 R d , dx o ˆ
R
d| v | p
| x | b p dx 2/p
≤ C a,b
ˆ
R
d|∇ v | 2
| x | 2 a dx ∀ v ∈ D a,b holds under conditions on a and b
p = 2 d
d − 2 + 2 (b − a) (critical case) B An optimal function among radial functions:
v ? (x ) =
1 + | x | (p−2) (a
c−a) −
p−22and C ? a,b = k | x | −b v ? k 2 p
k | x | −a ∇ v ? k 2 2
Question: C a,b = C ? a,b (symmetry) or C a,b > C ? a,b (symmetry breaking) ?
J. Dolbeault Magnetic ground state & symmetry
Critical CKN: range of the parameters
Figure: d = 3 ˆ
R
d| v | p
| x | b p dx 2/p
≤ C a,b
ˆ
R
d|∇ v | 2
| x | 2 a dx
a b
0 1
− 1
b = a
b = a+ 1
a =
d−22p
p = 2 d
d − 2 + 2 (b − a) a < a c := (d − 2)/2 a ≤ b ≤ a + 1 if d ≥ 3, a + 1/2 < b ≤ a + 1 if d = 1 and a < b ≤ a + 1 < 1,
p = 2/(b − a) if d = 2
[Il’in (1961)]
[Glaser, Martin, Grosse, Thirring (1976)]
[Caffarelli, Kohn, Nirenberg (1984)]
[F. Catrina, Z.-Q. Wang (2001)]
J. Dolbeault Magnetic ground state & symmetry
Linear instability of radial minimizers:
the Felli-Schneider curve
The Felli & Schneider curve b FS (a) := d ( a c − a )
2 p
(a c − a) 2 + d − 1 + a − a c
a b
0
[Smets], [Smets, Willem], [Catrina, Wang], [Felli, Schneider]
v 7→ C ? a,b ˆ
R
d|∇ v | 2
| x | 2 a dx − ˆ
R
d| v | p
| x | b p dx 2/p
is linearly instable at v = v ?
J. Dolbeault Magnetic ground state & symmetry
Symmetry versus symmetry breaking:
the sharp result in the critical case
[JD, Esteban, Loss (2016)]
a b
0
Theorem
Let d ≥ 2 and p < 2 ∗ . If either a ∈ [0, a c ) and b > 0, or a < 0 and b ≥ b FS (a), then the optimal functions for the critical
Caffarelli-Kohn-Nirenberg inequalities are radially symmetric
B
J. Dolbeault Magnetic ground state & symmetry
The symmetry proof in one slide
A change of variables: v( | x | α−1 x ) = w (x), D α v = α ∂v ∂s , 1 s ∇ ω v k v k L
2p,d−n( R
d) ≤ K α,n,p k D α v k ϑ L
2,d−n( R
d) k v k 1−ϑ L
p+1,d−n( R
d) ∀ v ∈ H p d−n,d−n (R d )
Concavity of the R´enyi entropy power: with L α = − D ∗ α D α = α 2 u 00 + n−1 s u 0
+ s 1
2∆ ω u and ∂u ∂t = L α u m
− dt d G [u(t , · )] ´
R
du m dµ 1−σ
≥ (1 − m) (σ − 1) ´
R
du m L α P −
´
Rd
´ u |D
αP|
2dµ
Rd
u
mdµ
2
dµ + 2 ´
R
dα 4 1 − 1 n
P 00 − P s
0− α
2(n−1) ∆
ωP s
22
+ 2α s
22∇ ω P 0 − ∇
ωs P
2 u m d µ + 2 ´
R
d(n − 2) α 2 FS − α 2
|∇ ω P | 2 + c(n , m , d) |∇ P
ω2P|
4u m d µ
Elliptic regularity and the Emden-Fowler transformation: justifying the integrations by parts
J. Dolbeault Magnetic ground state & symmetry
The variational problem on the cylinder
B With the Emden-Fowler transformation
v (r, ω) = r a−a
cϕ(s, ω) with r = | x | , s = − log r and ω = x r the variational problem becomes
Λ 7→ µ(Λ) := min
ϕ∈H
1(C)
k ∂ s ϕ k 2 L
2(C) + k∇ ω ϕ k 2 L
2(C) + Λ k ϕ k 2 L
2(C)
k ϕ k 2 L
p(C)
is a concave increasing function Restricted to symmetric functions, the variational problem becomes
µ ? (Λ) := min
ϕ∈H
1( R )
k ∂ s ϕ k 2 L
2( R
d) + Λ k ϕ k 2 L
2( R
d)
k ϕ k 2 L
p( R
d)
= µ ? (1) Λ α Symmetry means µ(Λ) = µ ? (Λ)
Symmetry breaking means µ(Λ) < µ ? (Λ)
J. Dolbeault Magnetic ground state & symmetry
Numerical results
20 40 60 80 100
10 20 30 40 50
symmetric non-symmetric asymptotic
bifurcation µ
Λ
αµ(Λ)
(Λ) = µ
(1) Λ
αParametric plot of the branch of optimal functions for p = 2.8, d = 5.
