inverse spectral problems for radial schrödinger operator ...serier/pdf/milan2006.pdf ·...
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PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Inverse spectral problems for radial Schrodingeroperator on [0,1].
Frederic Serier
LMJL-Laboratoire de Mathematiques Jean Leray, Universite de Nantes.Institut fur Mathematik, Universtat Zurich.
Milan 15/02/2006
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
1 PresentationObjectsMotivationOverview and results
2 The Direct spectral problemSpectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
3 The inverse spectral problemThe usual methodTransformation operatorsThe solutionIsospectral sets
4 Open problem : the real inverse problem for Schrodinger
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
1 PresentationObjectsMotivationOverview and results
2 The Direct spectral problemSpectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
3 The inverse spectral problemThe usual methodTransformation operatorsThe solutionIsospectral sets
4 Open problem : the real inverse problem for Schrodinger
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
� Radial Schrodinger operator(− d2
dx2+
a(a + 1)x2
+ q(x))
y = λy,
y(0) = y(1) = 0,(1)
where a ∈ N, λ ∈ C and q ∈ L2R(0, 1) is called potential.
Inverse spectral problem : for each integer a,
Construct a regular coordinate system λa × κa on L2R(0, 1) for
potentials q with spectral data from (1) ,
Describe the isospectral sets, which are the sets of potential withsame spectrum.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
� Radial Schrodinger operator(− d2
dx2+
a(a + 1)x2
+ q(x))
y = λy,
y(0) = y(1) = 0,(1)
where a ∈ N, λ ∈ C and q ∈ L2R(0, 1) is called potential.
Inverse spectral problem : for each integer a,
Construct a regular coordinate system λa × κa on L2R(0, 1) for
potentials q with spectral data from (1) ,
Describe the isospectral sets, which are the sets of potential withsame spectrum.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
� Radial Schrodinger operator(− d2
dx2+
a(a + 1)x2
+ q(x))
y = λy,
y(0) = y(1) = 0,(1)
where a ∈ N, λ ∈ C and q ∈ L2R(0, 1) is called potential.
Inverse spectral problem : for each integer a,
Construct a regular coordinate system λa × κa on L2R(0, 1) for
potentials q with spectral data from (1) ,
Describe the isospectral sets, which are the sets of potential withsame spectrum.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
I 3-D inverse acoustic scattering problem for inhomogeneous medium ofcompact support in the unit ball :
H = −∆ + q(‖X‖), X ∈ R3, ‖X‖ ≤ 1.
Splitting variables using spherical harmonics leads to consider a collectionof singular operators acting on L2
R(0, 1) :
Ha = − d2
dx2+
a(a + 1)x2
+ q(x), x ∈ (0, 1) a ∈ N.
Physical application : Helioseismology.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
I 3-D inverse acoustic scattering problem for inhomogeneous medium ofcompact support in the unit ball :
H = −∆ + q(‖X‖), X ∈ R3, ‖X‖ ≤ 1.
Splitting variables using spherical harmonics leads to consider a collectionof singular operators acting on L2
R(0, 1) :
Ha = − d2
dx2+
a(a + 1)x2
+ q(x), x ∈ (0, 1) a ∈ N.
Physical application : Helioseismology.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
I 3-D inverse acoustic scattering problem for inhomogeneous medium ofcompact support in the unit ball :
H = −∆ + q(‖X‖), X ∈ R3, ‖X‖ ≤ 1.
Splitting variables using spherical harmonics leads to consider a collectionof singular operators acting on L2
R(0, 1) :
Ha = − d2
dx2+
a(a + 1)x2
+ q(x), x ∈ (0, 1) a ∈ N.
Physical application : Helioseismology.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
J. Poschel and E. Trubowitz,Inverse spectral theory.Academic Press Inc., 1987.
Result for the regular case (a = 0)
λ0 × κ0 is a global real-analytic diffeomorphism on L2R(0, 1),
complete description of isospectral sets and all possible spectra.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
J. Poschel and E. Trubowitz,Inverse spectral theory.Academic Press Inc., 1987.
J.-C. Guillot and J. V. Ralston,
Inverse spectral theory for a singular Sturm-Liouville operator on [0, 1].J. Differential Equations, 76(2) :353–373, 1988.
