[PPT] - Inverse problems in a quantum waveguide Eric Soccorsi Aix PowerPoint Presentation

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Inverse problems in a quantum waveguide

´ Eric Soccorsi Aix Marseille Universit´ e In collaboration with Mourad Choulli (Lorraine), Otared Kavian (Versailles), Yavar Kian (Marseille) & Quang Sang Phan (Krakow) Workshop “New trends in modeling, control and inverse problems”, CIMI-Toulouse, June 16 - 19, 2014

´ Eric Soccorsi Inverse problems in a quantum waveguide

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Determining the scalar potential in a quantum waveguide

We consider the infinite cylindrical domain Ω = ω × R, where ω is a connected bounded open subset of RN−1, N ≥ 2. Given T > 0 we examine the IBVP    −i∂tu − ∆u + Vu = in Q = (0, T) × Ω, u(0, ·) = u0 in Ω, u = g

n Σ = (0, T) × Γ,

(1.1) where Γ := ∂ω × R. Here u0 (resp. g) is the initial (resp. boundary) condition associated to (1.1) and V is an unknown function of (t, x) ∈ Q. We aim to retrieve V from boundary observations of the solution u to (1.1).

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Estimating the electrostatic quantum disorder in nanotubes

(1.1) describes the quantum motion of a charged particle

constrained by the waveguide Ω, under the influence of the “electric” potential V .

Carbon nanotubes which have a length-to-diameter ratio up to

108/1 are commonly modelled by infinite cylindrical domains such as Ω.

Their physical properties are affected by the presence of

electrostatic quantum disorder V . This motivates for a closer look into the inverse problem of estimating the strength of the electric impurity potential V from the (partial) knowledge of the wave function u.

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Existing papers

There is a wide mathematical literature on uniqueness and

stability in inverse coefficient problems related to the Schr¨

dinger

equation in a bounded domain.

But there are only a few mathematical papers dealing with

inverse boundary value problems in an unbounded domain.

◮ Rakesh ’93: uniqueness of compactly supported scalar

potential in the wave equation in the half-plane from the ND map

◮ Li, Uhlmann ’10: uniqueness for a compactly supported scalar

potential in an infinite slab from partial DN map;

◮ Aktosun, Weder ’05 and Gestezy, Simon ’06: uniqueness of

the scalar potential in (0, +∞) from either the spectral measure or the Krein spectral shift function.

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Outline

Periodic scalar potential
1. Inverse spectral problem
2. Time dependent case: stability from Neumann data
More on the band functions of analytically fibered operators
1. Feynman-Hellman formula
2. Link with transport properties
Non periodic case

◮ Carleman estimate in an unbounded quantum waveguide ◮ Stability result

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

One-dimensional inverse spectral problem

Ω = (0, 1).
Sturm-Liouville operator associated with V ∈ L∞(Ω; R)

AV = − d2 dx2 + V (x) acting in L2(Ω) with DBC at x = 0, 1.

The spectrum of AV is purely discrete:

σ(AV ) = σd(AV ) = {λn(V ), n ≥ 1} with λ1(V ) < λ2(V ) < . . .

To each eigenvalue λn(V ) we associate a unique eigenfunction

ϕn(·, V ), normalized by ϕn(·, V )L2(Ω) = 1 and ϕ′

n(0, V ) > 0.

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Dirichlet inverse problem

Is V uniquely determined by σ(AV )?

◮ λ : L∞(Ω) ∋ V → (λn(V ))n≥1 is not injective. ◮ The Dirichlet spectrum cannot distinguish left from right:

˜

V (x) = V (1 − x), x ∈ Ω

⇒
λ( ˜

V ) = λ(V )

.
The case of “even” potentials (Borg ’46, Levinson ’49)

◮ E = {f ∈ L∞(Ω), ˜

f = f }.

◮ λ|E is injective.

General case (Gel’fand, Levitan ’51):

L∞(Ω) ∋ V → (λn(V ), ϕ′

n(1, V ))n≥1 is injective.

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Multi-dimensional case

Ω is a bounded open subset of RN, N ≥ 2, with C 2 boundary

and the scalar potential V ∈ L∞(Ω; R).

Self-adjoint Schr¨
dinger operators AV , acting in L2(Ω):
AV

= −∆ + V D(AV ) = H1

0(Ω) ∩ H2(Ω).