Non-symmetric solutions bifurcate from symmetric ones at a bifurcation point Λ
1computed by V. Felli and M. Schneider. The branch behaves for large values of Λ as shown by F. Catrina and Z.-Q. Wang
J. Dolbeault Magnetic ground state & symmetry
Three references
Lecture notes on Symmetry and nonlinear diffusion flows...
a course on entropy methods (see webpage)
[JD, Maria J. Esteban, and Michael Loss] Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs
... the elliptic point of view: Proc. Int. Cong. of Math., Rio de Janeiro, 3: 2279-2304, 2018.
[JD, Maria J. Esteban, and Michael Loss] Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization... the parabolic point of view
Journal of elliptic and parabolic equations, 2: 267-295, 2016.
J. Dolbeault Magnetic ground state & symmetry
Magnetic interpolation inequalities in the Euclidean space
B Three interpolation inequalities and their dual forms B Estimates in dimension d = 2 for constant magnetic fields
Lower estimates
Upper estimates and numerical results
A linear stability result (numerical) and an open question Warning: assumptions are not repeated
Estimates are given only in the case p > 2 but similar estimates hold in the other cases
Joint work with M.J. Esteban, A. Laptev and M. Loss
J. Dolbeault Magnetic ground state & symmetry
Magnetic Laplacian and spectral gap
In dimensions d = 2 and d = 3: the magnetic Laplacian is
− ∆ A ψ = − ∆ ψ − 2 i A · ∇ ψ + | A | 2 ψ − i (div A) ψ
where the magnetic potential (resp. field) is A (resp. B = curl A) and H 1 A (R d ) :=
ψ ∈ L 2 (R d ) : ∇ A ψ ∈ L 2 (R d ) , ∇ A := ∇ + i A Spectral gap inequality
k∇ A ψ k 2 L
2( R
d) ≥ Λ[B] k ψ k 2 L
2( R
d) ∀ ψ ∈ H 1 A ( R d ) Λ depends only on B = curl A
Assumption: equality holds for some ψ ∈ H 1 A ( R d ) If B is a constant magnetic field, Λ[B] = | B |
If d = 2, spec( − ∆ A ) = { (2j + 1) | B | : j ∈ N } is generated by the Landau levels. The Lowest Landau Level corresponds to j = 0
J. Dolbeault Magnetic ground state & symmetry
Magnetic interpolation inequalities
k∇ A ψ k 2 L
2( R
d) + α k ψ k 2 L
2( R
d) ≥ µ B (α) k ψ k 2 L
p( R
d) ∀ ψ ∈ H 1 A (R d ) for any α ∈ ( − Λ[B], + ∞ ) and any p ∈ (2, 2 ∗ ),
k∇ A ψ k 2 L
2( R
d) + β k ψ k 2 L
p( R
d) ≥ ν B (β) k ψ k 2 L
2( R
d) ∀ ψ ∈ H 1 A ( R d ) for any β ∈ (0, + ∞ ) and any p ∈ (1, 2)
k∇ A ψ k 2 L
2( R
d) ≥ γ ˆ
R
d| ψ | 2 log | ψ | 2 k ψ k 2 L
2( R
d)
!
dx + ξ B (γ) k ψ k 2 L
2( R
d)
(limit case corresponding to p = 2) for any γ ∈ (0, + ∞ )
C p :=
min u∈H
1( R
d)\{0}
k∇uk
2L2 (Rd)
+kuk
2L2 (Rd)
kuk
2Lp(Rd)
if p ∈ (2, 2 ∗ ) min u∈H
1( R
d)\{0}
k∇uk
2L2 (Rd)
+kuk
2Lp(Rd)
kuk
2L2 (Rd)
if p ∈ (1, 2) µ 0 (1) = C p if p ∈ (2, 2 ∗ ), ν 0 (1) = C p if p ∈ (1, 2)
ξ 0 (γ) = γ log π e 2 /γ
if p = 2
J. Dolbeault Magnetic ground state & symmetry
Technical assumptions
A ∈ L α loc ( R d ), α > 2 if d = 2 or α = 3 if d = 3 and
σ→+∞ lim σ d−2 ˆ
R
d| A(x) | 2 e −σ |x| dx = 0 if p ∈ (2, 2 ∗ )
σ→+∞ lim σ
d2−1
log σ ˆ
R
d| A(x) | 2 e −σ |x|
2dx = 0 if p = 2
σ→+∞ lim σ d−2 ˆ
|x|<1/σ | A(x ) | 2 dx if p ∈ (1, 2) These estimates can be found in [Esteban, Lions, 1989]