Result for the first singular problem (a = 1)
λ1 × κ1 is a still a global real-analytic diffeomorphism on L2R(0, 1),
description of isospectral sets and all possible spectra.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
J. Poschel and E. Trubowitz,Inverse spectral theory.Academic Press Inc., 1987.
J.-C. Guillot and J. V. Ralston,
Inverse spectral theory for a singular Sturm-Liouville operator on [0, 1].J. Differential Equations, 76(2) :353–373, 1988.
R. Carlson,
Inverse spectral theory for some singular Sturm-Liouville problemsJ. Differential Equations, 106(2) :121–140, 1993.
Result for any a ∈ N
Description of isospectral sets and all possible spectra.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
J. Poschel and E. Trubowitz,Inverse spectral theory.Academic Press Inc., 1987.
J.-C. Guillot and J. V. Ralston,
Inverse spectral theory for a singular Sturm-Liouville operator on [0, 1].J. Differential Equations, 76(2) :353–373, 1988.
R. Carlson,
Inverse spectral theory for some singular Sturm-Liouville problemsJ. Differential Equations, 106(2) :121–140, 1993.
L. A. Zhornitskaya and V. S. Serov,
Inverse eigenvalue problems for a singular Sturm-Liouville operator on [0, 1].Inverse Problems, 10(4) :975–987, 1994.
R. Carlson,
A Borg-Levinson theorem for Bessel operators.Pacific J. Math., 177(1) :1–26, 1997.
Result for a ≥ −1/2, a ∈ R)
λa × κa is one-to-one on L2R(0, 1) and continuous.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
J. Poschel and E. Trubowitz,Inverse spectral theory.Academic Press Inc., 1987.
J.-C. Guillot and J. V. Ralston,
Inverse spectral theory for a singular Sturm-Liouville operator on [0, 1].J. Differential Equations, 76(2) :353–373, 1988.
R. Carlson,
Inverse spectral theory for some singular Sturm-Liouville problemsJ. Differential Equations, 106(2) :121–140, 1993.
L. A. Zhornitskaya and V. S. Serov,
Inverse eigenvalue problems for a singular Sturm-Liouville operator on [0, 1].Inverse Problems, 10(4) :975–987, 1994.
R. Carlson,
A Borg-Levinson theorem for Bessel operators.Pacific J. Math., 177(1) :1–26, 1997.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
ObjectsMotivationOverview and results
J. Poschel and E. Trubowitz,Inverse spectral theory.Academic Press Inc., 1987.
J.-C. Guillot and J. V. Ralston,
Inverse spectral theory for a singular Sturm-Liouville operator on [0, 1].J. Differential Equations, 76(2) :353–373, 1988.
R. Carlson,
Inverse spectral theory for some singular Sturm-Liouville problemsJ. Differential Equations, 106(2) :121–140, 1993.
L. A. Zhornitskaya and V. S. Serov,
Inverse eigenvalue problems for a singular Sturm-Liouville operator on [0, 1].Inverse Problems, 10(4) :975–987, 1994.
R. Carlson,
A Borg-Levinson theorem for Bessel operators.Pacific J. Math., 177(1) :1–26, 1997.
Theorem
For all a ∈ N, λa × κa is a local (indeed global) real-analytic diffeomorphism on L2R(0, 1).
I The isospectral sets are submanifolds of L2R(0, 1) with explicit tangent and normal spaces.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
1 PresentationObjectsMotivationOverview and results
2 The Direct spectral problemSpectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
3 The inverse spectral problemThe usual methodTransformation operatorsThe solutionIsospectral sets
4 Open problem : the real inverse problem for Schrodinger
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Spectra and eigenvalues
Following and extending previous works, we getLocalization and identification of the spectra as a strictly increasingreal sequence : σ(Ha(q)) = {λa,n(q)}n≥1.