Purely discrete spectrum:

σ(AV ) = σd(AV ) = {λn(V ), n ≥ 1}, λ1(V ) ≤ λ2(V ) ≤ . . .

{ϕn(·, V ), n ≥ 1} = L2(Ω)-orthonormal basis of eigenfncs of AV :

AV ϕn(·, V ) = λn(V )ϕn(·, V ), n ≥ 1.

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Multi-dimensional Borg-Levinson theorem

Nachmann, Sylvester, Uhlmann ’88 and Novikov ’88:

L∞(Ω) ∋ V →

λn(V ), ∂νϕn(·, V )|∂Ω
n≥1 is injective,

where ∂νφ = ∇φ · ν and ν is the outward unit normal vector to ∂Ω.

Result improved by:

◮ Alessandrini, Sylvester ’ 90: stability; ◮ Isozaki ’91: partial spectral data; ◮ Canuto, Kavian ’04: conductivity; ◮ Choulli, Stefanov ’11: stability if the spectral data is known

asymptotically.

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3D inverse spectral problem

From now on ω is a connected bounded open subset of R2 with C 2-boundary and we consider the 3D unbounded waveguide: Ω = ω × R = {x = (x′, x3), x′ = (x1, x2) ∈ ω, x3 ∈ R}. The scalar potential V ∈ L∞(Ω; R) is 1-periodic wrt x3, V (x′, x3 + 1) = V (x′, x3), (x′, x3) ∈ Ω. and we introduce the self-adjoint operator AV = A acting in L2(Ω):

A

= −∆ + V D(A) = H1

0(Ω) ∩ H2(Ω).

Filonov, Kachkovskyi ’09: σ(A) = σac(A).
Question: Uniqueness of V from (partial) spectral data of A?

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Floquet-Bloch-Gel’fand transform

For f ∈ C ∞

0 (Ω) and θ ∈ [0, 1),

(Uf )θ(x) =

+∞

k=−∞

e−ikθf (x′, x3 + k), x′ ∈ ω, x3 ∈ R.

Quasi-periodicity: (Uf )θ(x′, x3 + 1) = eiθ(Uf )θ(x′, x3).
Elementary cell: ˜

Ω = ω × (0, 1).

U extends to a unitary transform from L2(Ω) onto

⊕

(0,1)

L2(˜ Ω)dθ = L2((0, 1)dθ; L2(˜ Ω)),

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Fiber decomposition of A

Main properties of U:

(i)

U ∂f

∂z

θ

(x) = ∂(Uf )θ ∂z (x), z = x1, x2, x3 (ii) (U(Vf ))θ(x) = V (x)(Uf )θ(x) since V (x′, x3 + 1) = V (x′, x3).

Consequence:

(UAf )θ = (−∆ + V )(Uf )θ, θ ∈ [0, 1).

Fiber operators A(θ), θ ∈ [0, 1), acting in L2(˜

Ω) as      A(θ) = −∆ + V D(A(θ)) =

u ∈ L2((0, 1)dx3; H1

0(ω)) ∩ H1(˜

Ω), ∆u ∈ L2(Y ), ∂ℓ

x3u(·, 1) = eiθ∂ℓ x3u(·, 0), ℓ = 0, 1

.

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Direct integral decomposition

This fiber decomposition may be reformulated as UAU∗ = ⊕

(0,1)

A(θ)dθ, which is equivalent to:        (i) ∀f ∈ D(A), (Uf )θ ∈ D(A(θ)), a.e. θ ∈ (0, 1); (ii) (UAf )θ = A(θ)(Uf )θ, a.e. θ ∈ (0, 1); (iii)

(0,1) A(θ)(Uf )θ2

L2(˜ Ω)dθ = Af 2 L2(Ω).

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Spectral decomposition of A(θ)

The resolvent of A(θ) is compact by the compactness of

H1

0(ω) ֒

→ L2(ω).

The spectrum of A(θ) is discrete:

σ(A(θ)) = σd(A(θ)) = {λn(θ), n ≥ 1}, λ1(θ) ≤ λ2(θ) ≤ . . .

{ϕn(θ), n ≥ 1}= L2(˜

Ω)-orthonormal basis of (real valued) eigenfncs of A(θ): A(θ)ϕn(θ) = λn(θ)ϕn(θ), n ≥ 1.

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Generalized eigenpairs of A

For θ ∈ [0, 1) and all n ≥ 1, put

φn(x′, x3 + k, θ) = eikx3ϕn(x′, x3), (x′, x3, θ) ∈ ˜ Ω, k ∈ Z.