J. Dolbeault Magnetic ground state & symmetry
A statement
Theorem
p ∈ (2, 2 ∗ ): µ B is monotone increasing on ( − Λ[B], + ∞ ), concave and
α→(−Λ[B]) lim
+µ B (α) = 0 and lim
α→+∞ µ B (α) α
d−22−
dp= C p
p ∈ (1, 2): ν B is monotone increasing on (0, + ∞ ), concave and
β→0 lim
+ν B (β ) = Λ[B] and lim
β→+∞ ν B (β) β −
2p+d2(2−p)p= C p
ξ B is continuous on (0, + ∞ ), concave, ξ B (0) = Λ[B] and ξ B (γ) = d 2 γ log π γ e
2(1 + o (1)) as γ → + ∞ Constant magnetic fields: equality is achieved
Nonconstant magnetic fields: only partial answers are known
J. Dolbeault Magnetic ground state & symmetry
2 4 6 8 10 2
4 6 8
Figure: Case d = 2, p = 3, B = 1: plot of α 7→ (2 π)
2p−1µ
B(α)
J. Dolbeault Magnetic ground state & symmetry
0 1 2 3 4 5 6 0
2 4 6 8
Figure: Case d = 2, p = 1.4, B = 1: plot of β 7→ ν
B(β) The horizontal axis is measured in units of (2 π)
1−2pβ
J. Dolbeault Magnetic ground state & symmetry
Magnetic Keller-Lieb-Thirring inequalities
λ A,V is the principal eigenvalue of − ∆ A + V
α B : (0, + ∞ ) → ( − Λ, + ∞ ) is the inverse function of α 7→ µ B (α) Corollary
(i) For any q = p/(p − 2) ∈ (d /2, + ∞ ) and any potential V ∈ L q + ( R d ) λ A,V ≥ − α B ( k V k L
q( R
d) )
lim µ→0
+α B (µ) = Λ and lim µ→+∞ α B (µ) µ
d−2−22 (q+1)q= − C
2 (q+1) d−2−2q
p
(ii) For any q = p/(2 − p) ∈ (1, + ∞ ) and any 0 < W −1 ∈ L q ( R d ) λ A,W ≥ ν B
k W −1 k −1 L
q( R
d)
(iii) For any γ > 0 and any W ≥ 0 s.t. e −W /γ ∈ L 1 (R d ) λ A,W ≥ ξ B (γ) − γ log ´
R
de −W /γ dx B
J. Dolbeault Magnetic ground state & symmetry
Proofs
J. Dolbeault Magnetic ground state & symmetry
Interpolation without magnetic field...
Assume that p > 2 and let C p denote the best constant in k∇ u k 2 L
2( R
d) + k u k 2 L
2( R
d) ≥ C p k u k 2 L
p( R
d) ∀ u ∈ H 1 ( R d ) By scaling, if we test the inequality by u · /λ
, we find that
k∇ u k 2 L
2( R
d) +λ 2 k u k 2 L
2( R
d) ≥ C p λ 2− d (1−
2p) k u k 2 L
p( R
d) ∀ u ∈ H 1 (R d ) ∀ λ > 0 An optimization on λ > 0 shows that the best constant in the
scale-invariant inequality k∇ u k d (1−
2 p
)
L
2( R
d) k u k 2−d (1−
2 p
)
L
2( R
d) ≥ S p k u k 2 L
p( R
d) ∀ u ∈ H 1 ( R d ) is given by
S p = 2 1 p (2 p − d (p − 2)) 1−d
p−22p(d (p − 2))
d(p−2)2pC p
J. Dolbeault Magnetic ground state & symmetry
... and with magnetic field
Proposition
Let d = 2 or 3. For any p ∈ (2, + ∞ ), any α > − Λ = − Λ[B] < 0
µ B (α) ≥ µ interp (α) :=
S p (α + Λ) Λ −d
p−22pif α ∈ h
− Λ, Λ (2p−d d (p−2) (p−2)) i C p α 1−d
p−22pif α ≥ Λ (2p−d d (p−2) (p−2))
Diamagnetic inequality: k∇| ψ |k L
2( R
d) ≤ k∇ A ψ k L
2( R
d)
Non-magnetic inequality with λ = α+Λ 1−t t , t ∈ [0, 1]
k∇ A ψ k 2 L
2( R
d) + α k ψ k 2 L
2( R
d) ≥ t
k∇ A ψ k 2 L
2( R
d) − Λ k ψ k 2 L
2( R
d)
+ (1 − t)
k∇| ψ |k L
2( R
d) + α + Λ t
1 − t k ψ k 2 L
2( R
d)
≥ C p (1 − t)
d(p−2)2p(α + t Λ) 1−d
p−22pk ψ k 2 L
p( R
d)
and optimize on t ∈ [max { 0, − α/Λ } , 1]
J. Dolbeault Magnetic ground state & symmetry
The special case of constant magnetic field in dimension d = 2
J. Dolbeault Magnetic ground state & symmetry
Constant magnetic field, d = 2...