Real-analyticity of each map q 7→ λa,n(q) on L2R(0, 1), expression
and estimate of its gradient :
∇qλa,n(t) = 2ja(ωa,nt)2 +O(
1n
), ωa,n =
√λa,n
Uniform estimation for the eigenvalues on bounded sets of L2R(0, 1)
λa,n(q) =(n +
a
2
)2
π2 +∫ 1
0
q(t)dt− a(a + 1) + `2(n).
Thus, the map λa is defined by
λa(q) =(∫ 1
0
q(t)dt,(λa,n(q)
)n≥1
)Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Spectra and eigenvalues
Following and extending previous works, we getLocalization and identification of the spectra as a strictly increasingreal sequence : σ(Ha(q)) = {λa,n(q)}n≥1.
Real-analyticity of each map q 7→ λa,n(q) on L2R(0, 1), expression
and estimate of its gradient :
∇qλa,n(t) = 2ja(ωa,nt)2 +O(
1n
), ωa,n =
√λa,n
Uniform estimation for the eigenvalues on bounded sets of L2R(0, 1)
λa,n(q) =(n +
a
2
)2
π2 +∫ 1
0
q(t)dt− a(a + 1) + `2(n).
Thus, the map λa is defined by
λa(q) =(∫ 1
0
q(t)dt,(λa,n(q)
)n≥1
)Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Spectra and eigenvalues
Following and extending previous works, we getLocalization and identification of the spectra as a strictly increasingreal sequence : σ(Ha(q)) = {λa,n(q)}n≥1.
Real-analyticity of each map q 7→ λa,n(q) on L2R(0, 1), expression
and estimate of its gradient :
∇qλa,n(t) = 2ja(ωa,nt)2 +O(
1n
), ωa,n =
√λa,n
Uniform estimation for the eigenvalues on bounded sets of L2R(0, 1)
λa,n(q) =(n +
a
2
)2
π2 +∫ 1
0
q(t)dt− a(a + 1) + `2(n).
Thus, the map λa is defined by
λa(q) =(∫ 1
0
q(t)dt,(λa,n(q)
)n≥1
)Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Spectra and eigenvalues
Following and extending previous works, we getLocalization and identification of the spectra as a strictly increasingreal sequence : σ(Ha(q)) = {λa,n(q)}n≥1.
Real-analyticity of each map q 7→ λa,n(q) on L2R(0, 1), expression
and estimate of its gradient :
∇qλa,n(t) = 2ja(ωa,nt)2 +O(
1n
), ωa,n =
√λa,n
Uniform estimation for the eigenvalues on bounded sets of L2R(0, 1)
λa,n(q) =(n +
a
2
)2
π2 +∫ 1
0
q(t)dt− a(a + 1) + `2(n).
Thus, the map λa is defined by
λa(q) =(∫ 1
0
q(t)dt,(λa,n(q)
)n≥1
)Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Additional data : terminal velocities
Definition (Terminal velocities)
κa,n(q) = ln∣∣∣∣y2
′(1, λa,n(q), q)y2′(1, λa,n(0), 0)
∣∣∣∣ , n ∈ N, n ≥ 1.
q 7→ κa,n(q) is real-analytic on L2R(0, 1) and has a gradient :
∇qκa,n(t) =1
ωa,nja(ωa,nt)ηa(ωa,nt) +O
(1n2
),
Uniformly on bounded sets of L2R(0, 1), we have
nκa,n(q) = `2(n).
We define the map κa by :
κa(q) = (nκa,n(q))n≥1
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Additional data : terminal velocities
Definition (Terminal velocities)
κa,n(q) = ln∣∣∣∣y2
′(1, λa,n(q), q)y2′(1, λa,n(0), 0)
∣∣∣∣ , n ∈ N, n ≥ 1.
q 7→ κa,n(q) is real-analytic on L2R(0, 1) and has a gradient :
∇qκa,n(t) =1
ωa,nja(ωa,nt)ηa(ωa,nt) +O
(1n2
),
Uniformly on bounded sets of L2R(0, 1), we have
nκa,n(q) = `2(n).