φn(·, θ) is a generalized eigenfunction of A associated with the

generalized eigenvalue λn(θ): φn(·, θ) ∈ Dloc(A) = {u ∈ L2

loc(Ω), ∀χ ∈ C ∞ 0 (R), χ(x3)u ∈ D(A)};

(−∆ + V )φn(·, θ) = λn(θ)φn(·, θ) in (C ∞

0 (Ω))′.

Generalized Fourier coefficients of f ∈ L2(Ω):

fn(θ) =

Ω

f (x)φn(x, θ)dx, (n, θ) ∈ N∗ × [0, 1).

For all f ∈ D(A) it holds true that

(Af )n(θ) = λn(θ)fn(θ), (n, θ) ∈ N∗ × [0, 1).

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Spectral decomposition of A

{φn(·, θ), (n, θ) ∈ N∗ × [0, 1)} is a complete system of

generalized eigenfunctions of A in the sense that the mapping f → (fn)n≥1, is unitary from L2(Ω) onto

n≥1 L2(0, 1).

A ≃ multiplication operator by the family of functions (λn)n≥1:

Λ :

n≥1 L2(0, 1)

→

n≥1 L2(0, 1)

(fn)n≥1 → (λnfn)n≥1

Since the θ → λn(θ) are non constant, the spectrum of Λ (and

thus of A) is absolutely continuous: σ(A) = σ(Λ) =

n≥1

λn([0, 1)).

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Uniqueness result

Consider two potentials Vj ∈ L∞(Ω; R), j = 1, 2, such that

Vj(x′, x3 + 1) = Vj(x′, x3), (x′, x3) ∈ Ω.

◮ Aj is the Dirichlet Laplacian in L2(Ω), perturbed by Vj. ◮ {(λj,n(θ), φj,n(·, θ)), n ≥ 1, θ ∈ [0, 1)} is the set of

generalized eigenpairs of Aj.

Theorem (Kavian, Kian, ES):

For θ0 ∈ [0, 1) and N ∈ N∗ fixed, assume that

λ1,n(θ0) = λ2,n(θ0), n ≥ N,

∂νφ1,n(·, θ0)|∂ω×(0,1) = ∂νφ2,n(·, θ0)|∂ω×(0,1), n ≥ N, where ν is the outgoing normal unit to ∂ω. Then we have V1 = V2.

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Comments

Let V ∈ L∞(RN), N ≥ 1, obey

V (x + d) = V (x), x ∈ RN, d ∈ L = ⊕N

j=1Zbj,

where b1, . . . , bN are N independent vectors in RN.

A = −∆ + V , acting in L2(RN), is ≃

⊕

(0,1)N A(θ)dθ.

Each A(θ) = −∆ + V acts in L2(˜

Ω), with ˜ Ω =   

N

j=1

tjbj, 0 ≤ tj < 1    .

The domain of A(θ) is made of fncs obeying for j = 1, . . . , N:

∂ℓ

xju(x + bj) = eiθj∂ℓ xju(x), x ∈ ∂ ˜

Ω, x + bj ∈ ∂ ˜ Ω, ℓ = 0, 1.

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Comments, continued

Let Aj, j = 1, 2, be the above operator with V = Vj.
Then we have the implication:

(σ(A1(0)) = σ(A2(0))) ⇒

σ(A1(θ)) = σ(A2(θ)), ∀θ ∈ [0, 1)N

;

◮ Mac Kean, Trubowitz ’76: case N = 1 ◮ Eskin, Ralston, Trubowitz ’84: case N ≥ 2 provided V is real

analytic + some “geometric” condition: ∀d, d′ ∈ L, |d| = |d′| ⇒ d = ±d′.

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Sketch of the proof

For θ ∈ [0, 1), j = 1, 2, and λ ∈ C \ [−M, +∞), with

M = maxj=1,2 VjL∞(Ω), we consider the BVP    (−∆ + Vj − λ)uj = 0, in ˜ Ω, uj = f

n ∂ω × (0, 1),

∂ℓ

x3uj(x′, 1)

= eiθ∂ℓ

x3uj(x′, 0),

x′ ∈ ω, ℓ = 0, 1, where f ∈ H1(∂ω × (0, 1)) obeys ∂ℓ

x3f (x′, 1) = eiθ∂ℓ x3f (x′, 0), x′ ∈ ∂ω, ℓ = 0, 1.