Assume that B = (0, B) is constant, d = 2 and choose A 1 = B 2 x 2 , A 2 = − B 2 x 1 ∀ x = (x 1 , x 2 ) ∈ R 2 Proposition
[Loss, Thaller, 1997] Consider a constant magnetic field with field strength B in two dimensions. For every c ∈ [0, 1], we have
ˆ
R
2|∇ A ψ | 2 dx ≥ 1 − c 2 ˆ
R
2|∇ ψ | 2 dx + c B ˆ
R
2ψ 2 dx and equality holds with ψ = u e iS and u > 0 if and only if
− ∂ 2 u 2 , ∂ 1 u 2
= 2 u 2
c (A + ∇ S)
J. Dolbeault Magnetic ground state & symmetry
... a computation (d = 2, constant magnetic field)
ˆ
R
2|∇ A ψ | 2 dx = ˆ
R
2|∇ u | 2 dx + ˆ
R
2| A + ∇ S | 2 u 2 dx
= 1 − c 2 ˆ
R
2|∇ u | 2 dx + ˆ
R
2c 2 |∇ u | 2 + | A + ∇ S | 2 u 2 dx
| {z }
≥ ´
R2
2 c |∇u| |A+∇S|u dx
with equality only if c |∇ u | = | A + ∇ S | u
2 |∇ u | | A + ∇ S | u = |∇ u 2 | | A + ∇ S | ≥ ∇ u 2 ⊥
· (A + ∇ S) where ∇ u 2 ⊥
:= − ∂ 2 u 2 , ∂ 1 u 2 Equality case: − ∂ 2 u 2 , ∂ 1 u 2
= γ (A + ∇ S) for γ = 2 u 2 /c Integration by parts yields
ˆ
R
2c 2 |∇ u | 2 + | A + ∇ S | 2 u 2
dx ≥ B c ˆ
R
2u 2 dx
J. Dolbeault Magnetic ground state & symmetry
... a lower estimate (d = 2, constant magnetic field)
Proposition
Consider a constant magnetic field with field strength B in two dimensions. Given any p ∈ (2, + ∞ ), and any α > − B, we have
µ B (α) ≥ C p 1 − c 2 1−
p2(α + c B )
2p=: µ LT (α) with
c = c(p, η) =
p η 2 + p − 1 − η
p − 1 = 1
η + p
η 2 + p − 1 ∈ (0, 1) and η = α (p − 2)/(2 B)
J. Dolbeault Magnetic ground state & symmetry
Upper estimate (1): d = 2, constant magnetic field
For every integer k ∈ N we introduce the special symmetry class ψ(x) =
x
2+i x
1|x|
k
v( | x | ) ∀ x = (x 1 , x 2 ) ∈ R 2 ( C k ) [Esteban, Lions, 1989]: if ψ ∈ C k , then
1 2 π
ˆ
R
2|∇ A ψ | 2 dx = ˆ +∞
0 | v 0 | 2 r dr + ˆ +∞
0 k r − B r 2
2
| v | 2 r dr and optimality is achieved in C k
Test function v σ (r) = e − r
2/(2 σ) : an optimization on σ > 0 provides an explicit expression of µ Gauss (α) such that
Proposition If p > 2, then
µ B (α) ≤ µ Gauss (α) ∀ α > − Λ[B]
This estimate is not optimal because v σ does not solve the Euler-Lagrange equations
J. Dolbeault Magnetic ground state & symmetry
Upper estimate (2): d = 2, constant magnetic field
A more numerical point of view . The Euler-Lagrange equation in C 0 is
− v 00 − v 0 r +
B
24 r 2 + α
v = µ EL (α) ˆ +∞
0 | v | p r dr
p2−1
| v | p−2 v We can restrict the problem to positive solutions such that
µ EL (α) =
ˆ +∞
0 | v | p r dr 1−
p2and then we have to solve the reduced problem
− v 00 − v 0 r +
B
24 r 2 + α
v = | v | p−2 v
J. Dolbeault Magnetic ground state & symmetry
Numerical results and the symmetry issue
J. Dolbeault Magnetic ground state & symmetry
2 4 6 8 10
-1 1 2 3
0.004 0.003 0.002 0.001 0.001 0.002 0.003
-0.02
-0.04
-0.06
-0.08
2 4
-1
6
Figure: Case d = 2, p = 3, B = 1 Upper estimates: α 7→ µ
Gauss(α), µ
EL(α) Lower estimates: α 7→ µ
interp(α), µ
LT(α)
The exact value associated with µ
Blies in the grey area.
Plots represent the curves log
10(µ/µ
EL) B
J. Dolbeault Magnetic ground state & symmetry
Asymptotics (1): Lowest Landau Level
Proposition
Let d = 2 and consider a constant magnetic field with field strength B. If ψ α is a minimizer for µ B (α) such that k ψ α k L
p( R
d) = 1, then there exists a non trivial ϕ α ∈ LLL such that
α→(−B) lim
+k ψ α − ϕ α k H
1A( R
2) = 0
Let ψ α ∈ H 1 A (R 2 ) be an optimal function such that k ψ α k L
p( R
d) = 1 and let us decompose it as ψ α = ϕ α + χ α , where ϕ α ∈ LLL and χ α is in the orthogonal of LLL
µ B (α) ≥ (α+B) k ϕ α k 2 L
2( R
d) +(α+3B) k χ α k 2 L
2( R
d) ≥ (α+3B) k χ α k 2 L
2( R
d) ∼ 2B k χ α k 2 L
2( R
d)
as α → ( − B) + because k∇ χ α k 2 L
2( R
d) ≥ 3B k χ α k 2 L
2( R
d)
Since lim α→(−B)
+µ B (α) = 0, lim α→(−B)
+k χ α k L
2( R
d) = 0 and
µ B (α) = (α+B) k ϕ α k 2 L
2( R
d) + k∇ A χ α k 2 L
2( R
d) +α k χ α k 2 L
2( R
d) ≥ 2 3 k∇ A χ α k 2 L
2( R
d)
concludes the proof
J. Dolbeault Magnetic ground state & symmetry
Asymptotics (2): semi-classical regime
Let us consider the small magnetic field regime. We assume that the magnetic potential is given by
A 1 = B 2 x 2 , A 2 = − B 2 x 1 ∀ x = (x 1 , x 2 ) ∈ R 2
if d = 2. In dimension d = 3, we choose A = B 2 ( − x 2 , x 1 , 0) and observe that the constant magnetic field is B = (0, 0, B), while the spectral gap is Λ[B] = B .
