Control Theory for linear Schrödinger
equations
Fabricio Macià
Universidad Politécnica de Madrid
Marrakesh Workshop On
Control, Inverse Problems and Stabilization of Infinite dimensional Systems
12-16 December 2016
Concurso ETSIN
The Schrödinger equation

 i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0, (t, x) ∈ R × Ω,

ψ(0, ·) = ψ 0 ,
||ψ 0 ||L2 (Ω) = 1,
where:

is a compact Riemannian manifold and




∆x is the Laplace-Beltrami operator,



or
Ω=




is a bounded smooth domain of Rd



with Dirichlet BC: ψ|R×Ω = 0,
and
V ∈ L∞ (R × Ω),
V real-valued.
Concurso ETSIN
The Schrödinger equation

 i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0, (t, x) ∈ R × Ω,

ψ(0, ·) = ψ 0 ,
||ψ 0 ||L2 (Ω) = 1,
where:

is a compact Riemannian manifold and




∆x is the Laplace-Beltrami operator,



or
Ω=




is a bounded smooth domain of Rd



with Dirichlet BC: ψ|R×Ω = 0,
and
V ∈ L∞ (R × Ω),
V real-valued.
Concurso ETSIN
Physical meaning of ψ (t, x)
The probability that an electron e(t) moving in Ω under the influence
of the potential V (t, x) can be found in some region U ⊂ Ω at time t is
given by:
Z
|ψ (t, x)|2 dx.
Prob (e (t) ∈ U ) =
U
This is consistent with the model because the L2 -norm is preserved
under the evolution:
Z
|ψ (t, x)|2 dx = ||ψ 0 ||2L2 (Ω) = 1.
Ω
Concurso ETSIN
Physical meaning of ψ (t, x)
The probability that an electron e(t) moving in Ω under the influence
of the potential V (t, x) can be found in some region U ⊂ Ω at time t is
given by:
Z
|ψ (t, x)|2 dx.
Prob (e (t) ∈ U ) =
U
This is consistent with the model because the L2 -norm is preserved
under the evolution:
Z
|ψ (t, x)|2 dx = ||ψ 0 ||2L2 (Ω) = 1.
Ω
Concurso ETSIN
Controllability problem
Chose ω ⊂ Ω open and T > 0.
(Null) Controllability from ω at time T
Given any Ψ0 ∈ L2 (Ω) of norm one, is it always possible to find a
forcing term
u ∈ L2 ((0, T ) × ω) (the control),
such that the solution Ψ of:
(
i∂t Ψ (t, x) + ∆x Ψ (t, x) − V (t, x) Ψ (t, x) = 1ω (x)u(t, x),
Ψ(0, ·) = Ψ0 ,
satisfies
Ψ(T, ·) = 0?
Concurso ETSIN
Controllability is equivalent to Observability
Controllability from ω at time T is equivalent to the following
observability property:
Observability from ω at time T
There exist a constant C = CT,ω > 0 such that, for every ψ 0 ∈ L2 (Ω):
||ψ 0 ||2L2 (Ω)
Z
T
Z
≤C
0
|ψ(t, x)|2 dx dt,
ω
where ψ is the solution of:
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 .
Concurso ETSIN
Proof: J.-L. Lion’s H.U.M.
The result follows if one is able to prove that the operator:
Λ : L2 ((0, T ) × ω) −→ L2 (Ω)
u
7−→ Ψ(0, ·),
where
(
i∂t Ψ (t, x) + ∆x Ψ (t, x) − V (t, x) Ψ (t, x) = 1ω (x)u(t, x),
Ψ(T, ·) = 0,
is surjective.
Closed graph theorem: surjectivity of Λ is equivalent to
solvability with estimate of the adjoint equation.
Λ∗ ψ 0 = u,
||ψ 0 ||L2 (Ω) ≤ C||u||L2 ((0,T )×ω) .
Concurso ETSIN
Proof: J.-L. Lion’s H.U.M.
The result follows if one is able to prove that the operator:
Λ : L2 ((0, T ) × ω) −→ L2 (Ω)
u
7−→ Ψ(0, ·),
where
(
i∂t Ψ (t, x) + ∆x Ψ (t, x) − V (t, x) Ψ (t, x) = 1ω (x)u(t, x),
Ψ(T, ·) = 0,
is surjective.
Closed graph theorem: surjectivity of Λ is equivalent to
solvability with estimate of the adjoint equation.
Λ∗ ψ 0 = u,
||ψ 0 ||L2 (Ω) ≤ C||u||L2 ((0,T )×ω) .
Concurso ETSIN
But a direct computation shows that:
Λ∗ ψ 0 = iψ|(0,T )×ω ,
where
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 .
Hence controllability (surjectivity of Λ) is equivalent to the
existence of C > 0 such that:
||ψ 0 ||L2 (Ω) ≤ C||Λ∗ ψ 0 ||L2 ((0,T )×ω) = C||ψ||L2 ((0,T )×ω) .
Concurso ETSIN
But a direct computation shows that:
Λ∗ ψ 0 = iψ|(0,T )×ω ,
where
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 .
Hence controllability (surjectivity of Λ) is equivalent to the
existence of C > 0 such that:
||ψ 0 ||L2 (Ω) ≤ C||Λ∗ ψ 0 ||L2 ((0,T )×ω) = C||ψ||L2 ((0,T )×ω) .
Concurso ETSIN
Main goal
Understand under which conditions on:
the geometry of Ω;
the structure of the potential V (t, x);
the control (open) set ω and T ;
the observability estimate:
||ψ 0 ||2L2 (Ω)
Z
T
Z
≤C
0
|ψ(t, x)|2 dx dt,
(OV,ω,T )
ω
holds for some constant C > 0 and every solution to:
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 ∈ L2 (Ω).
Concurso ETSIN
Main goal
Understand under which conditions on:
the geometry of Ω;
the structure of the potential V (t, x);
the control (open) set ω and T ;
the observability estimate:
||ψ 0 ||2L2 (Ω)
Z
T
Z
≤C
0
|ψ(t, x)|2 dx dt,
(OV,ω,T )
ω
holds for some constant C > 0 and every solution to:
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 ∈ L2 (Ω).
Concurso ETSIN
Main goal
Understand under which conditions on:
the geometry of Ω;
the structure of the potential V (t, x);
the control (open) set ω and T ;
the observability estimate:
||ψ 0 ||2L2 (Ω)
Z
T
Z
≤C
0
|ψ(t, x)|2 dx dt,
(OV,ω,T )
ω
holds for some constant C > 0 and every solution to:
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 ∈ L2 (Ω).