DN operators Λj = Λj,θ,λ : f → (∂νuj)|∂ω×(0,1), where uj is the

solution to the above BVP.

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DN operators estimate

Fix f ∈ H3/2

θ

(∂ω × (0, 1)), θ ∈ [0, 1) and λ ∈ C \ [−M, +∞).

Put ˜

fj,n(θ) = f , ∂νϕj,n(·, θ)L2(∂ω×(0,1)), for n ≥ 1 and j = 1, 2: uj(x) = −

+∞

n=1

˜ fj,n(θ) λj,n(θ) − λϕj,n(x, θ), x ∈ ˜ Ω.

This entails for all x ∈ ∂ω × (0, 1) that

(Λ1,θ0,λ − Λ2,θ0,λ)f (x) = −

N

n=1
˜

f1,n(θ0) λ1,n(θ0) − λ∂νϕ1,n(x, θ0) − ˜ f2,n(θ0) λ2,n(θ0) − λ∂νϕ2,n(x, θ0)

,

and consequently that Λ1,θ0,λ − Λ2,θ0,λ∗ ≤ C

|λ| for some suitable

topology, where C > 0 is independent of λ.

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The Born approximation method, revisited

For k ∈ Z, λ ∈ C \ R, η1, η2 ∈ S1, put
f (x′, x3)

= ei(

√ λη1·x′+(θ0−k)x3)

g(x′, x3) = ei

√ λη2·x′+θ0x3

, x′ ∈ ω, x3 ∈ R,

and Sj(λ, η1, η2) = Λj,θ0,λf , gL2(∂ω×(0,1)), j = 1, 2.

For ξ ∈ R2 \ {0}, choose λ(r) ∈ S1 and η1(r), η2(r) ∈ S1 s. t.

√ λ(η1 − η2) → −ξ and Im (λ) → +∞ as r → +∞.

For j = 1, 2, that Sj(r) = Sj(η1(r), η2(r), λ(r)) satisfies

lim

r→+∞ Sj(r)

= −|ξ|2 2

˜

Ω

e−i(x′·ξ+kx3)dx′dx3 +

˜

Ω

θ2

0 + Vj

e−i(x′·ξ+kx3)dx′dx3.

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End of the proof

For any ξ ∈ R2 \ {0} and k ∈ Z, we have:

lim

r→+∞ (S1(r) − S2(r))

=

Y

(V1 − V2)e−i(x′·ξ+kx3)dx′dx3.

On the other hand, since |λ| → +∞ as r → +∞, and

|S1(r) − S2(r)| ≤ Λ1,θ0,λ − Λ2,θ0,λ∗f g ≤ C |λ|, C > 0 being independent of λ, we get that lim

r→+∞ (S1(r) − S2(r)) = 0.

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Going a little bit further...

Under the conditions of the above uniqueness result, pick θ1 and θ2 in [0, 1) (they are not necessarily the same) and N ∈ N∗. Assume that λ1

n(θ1) = λ2 n(θ2),

e−iθ1x3∂νφ1,n(·, θ1)
|∂ω×(0,1) =
e−iθ2x3∂νφ2,n(·, θ2)
|∂ω×(0,1) ,

for all n ≥ N. Then we have: V1(x) − V2(x) = θ2

2 − θ2 1, x ∈ Ω.

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The non-constant conductivity case

Given two real-valued conductivities aj fulfilling

aj(x) ≥ c > 0, x ∈ Ω, and aj(x′, x3 + 1) = aj(x′, x3), x′ ∈ ω, x3 ∈ R, we consider the operators Aj = −∇ · aj∇ + Vj acting in L2(Ω) with DBC conditions.

Similarly as before, we have Aj =

⊕

(0,1) Aj(θ)dθ, and the spectral

properties of Aj(θ) are unchanged: σ(Aj(θ)) = σd(Aj(θ)) = {λj,n(θ), n ≥ 1}.

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The non-constant conductivity case, continued

Admissible coefficients: aj ∈ W 2,∞(Ω) ∩ C 1(Ω), j = 1, 2, must

satisfy ∇a1 = ∇a2 on ∂Ω.

In this case the result reads:

For θ0 ∈ [0, 1) and N ∈ N∗ fixed, assume that λ1,n(θ0) = λ2,n(θ0), (a1∂νφ1,n(·, θ0))|∂ω×(0,1) = (a2∂νφ2,n(·, θ0))|∂ω×(0,1) , for all n ≥ N. Then we have a1 = a2 and V1 = V2.