Proposition
Let d = 2 or 3 and consider a constant magnetic field B of intensity B with magnetic potential A
For any p ∈ (2, 2 ∗ ) and any fixed α and µ > 0, we have
ε→0 lim
+µ εB (α) = C p α
dp−
d−22Consider any function ψ ∈ H 1 A (R d ) and let ψ(x ) = χ( √ ε x ),
√ ε A x / √ ε
= A(x) with our conventions on A
J. Dolbeault Magnetic ground state & symmetry
Numerical stability of radial optimal functions
Let us denote by ψ 0 an optimal function in ( C 0 ) such that
− ψ 0 00 − ψ 0 0 r +
B
24 r 2 + α
ψ 0 = | ψ 0 | p−2 ψ 0
and consider the test function
ψ ε = ψ 0 + ε e i θ v where v = v (r ) and e i θ = (x 1 + i x 2 )/r As ε → 0 + , the leading order term is
2 π ˆ
R
2| v 0 | 2 dx + ˆ
R
21 r − B r 2
2 + α
| v | 2 dx − p 2
ˆ +∞
0 | ψ 0 | p−2 v 2 r dr
ε 2
and we have to solve the eigenvalue problem
− v 00 − v 0 r +
1 r − B r 2
2 + α
v − p 2 | ψ 0 | p−2 v = µ v
J. Dolbeault Magnetic ground state & symmetry
2 4 6 8 1
2 3 4
Figure: Case p = 3 and B = 1: plot of the eigenvalue µ as a function of α A careful investigation shows that µ is always positive, including in the limiting case as α → (−B )
+, thus proving the numerical stability of the optimal function in C
0with respect to perturbations in C
1J. Dolbeault Magnetic ground state & symmetry
An open question of symmetry
[Bonheure, Nys, Van Schaftingen, 2016] for a fixed α > 0 and for B small enough, the optimal functions are radially symmetric functions, i.e., belong to C 0
This regime is equivalent to the regime as α → + ∞ for a given B, at least if the magnetic field is constant
Numerically our upper and lower bounds are (in dimension d = 2, for a constant magnetic field) numerically extremely close
The optimal function in C 0 is linearly stable with respect to perturbations in C 1
B Prove that the optimality case is achieved among radial function if d = 2 and B is a constant magnetic field
J. Dolbeault Magnetic ground state & symmetry
Reference
JD, M.J. Esteban, A. Laptev, M. Loss. Interpolation inequalities and spectral estimates for magnetic operators. Annales Henri Poincar´e, 19 (5): 1439-1463, May 2018
J. Dolbeault Magnetic ground state & symmetry
Magnetic rings
B A magnetic interpolation inequality on S 1 : with p > 2 k ψ 0 + i a ψ k 2 L
2( S
1) + α k ψ k 2 L
2( S
1) ≥ µ a,p (α) k ψ k 2 L
p( S
1)
B Consequences
A Keller-Lieb-Thirring inequality
A new Hardy inequality for Aharonov-Bohm magnetic fields in R 2 Joint work with M.J. Esteban, A. Laptev and M. Loss
J. Dolbeault Magnetic ground state & symmetry
Magnetic flux, a reduction
Assume that a : R → R is a 2π-periodic function such that its restriction to ( − π, π] ≈ S 1 is in L 1 (S 1 ) and define the space
X a :=
ψ ∈ C per ( R ) : ψ 0 + i a ψ ∈ L 2 ( S 1 )
A standard change of gauge (see e.g. [Ilyin, Laptev, Loss, Zelik, 2016])
ψ(s) 7→ e i
´
s−π
(a(s)−¯ a) dσ ψ(s) where a ¯ := ´ π
−π a(s) dσ is the magnetic flux, reduces the problem to a is a constant function
For any k ∈ Z , ψ by s 7→ e iks ψ(s) shows that µ a,p (α) = µ k+a,p (α) a ∈ [0, 1]
µ a,p (α) = µ 1−a,p (α) because
| ψ 0 + i a ψ | 2 = | χ 0 + i (1 − a) χ | 2 =
ψ 0 − i a ψ
2 if χ(s) = e −is ψ(s) a ∈ [0, 1/2]
J. Dolbeault Magnetic ground state & symmetry
Optimal interpolation
We want to characterize the optimal constant in the inequality k ψ 0 + i a ψ k 2 L
2( S
1) + α k ψ k 2 L
2( S
1) ≥ µ a,p (α) k ψ k 2 L
p( S
1)
written for any p > 2, a ∈ (0, 1/2], α ∈ ( − a 2 , + ∞ ), ψ ∈ X a µ a,p (α) := inf
ψ∈X
a\{0}
´ π
−π | ψ 0 + i a ψ | 2 + α | ψ | 2 dσ k ψ k 2 L
p( S
1)