Concurso ETSIN
Main goal
Understand under which conditions on:
the geometry of Ω;
the structure of the potential V (t, x);
the control (open) set ω and T ;
the observability estimate:
||ψ 0 ||2L2 (Ω)
Z
T
Z
≤C
0
|ψ(t, x)|2 dx dt,
(OV,ω,T )
ω
holds for some constant C > 0 and every solution to:
(
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
ψ(0, ·) = ψ 0 ∈ L2 (Ω).
Concurso ETSIN
Observability is strong unique continuation
The observability estimate (OV,ω,T ) implies:
Strong Unique Continuation on (0, T ) × ω
There exists C > 0 such that for every solution ψ(t, x) to the
Schrödinger equation issued from an initial datum ψ 0 with
||ψ 0 ||L2 (Ω) = 1 one has:
T
Z
0
Z
|ψ(t, x)|2 dx dt ≥ C −1 .
ω
This of course implies the usual Unique Continuation Property:
ψ|(0,T )×ω = 0 =⇒ ψ 0 = 0.
Concurso ETSIN
Observability is strong unique continuation
The observability estimate (OV,ω,T ) implies:
Strong Unique Continuation on (0, T ) × ω
There exists C > 0 such that for every solution ψ(t, x) to the
Schrödinger equation issued from an initial datum ψ 0 with
||ψ 0 ||L2 (Ω) = 1 one has:
T
Z
0
Z
|ψ(t, x)|2 dx dt ≥ C −1 .
ω
This of course implies the usual Unique Continuation Property:
ψ|(0,T )×ω = 0 =⇒ ψ 0 = 0.
Concurso ETSIN
A sufficient condition: a theorem by G. Lebeau (1992)
Suppose V ∈ C ∞ (Ω; R) does not depend on t.
Suppose the open set ω ⊂ Ω satisfies:
Geometric Control Condition (GCC)
Every (generalized) geodesic of Ω intersects the control set ω.
Then the observability estimate (OV,ω,T ) holds.
Notice:
Neither T nor V (x) play a role in the Geometric Control Condition.
Concurso ETSIN
A sufficient condition: a theorem by G. Lebeau (1992)
Suppose V ∈ C ∞ (Ω; R) does not depend on t.
Suppose the open set ω ⊂ Ω satisfies:
Geometric Control Condition (GCC)
Every (generalized) geodesic of Ω intersects the control set ω.
Then the observability estimate (OV,ω,T ) holds.
Notice:
Neither T nor V (x) play a role in the Geometric Control Condition.
Concurso ETSIN
A sufficient condition: a theorem by G. Lebeau (1992)
Suppose V ∈ C ∞ (Ω; R) does not depend on t.
Suppose the open set ω ⊂ Ω satisfies:
Geometric Control Condition (GCC)
Every (generalized) geodesic of Ω intersects the control set ω.
Then the observability estimate (OV,ω,T ) holds.
Notice:
Neither T nor V (x) play a role in the Geometric Control Condition.
Concurso ETSIN
Generalized geodesics on a domain Ω ⊂ Rd
Line segments that reflect on the boundary ∂Ω following the law of
Geometric Optics. Also called billiard trajectories.
Concurso ETSIN
Why observability is related to the geodesics of Ω?
The Quantum-Classical correspondence principle.
High-frequency solutions to the Schrödinger equation propagate
according to classical, Newtonian, mechanics. In other words, the
position probability densities:
|ψ(t, x)|2
obey a propagation law that depends on geodesics (or billiard
trajectories).
When Schrödinger introduced his equation in 1925 he did it in such a
way that it was compatible with Newtonian mechanics in the classical
limit.
Concurso ETSIN
Why observability is related to the geodesics of Ω?
The Quantum-Classical correspondence principle.
High-frequency solutions to the Schrödinger equation propagate
according to classical, Newtonian, mechanics. In other words, the
position probability densities:
|ψ(t, x)|2
obey a propagation law that depends on geodesics (or billiard
trajectories).
When Schrödinger introduced his equation in 1925 he did it in such a
way that it was compatible with Newtonian mechanics in the classical
limit.
Concurso ETSIN
Why observability is related to the geodesics of Ω?
The Quantum-Classical correspondence principle.
High-frequency solutions to the Schrödinger equation propagate
according to classical, Newtonian, mechanics. In other words, the
position probability densities:
|ψ(t, x)|2
obey a propagation law that depends on geodesics (or billiard
trajectories).
When Schrödinger introduced his equation in 1925 he did it in such a
way that it was compatible with Newtonian mechanics in the classical
limit.
Concurso ETSIN
Wave-packets on a domain Ω in R2
Take x0 ∈ Ω, ξ0 ∈ S1 , a function ρ ∈ Cc∞ (Ω) and a sequence hk → 0+ .
Consider initial data of the following form:
ξ
x − x0
1
i 0 ·x
0
e hk ,
ψk (x) := √ ρ √
hk
hk
The corresponding solution ψk (t, x) is called a wave-packet.
Concurso ETSIN
Wave-packets on a domain Ω in R2
Take x0 ∈ Ω, ξ0 ∈ S1 , a function ρ ∈ Cc∞ (Ω) and a sequence hk → 0+ .
Consider initial data of the following form:
ξ
x − x0
1
i 0 ·x
0
e hk ,
ψk (x) := √ ρ √
hk
hk
The corresponding solution ψk (t, x) is called a wave-packet.
Concurso ETSIN
Dynamics of wave paquets
Let x(t) denote the position of the billiard trajectory after time t
starting at x0 with velocity ξ0 .
Time-scale the solutions in time as: ψk (hk t, x).
Propagation of time-scaled wave packets
The position probability densities |ψk (hk t, ·)|2 converge to x(t). In
other words, for the weak convergence of probability measures:
|ψk (hk t, ·)|2 * δx(t) ,
as k → ∞.
Notice:
The time-scaled functions ψk (hk t, ·) are not solutions to our original
Schrödinger equation.
Concurso ETSIN
Dynamics of wave paquets
Let x(t) denote the position of the billiard trajectory after time t
starting at x0 with velocity ξ0 .
Time-scale the solutions in time as: ψk (hk t, x).
Propagation of time-scaled wave packets
The position probability densities |ψk (hk t, ·)|2 converge to x(t). In
other words, for the weak convergence of probability measures:
|ψk (hk t, ·)|2 * δx(t) ,
as k → ∞.