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Time dependent case

We consider the Schr¨

dinger equation

   −i∂tu − ∆u + Vu = in Q u(0, ·) = u0 in Ω, u = g

n Σ,

(5.2) where the scalar potential V : Q → R is 1-periodic wrt x3: V (t, x′, x3 + 1) = V (t, x′, x3), (t, x′, x3) ∈ Q. Question: Can V be retrieved stably from the DN operator ΛV : (g, u0) → (∂νu|Σ, u(T, ·)), where u = u(g, u0) is the solution to (5.2)?

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The DN operator

Boundary and initial data:

τ : H2(0, T; H2(Ω)) → L2((0, T) × R, H3/2(∂ω)) × L2(Ω) w →

w|Σ, w(0, ·)
.

and put X = τ(H2(0, T; H2(Ω))).

Assuming that V ∈ C([0, T], W 2,∞(Ω)) and (g, u0) ∈ X, the

IBVP (5.2) admits a unique solution in L2(0, T; H2(Ω)) ∩ H1(0, T; L2(Ω)), that depends continuously on the (g, u0), so that the linear

perator ΛV is bounded from X to Y = L2(Σ) × L2(Ω).

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Stability result (Choulli, Kian, ES)

Let V1, V2 ∈ W 2,∞(0, T; W 2,∞(Ω)) be 1-periodic wrt x3 and fulfill the two following conditions: (V2 − V1)(T, ·) = (V2 − V1)(0, ·) = 0 in ˜ Ω, V2 − V1 = 0 in ˜ Σ = (0, T) × ∂ω × (0, 1). Put M = max{VjW 2,∞(0,T;W 2,∞(˜

Ω)), j = 1, 2}. Then there are

two constants C > 0 and γ∗ > 0, depending only on T, ω and M, such that the estimate V2 − V1L2( ˜

Q) ≤ C

ln
1

ΛV2 − ΛV1B(X,Y ) − 2

5

, holds whenever 0 < ΛV2 − ΛV1B(X,Y ) < γ∗.

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Fibered IBVP

For θ ∈ [0, 1) we consider the IBVP        (−i∂t − ∆ + V )v = in (0, T) × ˜ Ω, v(0, ·) = v0 in ˜ Ω, v = h

n ˜

Σ, ∂ℓ

x3v(·, 1)

= eiθ∂ℓ

x3v(·, 0)

n (0, T) × ω, ℓ = 0, 1.

(5.3)

Fibered boundary and spectral data:

˜ τ : H2(0, T; H2(˜ Ω)) → L2((0, T) × (0, 1), H3/2(∂ω)) × L2(˜ Ω) w →

w|˜

Σ, w(0, ·)

.

and define ˜ Xθ = ˜ τ(H2(0, T; H2

θ (˜

Ω))), where H2

θ (˜

Ω) = {u ∈ H2(˜ Ω); ∂ℓ

x3u(·, ·, 1) = eiθ∂ℓ x3u(·, ·, 0), ℓ = 0, 1}.

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Fiber decomposition of the DN operators

Fibered DN operators:

ΛV ,θ : ˜ Xθ → ˜ Y = L2((0, T) × ∂ω × (0, 1)) × L2(˜ Ω) (v0, h) → (∂νvθ, vθ(T, ·)), where vθ is the L2(0, T; H2

θ (˜

Ω)) ∩ H1(0, T; L2(˜ Ω))-solution to (5.3) associated with (v0, h), is bounded.

Direct integral decomposition:

UΛV U−1 = ⊕

(0,1)

ΛV ,θdθ, which entails that ΛV B(X,Y ) = sup

θ∈(0,1)

ΛV ,θB( ˜

Xθ, ˜ Y ).

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Stability result in ˜ Q

The theorem follows from this and the corresponding stability estimate in ˜ Q: Let M and Vj, j = 1, 2, be the same as in the theorem. Then we may find two constants C > 0 and γ∗ > 0, depending on T, ω and M, such that we have V2 − V1L2( ˜

Q) ≤ C

ln
1

ΛV2,θ − ΛV1,θB( ˜

Xθ, ˜ Y )

− 2

5

, for any θ ∈ [0, 1), provided 0 < ΛV2,θ − ΛV1,θB( ˜

Xθ, ˜ Y ) < γ∗.