p = − 2 = 2 d /(d − 2) with d = 1 [Exner, Harrell, Loss, 1998]
p = + ∞ [Galunov, Olienik, 1995] [Ilyin, Laptev, Loss, Zelik, 2016]
lim α→− a
2µ a,p (α) = 0 [JD, Esteban, Laptev, Loss, 2016]
Using a Fourier series ψ(s) = P
k ∈ Z ψ k e iks , we obtain that k ψ 0 + i a ψ k 2 L
2( S
1) = X
k ∈ Z
(a + k) 2 | ψ k | 2 ≥ a 2 k ψ k 2 L
2( S
1)
ψ 7→ k ψ 0 + i a ψ k 2 L
2( S
1) + α k ψ k 2 L
2( S
1) is coercive for any α > − a 2
J. Dolbeault Magnetic ground state & symmetry
An interpolation result for the magnetic ring
Theorem
For any p > 2, a ∈ R , and α > − a 2 , µ a,p (α) is achieved and
(i) if a ∈ [0, 1/2] and a 2 (p + 2) + α (p − 2) ≤ 1, then µ a,p (α) = a 2 + α and equality is achieved only by the constant functions
(ii) if a ∈ [0, 1/2] and a 2 (p + 2) + α (p − 2) > 1, then µ a,p (α) < a 2 + α and equality is not achieved by the constant functions
If α > − a 2 , a 7→ µ a,p (α) is monotone increasing on (0, 1/2)
-0.2 -0.1 0.1 0.2 0.3 0.4
0.1 0.2 0.3 0.4 0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.2 0.4 0.6 0.8 1.0 1.2
Figure: α 7→ µ
a,p(α) with p = 4 and (left) a = 0.45 or (right) a = 0.2
J. Dolbeault Magnetic ground state & symmetry
The proof: how to eliminate the phase
J. Dolbeault Magnetic ground state & symmetry
Reformulations of the interpolation problem (1/3)
Any minimizer ψ ∈ X a of µ a,p (α) satisfies the Euler-Lagrange equation (H a + α) ψ = | ψ | p−2 ψ , H a ψ = −
d ds + i a
2
ψ (*)
up to a multiplication by a constant and v (s) = ψ(s) e ias satisfies the condition
v (s + 2π) = e 2iπa v (s) ∀ s ∈ R Hence
µ a,p (α) = min
v∈Y
a\{0} Q p,α [v]
where Y a :=
v ∈ C ( R ) : v 0 ∈ L 2 ( S 1 ) , (*) holds and Q p,α [v] := k v 0 k 2 L
2( S
1) + α k v k 2 L
2( S
1)
k v k 2 L
p( S
1)
J. Dolbeault Magnetic ground state & symmetry
Reformulations of the interpolation problem (2/3)
With v = u e iφ the boundary condition becomes
u(π) = u( − π) , φ(π) = 2π (a + k ) + φ( − π) (**) for some k ∈ Z , and k v 0 k 2 L
2( S
1) = k u 0 k 2 L
2( S
1) + k u φ 0 k 2 L
2( S
1)
Hence
µ a,p (α) = min
(u,φ)∈Z
a\{0}
k u 0 k 2 L
2( S
1) + k u φ 0 k 2 L
2( S
1) + α k u k 2 L
2( S
1)
k u k 2 L
p( S
1)
where Z a :=
(u , φ) ∈ C( R ) 2 : u 0 , u φ 0 ∈ L 2 ( S 1 ) , (**) holds
J. Dolbeault Magnetic ground state & symmetry
Reformulations of the interpolation problem (3/3)
We use the Euler-Lagrange equations
− u 00 + | φ 0 | 2 u + α u = | u | p−2 u and (φ 0 u 2 ) 0 = 0
Integrating the second equation, and assuming that u never vanishes , we find a constant L such that φ 0 = L / u 2 . Taking (*) into account, we deduce from
L ˆ π
−π
ds u 2 =
ˆ π
−π
φ 0 ds = 2π (a + k ) that
k u φ 0 k 2 L
2( S
1) = L 2 ˆ π
−π
dσ
u 2 = (a + k) 2 k u −1 k 2 L
2( S
1)
Hence
φ(s) − φ(0) = a + k k u −1 k 2 L
2( S
1)
ˆ s
−π
ds u 2
J. Dolbeault Magnetic ground state & symmetry
Let us define
Q a,p,α [u] := k u 0 k 2 L
2( S
1) + a 2 k u −1 k −2 L
2( S
1) + α k u k 2 L
2( S
1)
k u k 2 L
p( S
1)
Lemma
For any a ∈ (0, 1/2), p > 2, α > − a 2 , µ a,p (α) = min
u∈H
1( S
1)\{0} Q a,p,α [u]
is achieved by a function u > 0
J. Dolbeault Magnetic ground state & symmetry
Proofs
The existence proof is done on the original formulation of the problem using the diamagnetic inequality
To prove that | ψ | 6 = 0, we use ψ(s) e ias = v 1 (s) + i v 2 (s), solves
− v j 00 + α v j = (v 1 2 + v 2 2 )
p2−1 v j , j = 1 , 2
and the Wronskian w = (v 1 v 2 0 − v 1 0 v 2 ) is constant so that ψ(s) = 0 is incompatible with the twisted boundary condition
if a 2 (p + 2) + α (p − 2) ≤ 1, then µ a,p (α) = a 2 + α because
k u 0 k 2 L
2( S
1) +a 2 k u −1 k −2 L