Notice:
The time-scaled functions ψk (hk t, ·) are not solutions to our original
Schrödinger equation.
Concurso ETSIN
Dynamics of wave paquets
Let x(t) denote the position of the billiard trajectory after time t
starting at x0 with velocity ξ0 .
Time-scale the solutions in time as: ψk (hk t, x).
Propagation of time-scaled wave packets
The position probability densities |ψk (hk t, ·)|2 converge to x(t). In
other words, for the weak convergence of probability measures:
|ψk (hk t, ·)|2 * δx(t) ,
as k → ∞.
Notice:
The time-scaled functions ψk (hk t, ·) are not solutions to our original
Schrödinger equation.
Concurso ETSIN
Dynamics of wave paquets
Let x(t) denote the position of the billiard trajectory after time t
starting at x0 with velocity ξ0 .
Time-scale the solutions in time as: ψk (hk t, x).
Propagation of time-scaled wave packets
The position probability densities |ψk (hk t, ·)|2 converge to x(t). In
other words, for the weak convergence of probability measures:
|ψk (hk t, ·)|2 * δx(t) ,
as k → ∞.
Notice:
The time-scaled functions ψk (hk t, ·) are not solutions to our original
Schrödinger equation.
Concurso ETSIN
Dynamics of a wave-packet on a domain in R2
Left The point • represents the dynamics of the billiard flow with
elastic reflections on the boundary, the point • represents the
center of mass of the probability measure |ψ(t, x)|2 .
Right Dynamics of the position density |ψ(t, x)|2 .
Figure: courtesy of Roman Schubert (U. Bristol)
Concurso ETSIN
(Rough) Ideas of the proof of Lebeau’s theorem
Perform a Littlewood-Paley decomposition to write the solution as
a superposition of time-scaled solutions that are localized in
frequency.
These time-scaled solutions behave as a superposition of
wave-packets.
Prove that if ω satisfies the GCC then the observability estimate
holds uniformly for the time-scaled, frequency-localized solutions.
This is the hardest part. It’s based on microlocal analysis
techniques.
The general case follows from a compactness argument and the
unique continuation property on ω for eigenfunctions of
−∆ + V (x) (Calderón’s theorem).
This step requires that V does not depend on t.
Concurso ETSIN
(Rough) Ideas of the proof of Lebeau’s theorem
Perform a Littlewood-Paley decomposition to write the solution as
a superposition of time-scaled solutions that are localized in
frequency.
These time-scaled solutions behave as a superposition of
wave-packets.
Prove that if ω satisfies the GCC then the observability estimate
holds uniformly for the time-scaled, frequency-localized solutions.
This is the hardest part. It’s based on microlocal analysis
techniques.
The general case follows from a compactness argument and the
unique continuation property on ω for eigenfunctions of
−∆ + V (x) (Calderón’s theorem).
This step requires that V does not depend on t.
Concurso ETSIN
(Rough) Ideas of the proof of Lebeau’s theorem
Perform a Littlewood-Paley decomposition to write the solution as
a superposition of time-scaled solutions that are localized in
frequency.
These time-scaled solutions behave as a superposition of
wave-packets.
Prove that if ω satisfies the GCC then the observability estimate
holds uniformly for the time-scaled, frequency-localized solutions.
This is the hardest part. It’s based on microlocal analysis
techniques.
The general case follows from a compactness argument and the
unique continuation property on ω for eigenfunctions of
−∆ + V (x) (Calderón’s theorem).
This step requires that V does not depend on t.
Concurso ETSIN
(Rough) Ideas of the proof of Lebeau’s theorem
Perform a Littlewood-Paley decomposition to write the solution as
a superposition of time-scaled solutions that are localized in
frequency.
These time-scaled solutions behave as a superposition of
wave-packets.
Prove that if ω satisfies the GCC then the observability estimate
holds uniformly for the time-scaled, frequency-localized solutions.
This is the hardest part. It’s based on microlocal analysis
techniques.
The general case follows from a compactness argument and the
unique continuation property on ω for eigenfunctions of
−∆ + V (x) (Calderón’s theorem).
This step requires that V does not depend on t.
Concurso ETSIN
(Rough) Ideas of the proof of Lebeau’s theorem
Perform a Littlewood-Paley decomposition to write the solution as
a superposition of time-scaled solutions that are localized in
frequency.
These time-scaled solutions behave as a superposition of
wave-packets.
Prove that if ω satisfies the GCC then the observability estimate
holds uniformly for the time-scaled, frequency-localized solutions.
This is the hardest part. It’s based on microlocal analysis
techniques.
The general case follows from a compactness argument and the
unique continuation property on ω for eigenfunctions of
−∆ + V (x) (Calderón’s theorem).
This step requires that V does not depend on t.
Concurso ETSIN
(Rough) Ideas of the proof of Lebeau’s theorem
Perform a Littlewood-Paley decomposition to write the solution as
a superposition of time-scaled solutions that are localized in
frequency.
These time-scaled solutions behave as a superposition of
wave-packets.
Prove that if ω satisfies the GCC then the observability estimate
holds uniformly for the time-scaled, frequency-localized solutions.
This is the hardest part. It’s based on microlocal analysis
techniques.
The general case follows from a compactness argument and the
unique continuation property on ω for eigenfunctions of
−∆ + V (x) (Calderón’s theorem).
This step requires that V does not depend on t.
Concurso ETSIN
Is the GCC also a necessary condition?
Intuitively: one expects the answer to be NO, because of the strong
dispersive character of the Schrödinger equation.
I will make this intuition precise later on.
However, there are exceptions...
Concurso ETSIN
Is the GCC also a necessary condition?
Intuitively: one expects the answer to be NO, because of the strong
dispersive character of the Schrödinger equation.
I will make this intuition precise later on.
However, there are exceptions...
Concurso ETSIN
Is the GCC also a necessary condition?
Intuitively: one expects the answer to be NO, because of the strong
dispersive character of the Schrödinger equation.
I will make this intuition precise later on.
However, there are exceptions...
Concurso ETSIN
Failure of the observability property
The observability estimate (OV,ω,T ) fails if and only if one can
construct a sequence of initial data (ψk0 ), with:
||ψk0 ||L2 (Ω) = 1,
and such that the corresponding solutions ψk (t, x) satisfy:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0.
ω
Equivalently:
(OV,ω,T ) holds if and only if the above construction is impossible.