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Half-plane quantum Hall Hamiltonian

Charged particle in Ω := R∗

+ × R ⊂ R2 submitted to a constant

magnetic field with strength b > 0, orthogonal to Ω.

Hamiltonian H(b) := (−i∇ − a)2 in Ω with magnetic potential

a(x, y) := (0, −bx) and DBC at x = 0.

Scaling: VbH(b)V∗

b = bH(1), where the transform

(Vbψ)(x, y) := b−1/4ψ

x

b1/2 , y b1/2

is unitary in L2(Ω).
Chose b = 1 and focus on the operator

H := −∂2

x + (−i∂y − x)2,

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Fiber decomposition

Fy is the partial Fourier transform with respect to y, i.e.

ˆ ϕ(x, k) = (Fyϕ)(x, k) := 1 √ 2π

R

e−ikyϕ(x, y)dy, ϕ ∈ L2(Ω).

Direct integral decomposition

FyHF∗

y =

⊕

R

h(k)dk, where the 1D operator h(k) := −∂2

x + V (x, k), V (x, k) := (x − k)2, x > 0, k ∈ R,

acts in L2(R+) with DBC at x = 0.

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Band functions

For all k ∈ R, h(k)−1 is compact since V (x, k)

|x|→+∞

− → +∞.

Discrete spectrum: σ(h(k)) = {λn(k), n ∈ N∗},

h(k)ψn(x, k) = λn(k)ψn(x, k), x ∈ R∗

+, n ∈ N∗,

where ψn(·, k) is real and L2-normalized.

R ∋ k → λn(k), n ∈ N∗, are the band functions of H:

◮ Simple eigenvalues that are analytic in k; ◮ λ′ n(k) < 0 for all k ∈ R; ◮ limk→−∞ λn(k) = +∞ and limk→+∞ λn(k) = En := 2n − 1.

The spectrum of H is a.c.:

σ(H) = ∪n≥1λn(R) = [E1, +∞).

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Approximate shape of the band functions

−1 −0.5 0.5 1 1.5 2 2.5 3 3.5 4 2 4 6 8 10 12 k ´ Eric Soccorsi Inverse problems in a quantum waveguide

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

The current operator

Time evolution of y (the multiplier by the coordinate y) in L2(Ω):

y(t) := e−itHyeitH, t ∈ R.

Second component of the velocity:

dy(t) dt = −i[H, y(t)] = −ie−itH[H, y]eitH.

The current operator:

Jy := −i[H, y] = −i∂y − x.

Current carried by a state ϕ is Jyϕ, ϕL2(Ω).

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Connection with the velocity operator

Λ′ : {fn, n ∈ N∗} → {λ′

nfn, n ∈ N∗} acting in

n∈N∗

L2(R).

Any ϕ ∈ L2(Ω) decomposes as ϕ =

n≥1 πn(ϕ), where

πn(ϕ)(x, y) :=

1 √ 2π

R eiykϕn(k)ψn(x, k)dk, (x, y) ∈ Ω

ϕn(k) := ˆ ϕ(·, k), ψn(·, k)L2(R+).

Feynman-Hellman formula:

Jyπn(ϕ), πn(ϕ)L2(Ω) =

R

λ′

n(k)|ϕn(k)|2dk, ϕ ∈ L2(Ω).

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Energy concentration

Let I ⊂ R be a bounded interval.
We note PI the spectral projection of H associated with I and

we consider ϕ ∈ L2(Ω) obeying PIϕ = ϕ, ie ∀n ∈ N∗, supp(ϕn) ⊂ λ−1

n (I).

Assume for simplicity that I ⊂ (E1, E2) so that ϕ = π1(ϕ), or

equivalently, ˆ ϕ(·, k) = ϕ1(k)ψ1(·, k).

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Edge velocity

Then we have d dt y(t)ϕ, ϕL2(Ω) = JyeitHϕ, eitHϕL2(Ω) = (k − x)eitλ1(k) ˆ ϕ(k), eitλ1(k) ˆ ϕ(k)L2(Ω) =

R

|ϕ1(k)|2

R+

(k − x)|ψ1(x, k)|2dx

dk.

This entails d dt y(t)ϕ, ϕL2(Ω) =

R

λ′

1(k)|ϕ1(k)|2dk,

with the aid of the Feyman-Hellman formula.

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Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions

Classical trajectories

b > 0

´ Eric Soccorsi Inverse problems in a quantum waveguide