We define the map κa by :
κa(q) = (nκa,n(q))n≥1
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Additional data : terminal velocities
Definition (Terminal velocities)
κa,n(q) = ln∣∣∣∣y2
′(1, λa,n(q), q)y2′(1, λa,n(0), 0)
∣∣∣∣ , n ∈ N, n ≥ 1.
q 7→ κa,n(q) is real-analytic on L2R(0, 1) and has a gradient :
∇qκa,n(t) =1
ωa,nja(ωa,nt)ηa(ωa,nt) +O
(1n2
),
Uniformly on bounded sets of L2R(0, 1), we have
nκa,n(q) = `2(n).
We define the map κa by :
κa(q) = (nκa,n(q))n≥1
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Additional data : terminal velocities
Definition (Terminal velocities)
κa,n(q) = ln∣∣∣∣y2
′(1, λa,n(q), q)y2′(1, λa,n(0), 0)
∣∣∣∣ , n ∈ N, n ≥ 1.
q 7→ κa,n(q) is real-analytic on L2R(0, 1) and has a gradient :
∇qκa,n(t) =1
ωa,nja(ωa,nt)ηa(ωa,nt) +O
(1n2
),
Uniformly on bounded sets of L2R(0, 1), we have
nκa,n(q) = `2(n).
We define the map κa by :
κa(q) = (nκa,n(q))n≥1
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Definition (Spectral map)
λa × κa : L2R(0, 1) → R× `2R × `2R
q 7→(∫ 1
0
q(t)dt,{
λa,n(q)}
n≥1, {nκa,n(q)}n≥1
)
Proposition
For all a ∈ N,
λa × κa is real-analytic on L2R(0, 1),
Its differential dq(λa × κa) is given by
dq(λa × κa)[v]=(∫ 1
0
v(t)dt,{⟨∇qλa,n(q), v
⟩}n≥1
,{⟨
n∇qκa,n(q), v⟩}
n≥1
)Solving the I.S.P means
Prove the invertibility of dq (λa × κa).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Definition (Spectral map)
λa × κa : L2R(0, 1) → R× `2R × `2R
q 7→(∫ 1
0
q(t)dt,{
λa,n(q)}
n≥1, {nκa,n(q)}n≥1
)
Proposition
For all a ∈ N,
λa × κa is real-analytic on L2R(0, 1),
Its differential dq(λa × κa) is given by
dq(λa × κa)[v]=(∫ 1
0
v(t)dt,{⟨∇qλa,n(q), v
⟩}n≥1
,{⟨
n∇qκa,n(q), v⟩}
n≥1
)Solving the I.S.P means
Prove the invertibility of dq (λa × κa).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
Spectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
Definition (Spectral map)
λa × κa : L2R(0, 1) → R× `2R × `2R
q 7→(∫ 1
0
q(t)dt,{
λa,n(q)}
n≥1, {nκa,n(q)}n≥1
)
Proposition
For all a ∈ N,
λa × κa is real-analytic on L2R(0, 1),
Its differential dq(λa × κa) is given by
dq(λa × κa)[v]=(∫ 1
0
v(t)dt,{⟨∇qλa,n(q), v
⟩}n≥1
,{⟨
n∇qκa,n(q), v⟩}
n≥1
)Solving the I.S.P means
Prove the invertibility of dq (λa × κa).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
1 PresentationObjectsMotivationOverview and results
2 The Direct spectral problemSpectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
3 The inverse spectral problemThe usual methodTransformation operatorsThe solutionIsospectral sets
4 Open problem : the real inverse problem for Schrodinger
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
dq(λa × κa) is expressed as an operator sending functions in L2R(0, 1)
onto their Fourier coefficients with respect to the family{1,
{∇qλa,n(q)
}n≥1
, {n∇qκa,n(q)}n≥1
}.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
dq(λa × κa) is expressed as an operator sending functions in L2R(0, 1)
onto their Fourier coefficients with respect to the family{1,
{∇qλa,n(q)
}n≥1
, {n∇qκa,n(q)}n≥1
}.