Concurso ETSIN
Failure of the observability property
The observability estimate (OV,ω,T ) fails if and only if one can
construct a sequence of initial data (ψk0 ), with:
||ψk0 ||L2 (Ω) = 1,
and such that the corresponding solutions ψk (t, x) satisfy:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0.
ω
Equivalently:
(OV,ω,T ) holds if and only if the above construction is impossible.
Concurso ETSIN
A simple example with V = 0
Consider the sphere:
S2 := {x = (x1 , x2 , x3 ) : |x1 |2 + |x2 |2 + |x3 |2 = 1}.
Let
ψk0 (x) = ck (x1 + ix2 )k ,
with ck such that ||ψk0 ||L2 (S2 ) = 1.
This function is a spherical harmonic (eigenfunction of the Laplacian):
−∆ψk0 (x) = k(k + 1)ψk0 (x),
x ∈ S2 .
Therefore the corresponding solution to the free Schrödinger equation
(V = 0) is:
ψk (t, x) = e−itk(k+1) ψk0 (x),
and
|ψk (t, x)|2 = |ψk0 (x)|2 = (ck )2 (|x1 |2 + |x2 |2 )k = (ck )2 (1 − |x3 |2 )k .
Clearly |ψk (t, ·)|2 concentrates on the equator {x3 = 0}.
Concurso ETSIN
A simple example with V = 0
Consider the sphere:
S2 := {x = (x1 , x2 , x3 ) : |x1 |2 + |x2 |2 + |x3 |2 = 1}.
Let
ψk0 (x) = ck (x1 + ix2 )k ,
with ck such that ||ψk0 ||L2 (S2 ) = 1.
This function is a spherical harmonic (eigenfunction of the Laplacian):
−∆ψk0 (x) = k(k + 1)ψk0 (x),
x ∈ S2 .
Therefore the corresponding solution to the free Schrödinger equation
(V = 0) is:
ψk (t, x) = e−itk(k+1) ψk0 (x),
and
|ψk (t, x)|2 = |ψk0 (x)|2 = (ck )2 (|x1 |2 + |x2 |2 )k = (ck )2 (1 − |x3 |2 )k .
Clearly |ψk (t, ·)|2 concentrates on the equator {x3 = 0}.
Concurso ETSIN
A simple example with V = 0
Consider the sphere:
S2 := {x = (x1 , x2 , x3 ) : |x1 |2 + |x2 |2 + |x3 |2 = 1}.
Let
ψk0 (x) = ck (x1 + ix2 )k ,
with ck such that ||ψk0 ||L2 (S2 ) = 1.
This function is a spherical harmonic (eigenfunction of the Laplacian):
−∆ψk0 (x) = k(k + 1)ψk0 (x),
x ∈ S2 .
Therefore the corresponding solution to the free Schrödinger equation
(V = 0) is:
ψk (t, x) = e−itk(k+1) ψk0 (x),
and
|ψk (t, x)|2 = |ψk0 (x)|2 = (ck )2 (|x1 |2 + |x2 |2 )k = (ck )2 (1 − |x3 |2 )k .
Clearly |ψk (t, ·)|2 concentrates on the equator {x3 = 0}.
Concurso ETSIN
A simple example with V = 0
Consider the sphere:
S2 := {x = (x1 , x2 , x3 ) : |x1 |2 + |x2 |2 + |x3 |2 = 1}.
Let
ψk0 (x) = ck (x1 + ix2 )k ,
with ck such that ||ψk0 ||L2 (S2 ) = 1.
This function is a spherical harmonic (eigenfunction of the Laplacian):
−∆ψk0 (x) = k(k + 1)ψk0 (x),
x ∈ S2 .
Therefore the corresponding solution to the free Schrödinger equation
(V = 0) is:
ψk (t, x) = e−itk(k+1) ψk0 (x),
and
|ψk (t, x)|2 = |ψk0 (x)|2 = (ck )2 (|x1 |2 + |x2 |2 )k = (ck )2 (1 − |x3 |2 )k .
Clearly |ψk (t, ·)|2 concentrates on the equator {x3 = 0}.
Concurso ETSIN
GCC is almost necessary on the sphere
Theorem
Suppose V = 0. If ω ∩ {x3 = 0} = ∅ then (OV,ω,T ) fails, since:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0,
and
||ψk0 ||L2 (S2 ) = 1.
ω
Since any other geodesic of S2 can be obtained by applying a rotation
to {x3 = 0}, and the composition of a spherical harmonic with a
rotation is again a spherical harmonic, we obtain a necessary condition
for the observability estimate:
Theorem
Suppose V = 0. If ω ∩ γ = ∅ for some geodesic γ of S2 then (OV,ω,T )
fails.
Concurso ETSIN
GCC is almost necessary on the sphere
Theorem
Suppose V = 0. If ω ∩ {x3 = 0} = ∅ then (OV,ω,T ) fails, since:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0,
and
||ψk0 ||L2 (S2 ) = 1.
ω
Since any other geodesic of S2 can be obtained by applying a rotation
to {x3 = 0}, and the composition of a spherical harmonic with a
rotation is again a spherical harmonic, we obtain a necessary condition
for the observability estimate:
Theorem
Suppose V = 0. If ω ∩ γ = ∅ for some geodesic γ of S2 then (OV,ω,T )
fails.
Concurso ETSIN
More general examples
The preceding result holds in a much more general setting.
Theorem (M. (2009))
Suppose Ω is a compact manifold such that all its geodesics are closed
curves and that V ∈ C ∞ (R × Ω).
If ω ∩ γ = ∅ for some geodesic γ of Ω then (OV,ω,T ) fails.
There are infinitely many isometry types of surfaces of revolution
satisfying the hypothesis.
The proof is based on the averaging method of Moser and Weinstein.
The sequences violating the estimate are not formed by eigenfunctions
of −∆ + V (x) (when V does not depend on t).
In fact, we have shown that eigenfunctions may fail to concentrate on
almost every geodesic (M.-Rivière (2016)).
Concurso ETSIN
More general examples
The preceding result holds in a much more general setting.
Theorem (M. (2009))
Suppose Ω is a compact manifold such that all its geodesics are closed
curves and that V ∈ C ∞ (R × Ω).
If ω ∩ γ = ∅ for some geodesic γ of Ω then (OV,ω,T ) fails.
There are infinitely many isometry types of surfaces of revolution
satisfying the hypothesis.
The proof is based on the averaging method of Moser and Weinstein.
The sequences violating the estimate are not formed by eigenfunctions
of −∆ + V (x) (when V does not depend on t).