2( S
1) +α k u k 2 L
2( S
1) = (1 − 4 a 2 ) k u 0 k 2 L
2( S
1) +α k u k 2 L
2( S
1)
+ 4 a 2
k u 0 k 2 L
2( S
1) + 1 4 k u −1 k 2 L
2( S
1)
if a 2 (p + 2) + α (p − 2) > 1, the test function u ε := 1 + ε w 1
Q a,p,α [u ε ] = a 2 + α + 1 − a 2 (p + 2) − α (p − 2)
ε 2 + o (ε 2 ) proves the linear instability of the constants and µ a,p (α) < a 2 + α
J. Dolbeault Magnetic ground state & symmetry
Q a,p,α [u] := ku
0
k
2L2 (S1 )+a
2ku
−1k
−2L2 (S1 )+α kuk
2L2 (S1 )kuk
2Lp(S1 )
,
µ a,p (α) = min
u∈H
1( S
1)\{0} Q a,p,α [u]
Q p,α [u] = Q a=0,p,α [u] , ν p (α) := inf
v ∈H
10( S
1)\{0} Q p,α [v ] Proposition
∀ p > 2, α > − a 2 , we have µ a,p (α) < µ 1/2,p (α) ≤ ν p (α) = µ 1/2,p (α)
0.5 1.0 1.5 2.0 2.5 3.0
0.1 0.2 0.3 0.4 0.5 0.6
Figure: p = 4, α = 0, a = 0.40, 0.41,. . . 0.49; u
00+ u
p−1= 0
J. Dolbeault Magnetic ground state & symmetry
Consequences: Keller-Lieb-Thirring inequalities and Hardy inequalities for Aharonov-Bohm magnetic fields
J. Dolbeault Magnetic ground state & symmetry
A Keller-Lieb-Thirring inequality
Magnetic Schr¨odinger operator H a − ϕ = − ds d + i a 2 ψ − ϕ The function α 7→ µ a,p (α) is monotone increasing, concave, and therefore has an inverse, denoted by α a,p : R + → ( − a 2 , + ∞ ), which is monotone increasing, and convex
Corollary
Let p > 2, a ∈ [0, 1/2], q = p/(p − 2) and assume that ϕ is a non-negative function in L q (S 1 ). Then
λ 1 (H a − ϕ) ≥ − α a,p k ϕ k L
q( S
1)
and α a,p (µ) = µ − a 2 iff 4 a 2 + µ (p − 2) ≤ 1 (optimal ϕ is constant) Equality is achieved
J. Dolbeault Magnetic ground state & symmetry
Aharonov-Bohm magnetic fields
On the two-dimensional Euclidean space R 2 , let us introduce the polar coordinates (r, ϑ) ∈ [0, + ∞ ) × S 1 of x ∈ R 2 and consider a magnetic potential a in a transversal (Poincar´e) gauge, or Poincar´e gauge
(a, e r ) = 0 and (a, e ϑ ) = a ϑ (r , ϑ) Magnetic Schr¨ odinger energy
ˆ
R
2| (i ∇ +a) Ψ | 2 dx = ˆ +∞
0
ˆ π
−π
| ∂ r Ψ | 2 + 1
r 2 | ∂ ϑ Ψ + i r a ϑ Ψ | 2
r dϑ d r Aharonov-Bohm magnetic fields: a ϑ (r, ϑ) = a/r for some constant a ∈ R (a is the magnetic flux), with magnetic field b = curl a ˆ
R
2| (i ∇ +a) Ψ | 2 dx ≥ τ ˆ
R
2ϕ(x/ | x | )
| x | 2 | Ψ | 2 dx ∀ ϕ ∈ L q (S 1 ) , q ∈ (1, + ∞ )
= ⇒ τ = τ a, k ϕ k L
q( S
1)
?
J. Dolbeault Magnetic ground state & symmetry
Hardy inequalities
[Hoffmann-Ostenhof, Laptev, 2015] proved Hardy’s inequality ˆ
R
d|∇ Ψ | 2 dx ≥ τ ˆ
R
dϕ(x/ | x | )
| x | 2 | Ψ | 2 dx
where the constant τ depends on the value of k ϕ k L
q( S
d−1) and d ≥ 3 Aharonov-Bohm vector potential in dimension d = 2
a(x) = a x 2
| x | 2 , − x 1
| x | 2
, x = (x 1 , x 2 ) ∈ R 2 , a ∈ R and recall the inequality [Laptev, Weidl, 1999]
ˆ
R
2| (i ∇ + a) Ψ | 2 dx ≥ min
k ∈ Z
(a − k ) 2 ˆ
R
2| Ψ |
| x | 2 dx
J. Dolbeault Magnetic ground state & symmetry
A new Hardy inequality
ˆ
R
2| (i ∇ +a) Ψ | 2 dx ≥ τ ˆ
R
2ϕ(x/ | x | )
| x | 2 | Ψ | 2 d x ∀ ϕ ∈ L q ( S 1 ) , q ∈ (1, + ∞ ) Corollary
Let p > 2, a ∈ [0, 1/2], q = p/(p − 2) and assume that ϕ is a
non-negative function in L q ( S 1 ). Then the inequality holds with τ > 0 given by
α a,p τ k ϕ k L
q( S
1)
= 0
Moreover, τ = a 2 / k ϕ k L
q( S
1) if 4 a 2 + k ϕ k L
q( S
1) (p − 2) ≤ 1
For any a ∈ (0, 1/2), by taking ϕ constant, small enough in order that 4 a 2 + k ϕ k L
q( S
1) (p − 2) ≤ 1, we recover the inequality
ˆ
R
2| (i ∇ + a) Ψ | 2 dx ≥ a 2 ˆ
R
2| Ψ | 2
| x | 2 dx
J. Dolbeault Magnetic ground state & symmetry
Proofs (Keller-Lieb-Thirring inequality)
B H¨ older’s inequality
k ψ 0 + i a ψ k 2 L
2( S
d) − ˆ π
−π