Recall
dq(λa × κa)[v]=(∫ 1
0
v(t)dt,{⟨∇qλa,n(q), v
⟩}n≥1
,{⟨
n∇qκa,n(q), v⟩}
n≥1
).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
dq(λa × κa) is expressed as an operator sending functions in L2R(0, 1)
onto their Fourier coefficients with respect to the family{1,
{∇qλa,n(q)
}n≥1
, {n∇qκa,n(q)}n≥1
}.
We would like to apply the following result :
Lemma (Almost-orthogonality)
Let F := {fn}n≥1 be a sequence in an Hilbert space H s.t.
1 F is free, i.e. none of the fn is in the closed span of the others.
2 There exists an orthonormal basis E := {en}n≥1 of H close to F ,
i.e.∑‖fn − en‖2
2 < ∞,
Then, F is a basis of H and F : x 7→ {〈fn, x〉}n≥1 is an invertible mapbetween H and `2.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Spectra and eigenvalues
Following and extending previous works, we getLocalization and identification of the spectra as a strictly increasingreal sequence : σ(Ha(q)) = {λa,n(q)}n≥1.
Real-analyticity of each map q 7→ λa,n(q) on L2R(0, 1), expression
and estimate of its gradient :
∇qλa,n(t) = 2ja(ωa,nt)2 +O(
1n
), ωa,n =
√λa,n
Uniform estimation for the eigenvalues on bounded sets of L2R(0, 1)
λa,n(q) =(n +
a
2
)2
π2 +∫ 1
0
q(t)dt− a(a + 1) + `2(n).
Thus, the map λa is defined by
λa(q) =(∫ 1
0
q(t)dt,(λa,n(q)
)n≥1
)Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Additional data : terminal velocities
Definition (Terminal velocities)
κa,n(q) = ln∣∣∣∣y2
′(1, λa,n(q), q)y2′(1, λa,n(0), 0)
∣∣∣∣ , n ∈ N, n ≥ 1.
q 7→ κa,n(q) is real-analytic on L2R(0, 1) and has a gradient :
∇qκa,n(t) =1
ωa,nja(ωa,nt)ηa(ωa,nt) +O
(1n2
),
Uniformly on bounded sets of L2R(0, 1), we have
nκa,n(q) = `2(n).
We define the map κa by :
κa(q) = (nκa,n(q))n≥1
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
dq(λa × κa) is expressed as an operator sending functions in L2R(0, 1)
onto their Fourier coefficients with respect to the family{1,
{∇qλa,n(q)
}n≥1
, {n∇qκa,n(q)}n≥1
}.
We would like to apply the following result :
Lemma (Almost-orthogonality)
Let F := {fn}n≥1 be a sequence in an Hilbert space H s.t.
1 F is free, i.e. none of the fn is in the closed span of the others.
2 There exists an orthonormal basis E := {en}n≥1 of H close to F ,
i.e.∑‖fn − en‖2
2 < ∞,
Then, F is a basis of H and F : x 7→ {〈fn, x〉}n≥1 is an invertible mapbetween H and `2.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Elementary transformations
Key point : Reduce the singularity stepwise by sending gradients for Ha
to those related to Ha−1.Let Φa(x) = ja(x)2 and Ψa(x) = ja(x)ηa(x).
Lemma (Guillot & Ralston a = 1 (1988), Rundell & Sacks a ∈ N (2001) )
For all a ∈ N, a ≥ 1 let Sa : L2C(0, 1) −→ L2
C(0, 1) be defined by
Sa[f ](x) = f(x)− 4a x2a−1
∫ 1
x
f(t)t2a
dt.
Then,
1 Sa is a Banach isomorphism between L2C(0, 1) and
(x 7→ x2a
)⊥.
2 Functions Φa and Ψa have the properties
Φa = −Sa∗[Φa−1], Ψa = −Sa
∗[Ψa−1],
Φ′a = −S−1a [Φ′a−1], Ψ′
a = −S−1a [Ψ′
a−1].
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Elementary transformations
Key point : Reduce the singularity stepwise by sending gradients for Ha
to those related to Ha−1.Let Φa(x) = ja(x)2 and Ψa(x) = ja(x)ηa(x).