In fact, we have shown that eigenfunctions may fail to concentrate on
almost every geodesic (M.-Rivière (2016)).
Concurso ETSIN
More general examples
The preceding result holds in a much more general setting.
Theorem (M. (2009))
Suppose Ω is a compact manifold such that all its geodesics are closed
curves and that V ∈ C ∞ (R × Ω).
If ω ∩ γ = ∅ for some geodesic γ of Ω then (OV,ω,T ) fails.
There are infinitely many isometry types of surfaces of revolution
satisfying the hypothesis.
The proof is based on the averaging method of Moser and Weinstein.
The sequences violating the estimate are not formed by eigenfunctions
of −∆ + V (x) (when V does not depend on t).
In fact, we have shown that eigenfunctions may fail to concentrate on
almost every geodesic (M.-Rivière (2016)).
Concurso ETSIN
More general examples
The preceding result holds in a much more general setting.
Theorem (M. (2009))
Suppose Ω is a compact manifold such that all its geodesics are closed
curves and that V ∈ C ∞ (R × Ω).
If ω ∩ γ = ∅ for some geodesic γ of Ω then (OV,ω,T ) fails.
There are infinitely many isometry types of surfaces of revolution
satisfying the hypothesis.
The proof is based on the averaging method of Moser and Weinstein.
The sequences violating the estimate are not formed by eigenfunctions
of −∆ + V (x) (when V does not depend on t).
In fact, we have shown that eigenfunctions may fail to concentrate on
almost every geodesic (M.-Rivière (2016)).
Concurso ETSIN
More general examples
The preceding result holds in a much more general setting.
Theorem (M. (2009))
Suppose Ω is a compact manifold such that all its geodesics are closed
curves and that V ∈ C ∞ (R × Ω).
If ω ∩ γ = ∅ for some geodesic γ of Ω then (OV,ω,T ) fails.
There are infinitely many isometry types of surfaces of revolution
satisfying the hypothesis.
The proof is based on the averaging method of Moser and Weinstein.
The sequences violating the estimate are not formed by eigenfunctions
of −∆ + V (x) (when V does not depend on t).
In fact, we have shown that eigenfunctions may fail to concentrate on
almost every geodesic (M.-Rivière (2016)).
Concurso ETSIN
The converse also holds
Theorem (M. (2009))
Suppose Ω is a compact manifold such that all its geodesics are closed
curves and that V ∈ C ∞ (R × Ω; R).
If ω satisfies the GCC and the solutions to
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
satisty the unique continuation property on (0, T ) × ω then the
observability estimate (OV,ω,T ) holds.
If V = V (x) does not depend on the time variable t then we do not
need to assume that the solutions satisfy the unique continuation
property to obtain the conclusion.
Concurso ETSIN
Periodic Boundary Conditions
Consider the Schrödinger equation on Rd with 2πZd -periodic boundary
conditions. This is the same as taking Ω = Td = Rd /2πZd , the flat
torus.
Theorem (Anantharaman, M. (2010))
Let ω ⊂ Td be an open set and T > 0.
If V ∈ L∞ (Td ; R) does not depent on t and is continuous outside
of a set of Lebesgue measure zero then (OV,ω,T ) holds.
If V ∈ L∞ (R × Td ; R) has the same regularity as above and the
solutions of
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
satisty the unique continuation property on (0, T ) × ω then,
provided d = 2, (OV,ω,T ) holds.
If d ≥ 3, one has also to assume that UCP holds for the solutions
to a certain family of lower dimensional Schrödinger equations
Concurso ETSIN
involving space-averages of V .
Periodic Boundary Conditions
Consider the Schrödinger equation on Rd with 2πZd -periodic boundary
conditions. This is the same as taking Ω = Td = Rd /2πZd , the flat
torus.
Theorem (Anantharaman, M. (2010))
Let ω ⊂ Td be an open set and T > 0.
If V ∈ L∞ (Td ; R) does not depent on t and is continuous outside
of a set of Lebesgue measure zero then (OV,ω,T ) holds.
If V ∈ L∞ (R × Td ; R) has the same regularity as above and the
solutions of
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
satisty the unique continuation property on (0, T ) × ω then,
provided d = 2, (OV,ω,T ) holds.
If d ≥ 3, one has also to assume that UCP holds for the solutions
to a certain family of lower dimensional Schrödinger equations
Concurso ETSIN
involving space-averages of V .
Periodic Boundary Conditions
Consider the Schrödinger equation on Rd with 2πZd -periodic boundary
conditions. This is the same as taking Ω = Td = Rd /2πZd , the flat
torus.
Theorem (Anantharaman, M. (2010))
Let ω ⊂ Td be an open set and T > 0.
If V ∈ L∞ (Td ; R) does not depent on t and is continuous outside
of a set of Lebesgue measure zero then (OV,ω,T ) holds.
If V ∈ L∞ (R × Td ; R) has the same regularity as above and the
solutions of
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
satisty the unique continuation property on (0, T ) × ω then,
provided d = 2, (OV,ω,T ) holds.
If d ≥ 3, one has also to assume that UCP holds for the solutions
to a certain family of lower dimensional Schrödinger equations
Concurso ETSIN
involving space-averages of V .
Periodic Boundary Conditions
Consider the Schrödinger equation on Rd with 2πZd -periodic boundary
conditions. This is the same as taking Ω = Td = Rd /2πZd , the flat
torus.
Theorem (Anantharaman, M. (2010))
Let ω ⊂ Td be an open set and T > 0.
If V ∈ L∞ (Td ; R) does not depent on t and is continuous outside
of a set of Lebesgue measure zero then (OV,ω,T ) holds.
If V ∈ L∞ (R × Td ; R) has the same regularity as above and the
solutions of
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
satisty the unique continuation property on (0, T ) × ω then,
provided d = 2, (OV,ω,T ) holds.
If d ≥ 3, one has also to assume that UCP holds for the solutions
to a certain family of lower dimensional Schrödinger equations
Concurso ETSIN
involving space-averages of V .
Some remarks on this result
The result holds without assuming any condition on ω. Therefore
GCC is not necessary in this case.
The unique continuation property holds in many cases (Holmgen,
Hörmander, Tataru, Robbiano-Zuily...). It involves some
analyticity on V .
There is a major difficulty in passing from d = 2 to d ≥ 3. It
involves a non-trivial recursion procedure.
We do not need an explicit expression of the solution in terms of
Fourier series. This allows us to deal with potentials.
Concurso ETSIN
Some remarks on this result
The result holds without assuming any condition on ω. Therefore
GCC is not necessary in this case.
