Inverse spectral problems for radial Schrödinger operator ...serier/pdf/Milan2006.pdf ·...

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).

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

ObjectsMotivationOverview and results

� Radial Schrodinger operator(− d2

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.

a(a + 1)x2

+ q(x))

y = λy,

y(0) = y(1) = 0,(1)

a(a + 1)x2

+ q(x))

y = λy,

y(0) = y(1) = 0,(1)

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

a(a + 1)x2

+ q(x), x ∈ (0, 1) a ∈ N.

Physical application : Helioseismology.

H = −∆ + q(‖X‖), X ∈ R3, ‖X‖ ≤ 1.

R(0, 1) :

Ha = − d2

a(a + 1)x2

+ q(x), x ∈ (0, 1) a ∈ N.

H = −∆ + q(‖X‖), X ∈ R3, ‖X‖ ≤ 1.

R(0, 1) :

Ha = − d2

a(a + 1)x2

+ q(x), x ∈ (0, 1) a ∈ N.

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.

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.

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.

R. Carlson,

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.

R. Carlson,

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.

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(

), ωa,n =

√λa,n

Uniform estimation for the eigenvalues on bounded sets of L2R(0, 1)

λa,n(q) =(n +

π2 +∫ 1

q(t)dt− a(a + 1) + `2(n).

Thus, the map λa is defined by

λa(q) =(∫ 1

q(t)dt,(λa,n(q)

)n≥1

)Frederic Serier I.S.P. for radial Schrodinger operator on (0,1).

), ωa,n =

√λa,n

λa,n(q) =(n +

π2 +∫ 1

q(t)dt− a(a + 1) + `2(n).

λa(q) =(∫ 1

q(t)dt,(λa,n(q)

)n≥1

), ωa,n =

√λa,n

λa,n(q) =(n +

π2 +∫ 1

q(t)dt− a(a + 1) + `2(n).

λa(q) =(∫ 1

q(t)dt,(λa,n(q)

)n≥1

), ωa,n =

√λa,n

λa,n(q) =(n +

π2 +∫ 1

q(t)dt− a(a + 1) + `2(n).

λa(q) =(∫ 1

q(t)dt,(λa,n(q)

)n≥1

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

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

κa,n(q) = ln∣∣∣∣y2

′(1, λa,n(q), q)y2′(1, λa,n(0), 0)

∣∣∣∣ , n ∈ N, n ≥ 1.

∇qκa,n(t) =1

nκa,n(q) = `2(n).

κa,n(q) = ln∣∣∣∣y2

′(1, λa,n(q), q)y2′(1, λa,n(0), 0)

∣∣∣∣ , n ∈ N, n ≥ 1.

∇qκa,n(t) =1

nκa,n(q) = `2(n).

κa,n(q) = ln∣∣∣∣y2

′(1, λa,n(q), q)y2′(1, λa,n(0), 0)

∣∣∣∣ , n ∈ N, n ≥ 1.

∇qκa,n(t) =1

nκa,n(q) = `2(n).

Definition (Spectral map)

λa × κa : L2R(0, 1) → R× `2R × `2R

q 7→(∫ 1

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

v(t)dt,{⟨∇qλa,n(q), v

⟩}n≥1

n∇qκa,n(q), v⟩}

)Solving the I.S.P means

Prove the invertibility of dq (λa × κa).

λa × κa : L2R(0, 1) → R× `2R × `2R

q 7→(∫ 1

q(t)dt,{

λa,n(q)}

n≥1, {nκa,n(q)}n≥1

Proposition

For all a ∈ N,

dq(λa × κa)[v]=(∫ 1

⟩}n≥1

λa × κa : L2R(0, 1) → R× `2R × `2R

q 7→(∫ 1

q(t)dt,{

λa,n(q)}

n≥1, {nκa,n(q)}n≥1

Proposition

For all a ∈ N,

dq(λa × κa)[v]=(∫ 1

⟩}n≥1

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

{∇qλa,n(q)

}n≥1

Recall

dq(λa × κa)[v]=(∫ 1

⟩}n≥1

{∇qλa,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.

), ωa,n =

√λa,n

λa,n(q) =(n +

π2 +∫ 1

q(t)dt− a(a + 1) + `2(n).

λa(q) =(∫ 1

q(t)dt,(λa,n(q)

)n≥1

κa,n(q) = ln∣∣∣∣y2

′(1, λa,n(q), q)y2′(1, λa,n(0), 0)

∣∣∣∣ , n ∈ N, n ≥ 1.

∇qκa,n(t) =1

nκa,n(q) = `2(n).

{∇qλa,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.

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

f(t)t2a

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].

Sa[f ](x) = f(x)− 4a x2a−1

f(t)t2a

(x 7→ x2a

∗[Ψa−1],

Φ′a = −S−1a [Φ′a−1], Ψ′

a = −S−1a [Ψ′

a−1].

Sa[f ](x) = f(x)− 4a x2a−1

f(t)t2a

(x 7→ x2a

∗[Ψa−1],

Φ′a = −S−1a [Φ′a−1], Ψ′

a = −S−1a [Ψ′

a−1].

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)].

Proof of the main theorem

The family giving dq(λa × κa) can now be expressed as the imagethrough T ∗a of the family

1, {cos (2ωa,nt) + Rn(t)} ,

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.

1, {cos (2ωa,nt) + Rn(t)} ,

}, n ≥ 1

1, {cos (2ωa,nt) + Rn(t)} ,

}, n ≥ 1

1, {cos (2ωa,nt) + Rn(t)} ,

}, n ≥ 1

1, {cos (2ωa,nt) + Rn(t)} ,

}, n ≥ 1

Definition

For all q0 ∈ L2R(0, 1) and a ∈ N,

Iso(q0, a) ={q ∈ L2

R(0, 1) : λa(q) = λa(q0)}

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

dx∇qλa,n

]: ξ ∈ `2R

Nq Iso(q0, a) =

{η0 +

∑n≥1

ηn [∇qλa,n − 1] : (η0, η) ∈ R× `2R

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 :

ja(√

λa,nx)2}

n≥1,{

jb(√

λb,nx)2}

}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).

dx2 + a(a+1)x2 + q.

Question :

ja(√

λa,nx)2}

n≥1,{

jb(√

λb,nx)2}

}a basis for L2

R(0, 1) ?

λa,n ! (n + a2 )2π2

dx2 + a(a+1)x2 + q.

Question :

ja(√

λa,nx)2}

n≥1,{

jb(√

λb,nx)2}

}a basis for L2

R(0, 1) ?

λa,n ! (n + a2 )2π2

dx2 + a(a+1)x2 + q.

Question :

ja(√

λa,nx)2}

n≥1,{

jb(√

λb,nx)2}

}a basis for L2

R(0, 1) ?

λa,n ! (n + a2 )2π2

dx2 + a(a+1)x2 + q.

Question :

ja(√

λa,nx)2}

n≥1,{

jb(√

λb,nx)2}

}a basis for L2

R(0, 1) ?

λa,n ! (n + a2 )2π2

dx2 + a(a+1)x2 + q.

Question :

ja(√

λa,nx)2}

n≥1,{

jb(√

λb,nx)2}

}a basis for L2

R(0, 1) ?

λa,n ! (n + a2 )2π2

Inverse spectral problems for radial Schrödinger operator ...serier/pdf/Milan2006.pdf ·...

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