Lemma (Guillot & Ralston a = 1 (1988), Rundell & Sacks a ∈ N (2001) )
For all a ∈ N, a ≥ 1 let Sa : L2C(0, 1) −→ L2
C(0, 1) be defined by
Sa[f ](x) = f(x)− 4a x2a−1
∫ 1
x
f(t)t2a
dt.
Then,
1 Sa is a Banach isomorphism between L2C(0, 1) and
(x 7→ x2a
)⊥.
2 Functions Φa and Ψa have the properties
Φa = −Sa∗[Φa−1], Ψa = −Sa
∗[Ψa−1],
Φ′a = −S−1a [Φ′a−1], Ψ′
a = −S−1a [Ψ′
a−1].
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Elementary transformations
Key point : Reduce the singularity stepwise by sending gradients for Ha
to those related to Ha−1.Let Φa(x) = ja(x)2 and Ψa(x) = ja(x)ηa(x).
Lemma (Guillot & Ralston a = 1 (1988), Rundell & Sacks a ∈ N (2001) )
For all a ∈ N, a ≥ 1 let Sa : L2C(0, 1) −→ L2
C(0, 1) be defined by
Sa[f ](x) = f(x)− 4a x2a−1
∫ 1
x
f(t)t2a
dt.
Then,
1 Sa is a Banach isomorphism between L2C(0, 1) and
(x 7→ x2a
)⊥.
2 Functions Φa and Ψa have the properties
Φa = −Sa∗[Φa−1], Ψa = −Sa
∗[Ψa−1],
Φ′a = −S−1a [Φ′a−1], Ψ′
a = −S−1a [Ψ′
a−1].
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
The transformation operator
Theorem (Rundell & Sacks (2001))
For all a ∈ N∗, define Ta by
Ta = (−1)a+1SaSa−1 · · ·S1.
Alors,
1 Ta is an isomorphism between L2C(0, 1) and Vect
{x2, x4, . . . , x2a
}⊥.
2 For all λ ∈ C, we have
2Φa(λt)− 1 = T ∗a[cos(2λt)
], Ψa(λt) = −1
2T ∗a
[sin(2λt)
],
Φ′a(λt) = T−1a [− sin(2λt)], Ψ′
a(λt) = T−1a [cos(2λt)].
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Proof of the main theorem
The family giving dq(λa × κa) can now be expressed as the imagethrough T ∗a of the family
F ={
1, {cos (2ωa,nt) + Rn(t)} ,
{−n
2ωa,nsin (2ωa,nt) + Sn(t)
}, n ≥ 1
}where (Rn)n and (Sn)n are remainder uniformly in `2R.We obtain the chain :
dq (λa × κa) = F ◦ Ta
where F maps of function in L2R(0, 1) onto its Fourier coefficients with
respect to F .The proof is now “complete”, since :
1 Almost-orthogonality applies and gives the invertibility of F.
2 Transformation operator is constructed to be invertible.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Proof of the main theorem
The family giving dq(λa × κa) can now be expressed as the imagethrough T ∗a of the family
F ={
1, {cos (2ωa,nt) + Rn(t)} ,
{−n
2ωa,nsin (2ωa,nt) + Sn(t)
}, n ≥ 1
}where (Rn)n and (Sn)n are remainder uniformly in `2R.We obtain the chain :
dq (λa × κa) = F ◦ Ta
where F maps of function in L2R(0, 1) onto its Fourier coefficients with
respect to F .The proof is now “complete”, since :
1 Almost-orthogonality applies and gives the invertibility of F.
2 Transformation operator is constructed to be invertible.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Proof of the main theorem
The family giving dq(λa × κa) can now be expressed as the imagethrough T ∗a of the family
F ={
1, {cos (2ωa,nt) + Rn(t)} ,
{−n
2ωa,nsin (2ωa,nt) + Sn(t)
}, n ≥ 1
}where (Rn)n and (Sn)n are remainder uniformly in `2R.We obtain the chain :
dq (λa × κa) = F ◦ Ta
where F maps of function in L2R(0, 1) onto its Fourier coefficients with
respect to F .The proof is now “complete”, since :