The unique continuation property holds in many cases (Holmgen,
Hörmander, Tataru, Robbiano-Zuily...). It involves some
analyticity on V .
There is a major difficulty in passing from d = 2 to d ≥ 3. It
involves a non-trivial recursion procedure.
We do not need an explicit expression of the solution in terms of
Fourier series. This allows us to deal with potentials.
Concurso ETSIN
Some remarks on this result
The result holds without assuming any condition on ω. Therefore
GCC is not necessary in this case.
The unique continuation property holds in many cases (Holmgen,
Hörmander, Tataru, Robbiano-Zuily...). It involves some
analyticity on V .
There is a major difficulty in passing from d = 2 to d ≥ 3. It
involves a non-trivial recursion procedure.
We do not need an explicit expression of the solution in terms of
Fourier series. This allows us to deal with potentials.
Concurso ETSIN
Some remarks on this result
The result holds without assuming any condition on ω. Therefore
GCC is not necessary in this case.
The unique continuation property holds in many cases (Holmgen,
Hörmander, Tataru, Robbiano-Zuily...). It involves some
analyticity on V .
There is a major difficulty in passing from d = 2 to d ≥ 3. It
involves a non-trivial recursion procedure.
We do not need an explicit expression of the solution in terms of
Fourier series. This allows us to deal with potentials.
Concurso ETSIN
References to other work
Jaffard 1990) proved the result for V = 0 (using Fourier
expansions). There is also related previous work by Haraux.
Burq and Zworski gave a new proof (of microlocal nature) for
d = 2 and V = 0 (2003).
Burq and Zworski proved the result when d = 2, V 6= 0 is
continuous, and time-independent, (2011).
Bourgain, Burq and Zworski (2012) proved the estimate for rough
potentials (V ∈ L2 (T2 ), time-independent) when d = 2.
Bourgain (2013) extended this result to d = 3 and V ∈ L∞ (T3 ),
time-independent.
In recent work Anantharaman, Fermanian, M. (2013) we extended
this result to constant-coefficient, homogeneous second order
elliptic operators different from the Laplacian. We also show that
ellipticity is necessary.
Concurso ETSIN
The Disk with Dirichlet B.C.
If Ω = D = {z ∈ R2 : |z| < 1} we are able to prove the following.
Theorem (Anantharaman, Léautaud, M. (2014))
Let ω ⊂ D be an open set such that ω ∩ ∂D 6= ∅ and T > 0.
If V ∈ L∞ (D; R) does not depent on t and is continuous outside of
a set of Lebesgue measure zero then (OV,ω,T ) holds.
If V ∈ C ∞ (R × D; R) and the solutions of
i∂t ψ (t, x) + ∆x ψ (t, x) − V (t, x) ψ (t, x) = 0,
satisty the unique continuation property on (0, T ) × ω then,
(OV,ω,T ) holds.
Concurso ETSIN
Some remarks on this result
The geometric condition we assume on ω is also necessary. Note
that it is a much weaker condition than the GCC.
We also prove boundary observability estimates from any open set
of the boundary.
We, again, do not make use of explicit expression of the solution in
terms of Bessel functions, even when V = 0 for which the result
seems to be new.
Most importantly, we follow the same proof scheme as in the result
for the torus. Both results are based on the complete integrability
of the geodesic-billiard flow. Roughly speaking, this means that
the classical phase-spaces can be foliated by invariant torii that
are in turn foliated by families of periodic orbits.
There is however a major difficulty in this case, related to the
presence of the boundary. This introduces big singularities that
have to be dealt with.
Concurso ETSIN
Some remarks on this result
The geometric condition we assume on ω is also necessary. Note
that it is a much weaker condition than the GCC.
We also prove boundary observability estimates from any open set
of the boundary.
We, again, do not make use of explicit expression of the solution in
terms of Bessel functions, even when V = 0 for which the result
seems to be new.
Most importantly, we follow the same proof scheme as in the result
for the torus. Both results are based on the complete integrability
of the geodesic-billiard flow. Roughly speaking, this means that
the classical phase-spaces can be foliated by invariant torii that
are in turn foliated by families of periodic orbits.
There is however a major difficulty in this case, related to the
presence of the boundary. This introduces big singularities that
have to be dealt with.
Concurso ETSIN
Some remarks on this result
The geometric condition we assume on ω is also necessary. Note
that it is a much weaker condition than the GCC.
We also prove boundary observability estimates from any open set
of the boundary.
We, again, do not make use of explicit expression of the solution in
terms of Bessel functions, even when V = 0 for which the result
seems to be new.
Most importantly, we follow the same proof scheme as in the result
for the torus. Both results are based on the complete integrability
of the geodesic-billiard flow. Roughly speaking, this means that
the classical phase-spaces can be foliated by invariant torii that
are in turn foliated by families of periodic orbits.
There is however a major difficulty in this case, related to the
presence of the boundary. This introduces big singularities that
have to be dealt with.
Concurso ETSIN
Some remarks on this result
The geometric condition we assume on ω is also necessary. Note
that it is a much weaker condition than the GCC.
We also prove boundary observability estimates from any open set
of the boundary.
We, again, do not make use of explicit expression of the solution in
terms of Bessel functions, even when V = 0 for which the result
seems to be new.
Most importantly, we follow the same proof scheme as in the result
for the torus. Both results are based on the complete integrability
of the geodesic-billiard flow. Roughly speaking, this means that
the classical phase-spaces can be foliated by invariant torii that
are in turn foliated by families of periodic orbits.
There is however a major difficulty in this case, related to the
presence of the boundary. This introduces big singularities that
have to be dealt with.
Concurso ETSIN
Some remarks on this result
The geometric condition we assume on ω is also necessary. Note
that it is a much weaker condition than the GCC.
We also prove boundary observability estimates from any open set
of the boundary.
We, again, do not make use of explicit expression of the solution in
terms of Bessel functions, even when V = 0 for which the result
seems to be new.
Most importantly, we follow the same proof scheme as in the result
for the torus. Both results are based on the complete integrability
of the geodesic-billiard flow. Roughly speaking, this means that
the classical phase-spaces can be foliated by invariant torii that
are in turn foliated by families of periodic orbits.
There is however a major difficulty in this case, related to the
presence of the boundary. This introduces big singularities that
have to be dealt with.
Concurso ETSIN
Some remarks on this result
The geometric condition we assume on ω is also necessary. Note
that it is a much weaker condition than the GCC.
We also prove boundary observability estimates from any open set
of the boundary.