ϕ | ψ | 2 dσ ≥ k ψ 0 + i a ψ k 2 L
2( S
d) − µ k ψ k 2 L
p( S
d)
where µ = k ϕ k L
q( S
d) and 1 q + 2 p = 1: choose µ a,p (α) = µ k ψ 0 + i a ψ k 2 L
2( S
d) − µ k ψ k 2 L
p( S
d) ≥ − α k ψ k 2 L
2( S
d)
J. Dolbeault Magnetic ground state & symmetry
Proofs (Hardy inequality)
Let τ ≥ 0, x = (r , ϑ) ∈ R 2 be polar coordinates in R 2 ˆ
R
2| (i ∇ + a) Ψ | 2 − τ ϕ
| x | 2 | Ψ | 2
dx
= ˆ ∞
0
ˆ
S
1r | ∂ r Ψ | 2
| {z }
≥0
+ 1
r | ∂ ϑ Ψ + i a Ψ | 2 − τ ϕ r | Ψ | 2
dϑ dr
≥ λ 1 (H a − τ ϕ) ˆ ∞
0
ˆ
S
11
r | Ψ | 2 dϑ dr
≥ − α a,p (τ k ϕ k L
q( S
d) ) ˆ ∞
0
ˆ
S
11
r | Ψ | 2 dϑ If τ = 0, then α a,p (τ k ϕ k L
q( S
d) ) = α a,p (0) = − a 2
α a,p (τ k ϕ k L
q( S
d) ) > 0 for τ large
= ⇒ ∃ ! τ > 0 such that α a,p (τ k ϕ k L
q( S
d) ) = 0
J. Dolbeault Magnetic ground state & symmetry
Comments
B The region a 2 (p + 2) + α (p − 2) < 1 is exactly the set where the constant functions are linearly stable critical points
B The proof of the rigidity result is based
- neither on the carr´ e du champ method, at least directly
- nor on a Fourier representation of the operator as it was the case in earlier proofs (p = + ∞ , or p > 2 and α = 0)
B Magnetic rings: see [Bonnaillie-No¨el, H´erau, Raymond, 2017]
B Deducing Hardy’s inequality applied with Aharonov-Bohm magnetic fields from a Keller-Lieb-Thirring inequality is an extension of
[Hoffmann-Ostenhof, Laptev, 2015] to the magnetic case B Our results are not limited to the semi-classical regime
J. Dolbeault Magnetic ground state & symmetry
Symmetry in Aharonov-Bohm magnetic fields
Aharonov-Bohm effect
Interpolation and Keller-Lieb-Thirring inequalities in R 2 B Statements
B Constants and numerics
Symmetry and symmetry breaking
Joint work with D. Bonheure, M.J. Esteban, A. Laptev, & M. Loss
J. Dolbeault Magnetic ground state & symmetry
Aharonov-Bohm effect
A major difference between classical mechanics and quantum mechanics is that particles are described by a non-local object, the wave function. In 1959 Y. Aharonov and D. Bohm proposed a series of experiments intended to put in evidence such phenomena which are nowadays called Aharonov-Bohm effects
One of the proposed experiments relies on a long, thin solenoid which produces a magnetic field such that the region in which the magnetic field is non-zero can be approximated by a line in dimension d = 3 and by a point in dimension d = 2
B [Physics today, 2009] “The notion, introduced 50 years ago, that electrons could be affected by electromagnetic potentials without coming in contact with actual force fields was received with a skepticism that has spawned a flourishing of experimental tests and expansions of the original idea.” Problem solved by considering appropriate weak solutions !
B Is the wave function a physical object or is its modulus ? Decisive experiments have been done only 20 years ago
J. Dolbeault Magnetic ground state & symmetry
The interpolation inequality
Let us consider an Aharonov-Bohm vector potential A(x) = a
| x | 2 (x 2 , − x 1 ) , x = (x 1 , x 2 ) ∈ R 2 \ { 0 } , a ∈ R Magnetic Hardy inequality [Laptev, Weidl, 1999]
ˆ
R
2|∇ A ψ | 2 dx ≥ min
k∈Z (a − k ) 2 ˆ
R
2| ψ | 2
| x | 2 dx where ∇ A ψ := ∇ ψ + i A ψ, so that, with ψ = | ψ | e iS
ˆ
R
2|∇ A ψ | 2 dx = ˆ
R
2(∂ r | ψ | ) 2 + (∂ r S ) 2 | ψ | 2 + 1
r 2 (∂ θ S + A) 2 | ψ | 2
dx Magnetic interpolation inequality
ˆ
R
2|∇ A ψ | 2 dx + λ ˆ
R
2| ψ | 2
| x | 2 dx ≥ µ(λ) ˆ
R
2| ψ | p
| x | 2 dx 2/p
B Symmetrization: [Erd¨ os, 1996], [Boulenger, Lenzmann], [Lenzmann, Sok]
J. Dolbeault Magnetic ground state & symmetry