1 Almost-orthogonality applies and gives the invertibility of F.
2 Transformation operator is constructed to be invertible.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Proof of the main theorem
The family giving dq(λa × κa) can now be expressed as the imagethrough T ∗a of the family
F ={
1, {cos (2ωa,nt) + Rn(t)} ,
{−n
2ωa,nsin (2ωa,nt) + Sn(t)
}, n ≥ 1
}where (Rn)n and (Sn)n are remainder uniformly in `2R.We obtain the chain :
dq (λa × κa) = F ◦ Ta
where F maps of function in L2R(0, 1) onto its Fourier coefficients with
respect to F .The proof is now “complete”, since :
1 Almost-orthogonality applies and gives the invertibility of F.
2 Transformation operator is constructed to be invertible.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Proof of the main theorem
The family giving dq(λa × κa) can now be expressed as the imagethrough T ∗a of the family
F ={
1, {cos (2ωa,nt) + Rn(t)} ,
{−n
2ωa,nsin (2ωa,nt) + Sn(t)
}, n ≥ 1
}where (Rn)n and (Sn)n are remainder uniformly in `2R.We obtain the chain :
dq (λa × κa) = F ◦ Ta
where F maps of function in L2R(0, 1) onto its Fourier coefficients with
respect to F .The proof is now “complete”, since :
1 Almost-orthogonality applies and gives the invertibility of F.
2 Transformation operator is constructed to be invertible.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Definition
For all q0 ∈ L2R(0, 1) and a ∈ N,
Iso(q0, a) ={q ∈ L2
R(0, 1) : λa(q) = λa(q0)}
.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
The usual methodTransformation operatorsThe solutionIsospectral sets
Definition
For all q0 ∈ L2R(0, 1) and a ∈ N,
Iso(q0, a) ={q ∈ L2
R(0, 1) : λa(q) = λa(q0)}
.
Theorem (Isospectral sets)
For all q0 ∈ L2R(0, 1) and a ∈ N, we have
Iso(q0, a) is a real-analytic submanifold of L2R(0, 1).
For all q ∈ Iso(q0, a), we have
Tq Iso(q0, a) =
{ ∑n≥1
ξn
n
[2
d
dx∇qλa,n
]: ξ ∈ `2R
},
Nq Iso(q0, a) =
{η0 +
∑n≥1
ηn [∇qλa,n − 1] : (η0, η) ∈ R× `2R
}.
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
1 PresentationObjectsMotivationOverview and results
2 The Direct spectral problemSpectra and eigenvaluesAdditional data : terminal velocitiesThe spectral map
3 The inverse spectral problemThe usual methodTransformation operatorsThe solutionIsospectral sets
4 Open problem : the real inverse problem for Schrodinger
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
I Recall that Ha(q) = − d2
dx2 + a(a+1)x2 + q.
Question :
Does the knowledge of λa(q) and λb(q) determine q ?
R. Carlson and C. Shubin,
Spectral Rigidity for Radial Schrodinger OperatorsJ. Diff. Eq., 113(2) :338–354, 1994.
W. Rundell and P. E. Sacks,
Reconstruction of a Radially Potential from Two Spectral SequencesJ. Math. Anal. Appl., 264(2) :354–381, 2001.
Linearized problem around q = 0 :
Is{
1,{
ja(√
λa,nx)2}
n≥1,{
jb(√
λb,nx)2}
n≥1
}a basis for L2
R(0, 1) ?
λa,n ! (n + a2 )2π2
Conjecture for (a, b) = (a, a + 1) true for a = 0, 1, 2, 3 and (0, 2).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
I Recall that Ha(q) = − d2
dx2 + a(a+1)x2 + q.
Question :
Does the knowledge of λa(q) and λb(q) determine q ?
R. Carlson and C. Shubin,
Spectral Rigidity for Radial Schrodinger OperatorsJ. Diff. Eq., 113(2) :338–354, 1994.