We, again, do not make use of explicit expression of the solution in
terms of Bessel functions, even when V = 0 for which the result
seems to be new.
Most importantly, we follow the same proof scheme as in the result
for the torus. Both results are based on the complete integrability
of the geodesic-billiard flow. Roughly speaking, this means that
the classical phase-spaces can be foliated by invariant torii that
are in turn foliated by families of periodic orbits.
There is however a major difficulty in this case, related to the
presence of the boundary. This introduces big singularities that
have to be dealt with.
Concurso ETSIN
Some ideas of the proof in the simplest case
The circle: Ω = T1 = S1 , d = 1 and V time independent.
The circle has only one geodesic, the circle itself. Therefore Lebeau’s
theorem and our result coincide: if ω ⊂ S1 is open, automatically it
satisfies the GCC.
Disclaimer: The proof I am going to outline next is by no means the
simplest or sharpest. It however gives a small hint of how the more
general result is proved.
Concurso ETSIN
Some ideas of the proof in the simplest case
The circle: Ω = T1 = S1 , d = 1 and V time independent.
The circle has only one geodesic, the circle itself. Therefore Lebeau’s
theorem and our result coincide: if ω ⊂ S1 is open, automatically it
satisfies the GCC.
Disclaimer: The proof I am going to outline next is by no means the
simplest or sharpest. It however gives a small hint of how the more
general result is proved.
Concurso ETSIN
Some ideas of the proof in the simplest case
The circle: Ω = T1 = S1 , d = 1 and V time independent.
The circle has only one geodesic, the circle itself. Therefore Lebeau’s
theorem and our result coincide: if ω ⊂ S1 is open, automatically it
satisfies the GCC.
Disclaimer: The proof I am going to outline next is by no means the
simplest or sharpest. It however gives a small hint of how the more
general result is proved.
Concurso ETSIN
Step 1: Compactness
We start considering a sequence of initial data (ψk0 ) such that the
corresponding solutions satisfy:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0,
and
||ψk0 ||L2 (S1 ) = 1.
ω
We want to reach a contradiction.
After extracting a subsequence we can assume:
|ψk |2 * ν̃ weakly as positive measures on Rt × S1x .
We can moreover show that:
ν̃(dt, dx) = νt (dx)dt,
νt (ω) = 0 for a.e. t ∈ (0, T ),
and νt (dx) are probability measures on S1 .
Concurso ETSIN
Step 1: Compactness
We start considering a sequence of initial data (ψk0 ) such that the
corresponding solutions satisfy:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0,
and
||ψk0 ||L2 (S1 ) = 1.
ω
We want to reach a contradiction.
After extracting a subsequence we can assume:
|ψk |2 * ν̃ weakly as positive measures on Rt × S1x .
We can moreover show that:
ν̃(dt, dx) = νt (dx)dt,
νt (ω) = 0 for a.e. t ∈ (0, T ),
and νt (dx) are probability measures on S1 .
Concurso ETSIN
Step 1: Compactness
We start considering a sequence of initial data (ψk0 ) such that the
corresponding solutions satisfy:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0,
and
||ψk0 ||L2 (S1 ) = 1.
ω
We want to reach a contradiction.
After extracting a subsequence we can assume:
|ψk |2 * ν̃ weakly as positive measures on Rt × S1x .
We can moreover show that:
ν̃(dt, dx) = νt (dx)dt,
νt (ω) = 0 for a.e. t ∈ (0, T ),
and νt (dx) are probability measures on S1 .
Concurso ETSIN
Step 1: Compactness
We start considering a sequence of initial data (ψk0 ) such that the
corresponding solutions satisfy:
Z
T
Z
lim
k→∞ 0
|ψk (t, x)|2 dx dt = 0,
and
||ψk0 ||L2 (S1 ) = 1.
ω
We want to reach a contradiction.
After extracting a subsequence we can assume:
|ψk |2 * ν̃ weakly as positive measures on Rt × S1x .
We can moreover show that:
ν̃(dt, dx) = νt (dx)dt,
νt (ω) = 0 for a.e. t ∈ (0, T ),
and νt (dx) are probability measures on S1 .
Concurso ETSIN
Goal: Compute νt .
Using the Littlewood-Paley trick and unique continuation for
eigenfunctions, we can assume that there exist hk → 0+ such that
||1(1,2) (hk Dx )ψk0 ||L2 (S1 ) → 1.
The energy of ψk0 is concentrated on frequencies in an interval
(1/hk , 2/hk ).
Concurso ETSIN
Goal: Compute νt .
Using the Littlewood-Paley trick and unique continuation for
eigenfunctions, we can assume that there exist hk → 0+ such that
||1(1,2) (hk Dx )ψk0 ||L2 (S1 ) → 1.
The energy of ψk0 is concentrated on frequencies in an interval
(1/hk , 2/hk ).
Concurso ETSIN
Step 2: Lift to phase space S1 × R
Form the Wigner transforms (Wigner, 1932) of the solutions ψk :
Z
v v
dv
Wk (t, x, ξ) :=
ψk t, x − hk
ψk t, x + hk eiξv .
2
2
2π
R
One has
Z
Wk (t, x, dξ) = |ψk (t, x)|2 .
R
Concurso ETSIN
Step 2: Lift to phase space S1 × R
Form the Wigner transforms (Wigner, 1932) of the solutions ψk :
Z
v v
dv
Wk (t, x, ξ) :=
ψk t, x − hk
ψk t, x + hk eiξv .
2
2
2π
R
One has
Z
Wk (t, x, dξ) = |ψk (t, x)|2 .
R
Concurso ETSIN
After extracting a subsequence one has:
Wk * µt (dx, dξ)dt,
as k → ∞,
in the sense of distributions D0 (Rt × S1x × Rξ ).
It is possible to show µt (dx, dξ) are probability measures on
phase-space S1 × R (it is called a microlocal defect measure, P. Gérard
(1990)).
Moreover,
Z
νt =
µt (·, dξ),
µt ({ξ = 0}) = 0.
R
Concurso ETSIN
After extracting a subsequence one has:
Wk * µt (dx, dξ)dt,
as k → ∞,
in the sense of distributions D0 (Rt × S1x × Rξ ).
It is possible to show µt (dx, dξ) are probability measures on
phase-space S1 × R (it is called a microlocal defect measure, P. Gérard
(1990)).
Moreover,
Z
νt =
µt (·, dξ),
µt ({ξ = 0}) = 0.