W. Rundell and P. E. Sacks,
Reconstruction of a Radially Potential from Two Spectral SequencesJ. Math. Anal. Appl., 264(2) :354–381, 2001.
Linearized problem around q = 0 :
Is{
1,{
ja(√
λa,nx)2}
n≥1,{
jb(√
λb,nx)2}
n≥1
}a basis for L2
R(0, 1) ?
λa,n ! (n + a2 )2π2
Conjecture for (a, b) = (a, a + 1) true for a = 0, 1, 2, 3 and (0, 2).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
I Recall that Ha(q) = − d2
dx2 + a(a+1)x2 + q.
Question :
Does the knowledge of λa(q) and λb(q) determine q ?
R. Carlson and C. Shubin,
Spectral Rigidity for Radial Schrodinger OperatorsJ. Diff. Eq., 113(2) :338–354, 1994.
W. Rundell and P. E. Sacks,
Reconstruction of a Radially Potential from Two Spectral SequencesJ. Math. Anal. Appl., 264(2) :354–381, 2001.
Linearized problem around q = 0 :
Is{
1,{
ja(√
λa,nx)2}
n≥1,{
jb(√
λb,nx)2}
n≥1
}a basis for L2
R(0, 1) ?
λa,n ! (n + a2 )2π2
Conjecture for (a, b) = (a, a + 1) true for a = 0, 1, 2, 3 and (0, 2).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
I Recall that Ha(q) = − d2
dx2 + a(a+1)x2 + q.
Question :
Does the knowledge of λa(q) and λb(q) determine q ?
R. Carlson and C. Shubin,
Spectral Rigidity for Radial Schrodinger OperatorsJ. Diff. Eq., 113(2) :338–354, 1994.
W. Rundell and P. E. Sacks,
Reconstruction of a Radially Potential from Two Spectral SequencesJ. Math. Anal. Appl., 264(2) :354–381, 2001.
Linearized problem around q = 0 :
Is{
1,{
ja(√
λa,nx)2}
n≥1,{
jb(√
λb,nx)2}
n≥1
}a basis for L2
R(0, 1) ?
λa,n ! (n + a2 )2π2
Conjecture for (a, b) = (a, a + 1) true for a = 0, 1, 2, 3 and (0, 2).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
I Recall that Ha(q) = − d2
dx2 + a(a+1)x2 + q.
Question :
Does the knowledge of λa(q) and λb(q) determine q ?
R. Carlson and C. Shubin,
Spectral Rigidity for Radial Schrodinger OperatorsJ. Diff. Eq., 113(2) :338–354, 1994.
W. Rundell and P. E. Sacks,
Reconstruction of a Radially Potential from Two Spectral SequencesJ. Math. Anal. Appl., 264(2) :354–381, 2001.
Linearized problem around q = 0 :
Is{
1,{
ja(√
λa,nx)2}
n≥1,{
jb(√
λb,nx)2}
n≥1
}a basis for L2
R(0, 1) ?
λa,n ! (n + a2 )2π2
Conjecture for (a, b) = (a, a + 1) true for a = 0, 1, 2, 3 and (0, 2).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).
PresentationThe Direct spectral problemThe inverse spectral problem
Open problem : the real inverse problem for Schrodinger
I Recall that Ha(q) = − d2
dx2 + a(a+1)x2 + q.
Question :
Does the knowledge of λa(q) and λb(q) determine q ?
R. Carlson and C. Shubin,
Spectral Rigidity for Radial Schrodinger OperatorsJ. Diff. Eq., 113(2) :338–354, 1994.
W. Rundell and P. E. Sacks,
Reconstruction of a Radially Potential from Two Spectral SequencesJ. Math. Anal. Appl., 264(2) :354–381, 2001.
Linearized problem around q = 0 :
Is{
1,{
ja(√
λa,nx)2}
n≥1,{
jb(√
λb,nx)2}
n≥1
}a basis for L2
R(0, 1) ?
λa,n ! (n + a2 )2π2
Conjecture for (a, b) = (a, a + 1) true for a = 0, 1, 2, 3 and (0, 2).
Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).