R
Concurso ETSIN
Step 3: Geometric invariance
Direct computation shows:
ξ∂x Wk (t, x, ξ) = O(hk ).
After taking limits, this yields the transport equation:
ξ∂x µt = 0.
Note that we have not used so far that d = 1!!
If we use this fact, together with µt ({ξ = 0}) = 0 and the transport
equation, it implies:
Z
1
dx.
νt =
µt (·, dξ) =
2π
R
This contradics:
νt (ω) = 0,
as we assumed originally.
Concurso ETSIN
Step 3: Geometric invariance
Direct computation shows:
ξ∂x Wk (t, x, ξ) = O(hk ).
After taking limits, this yields the transport equation:
ξ∂x µt = 0.
Note that we have not used so far that d = 1!!
If we use this fact, together with µt ({ξ = 0}) = 0 and the transport
equation, it implies:
Z
1
νt =
µt (·, dξ) =
dx.
2π
R
This contradics:
νt (ω) = 0,
as we assumed originally.
Concurso ETSIN
Step 3: Geometric invariance
Direct computation shows:
ξ∂x Wk (t, x, ξ) = O(hk ).
After taking limits, this yields the transport equation:
ξ∂x µt = 0.
Note that we have not used so far that d = 1!!
If we use this fact, together with µt ({ξ = 0}) = 0 and the transport
equation, it implies:
Z
1
νt =
µt (·, dξ) =
dx.
2π
R
This contradics:
νt (ω) = 0,
as we assumed originally.
Concurso ETSIN
Step 3: Geometric invariance
Direct computation shows:
ξ∂x Wk (t, x, ξ) = O(hk ).
After taking limits, this yields the transport equation:
ξ∂x µt = 0.
Note that we have not used so far that d = 1!!
If we use this fact, together with µt ({ξ = 0}) = 0 and the transport
equation, it implies:
Z
1
νt =
µt (·, dξ) =
dx.
2π
R
This contradics:
νt (ω) = 0,
as we assumed originally.
Concurso ETSIN
Step 3: Geometric invariance
Direct computation shows:
ξ∂x Wk (t, x, ξ) = O(hk ).
After taking limits, this yields the transport equation:
ξ∂x µt = 0.
Note that we have not used so far that d = 1!!
If we use this fact, together with µt ({ξ = 0}) = 0 and the transport
equation, it implies:
Z
1
νt =
µt (·, dξ) =
dx.
2π
R
This contradics:
νt (ω) = 0,
as we assumed originally.
Concurso ETSIN
The proof can be easily adapted to time-dependent potentials,
assuming unique continuation, which in one dimension is true due to
work of Isakov.
However, for higher dimensional tori the geometric invariance
properties of the measures µt are much more involved. Reaching a
contradiction requires introducing new objects that are more precise
that Wigner distributions, the so-called two-microlocal Wigner
transforms around invariant torii in phase space.
In the case of spheres the geometric description of the measures µt is
much simpler, and only requires adding a few extra steps to the proof
presented above.
Concurso ETSIN
The proof can be easily adapted to time-dependent potentials,
assuming unique continuation, which in one dimension is true due to
work of Isakov.
However, for higher dimensional tori the geometric invariance
properties of the measures µt are much more involved. Reaching a
contradiction requires introducing new objects that are more precise
that Wigner distributions, the so-called two-microlocal Wigner
transforms around invariant torii in phase space.
In the case of spheres the geometric description of the measures µt is
much simpler, and only requires adding a few extra steps to the proof
presented above.
Concurso ETSIN
The proof can be easily adapted to time-dependent potentials,
assuming unique continuation, which in one dimension is true due to
work of Isakov.
However, for higher dimensional tori the geometric invariance
properties of the measures µt are much more involved. Reaching a
contradiction requires introducing new objects that are more precise
that Wigner distributions, the so-called two-microlocal Wigner
transforms around invariant torii in phase space.
In the case of spheres the geometric description of the measures µt is
much simpler, and only requires adding a few extra steps to the proof
presented above.
Concurso ETSIN
Bibliograhpy: around Lebeau’s result
B. Dehman, P. Gérard, and G. Lebeau. Stabilization and control
for the nonlinear Schrödinger equation on a compact surface.
Mathematische Zeitschrift 254(4) (2006), 729–749.
C. Laurent. Internal control of the Schrödinger equation.
Mathematics of Control and Related. Fields 4(2) (2014), 161–186.
G. Lebeau. Contrôle de l’équation de Schrödinger. Journal de
Mathématiques Pures Appliquées 71 (1992), 267–291.
Concurso ETSIN
Bibliography: Spheres and Zoll manifolds
F. Macià. Semiclassical measures and the Schrödinger flow on
Riemannian manifolds, Nonlinearity 22 (2009) , 1003–1020.
F. Macià and G. Rivière. Concentration and non-concentration for
the Schrödinger evolution on Zoll manifolds. Communications in
Mathematical Physics 345(3) (2016), 1019–1054.
F. Macià and G. Rivière. Observability and Quantum Limits for
the Schrödinger equation on Sd . Preprint (2016).
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Bibliograhpy: The torus and the disk
N. Anantharaman, C. Fermanian-Kammerer and F. Macià.
Semiclassical Completely Integrable Systems : Long-Time
Dynamics And Observability Via Two-Microlocal Wigner
Measures. American Journal of Mathematics 137(3) (2015),
577–638.
N. Anantharaman, M. Léautaud and F. Macià. Wigner measures
and observability for the Schrödinger equation on the disk.
Inventiones Mathematicae 206(2) (2016), 485–599.
N. Anantharaman and F. Macià. Semiclassical measures for the
Schrödinger equation on the torus. Journal of the European
Mathematical Society 16(6) (2014), 1253–1288.
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Bibliography: Surveys and Lecture notes
Lecture Notes.
F. Macià. High-frequency dynamics for the Schrödinger equation,
with applications to dispersion and observability. Nonlinear
Optical and Atomic Systems. Lecture Notes in Mathematics, 2146
(2015), 275–335.
Survey articles.
N. Anantharaman and F. Macià. The dynamics of the Schrödinger
flow from the point of view of semiclassical measures. Spectral
Geometry, Proceedings of Symposia in Pure Mathematics, Vol.
84, Amer. Math. Soc., Providence, RI, 2012, 93–116.
F. Macià. The Schrödinger flow in a compact manifold:
High-frequency dynamics and dispersion, Modern Aspects of the
Theory of Partial Differential Equations, Operator Theory:
Advances and Applications, Vol. 216, Springer, Basel, 2011,
275–289.
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