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
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 R N − 1 , N ≥ 2. Given T > 0 we examine the IBVP − i ∂ t u − ∆ u + Vu = 0 in Q = (0 , T ) × Ω , u (0 , · ) = u 0 in Ω , (1.1) = on Σ = (0 , T ) × Γ , u g where Γ := ∂ω × R . Here u 0 (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). ´ Eric Soccorsi Inverse problems in a quantum waveguide
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 10 8 / 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 . ´ Eric Soccorsi Inverse problems in a quantum waveguide
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¨ odinger 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. ´ Eric Soccorsi Inverse problems in a quantum waveguide
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 ´ Eric Soccorsi Inverse problems in a quantum waveguide
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 ) A V = − d 2 dx 2 + V ( x ) acting in L 2 (Ω) with DBC at x = 0 , 1 . • The spectrum of A V is purely discrete: σ ( A V ) = σ d ( A V ) = { λ n ( V ) , n ≥ 1 } with λ 1 ( V ) < λ 2 ( V ) < . . . • To each eigenvalue λ n ( V ) we associate a unique eigenfunction ϕ n ( · , V ), normalized by � ϕ n ( · , V ) � L 2 (Ω) = 1 and ϕ ′ n (0 , V ) > 0 . ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions Dirichlet inverse problem • Is V uniquely determined by σ ( A V )? ◮ λ : L ∞ (Ω) ∋ V �→ ( λ n ( V )) n ≥ 1 is not injective. ◮ The Dirichlet spectrum cannot distinguish left from right: � � � � ˜ λ ( ˜ V ( x ) = V (1 − x ) , x ∈ Ω ⇒ 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 �→ ( λ n ( V ) , ϕ ′ n (1 , V )) n ≥ 1 is injective . ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions Multi-dimensional case • Ω is a bounded open subset of R N , N ≥ 2, with C 2 boundary and the scalar potential V ∈ L ∞ (Ω; R ). odinger operators A V , acting in L 2 (Ω): • Self-adjoint Schr¨ � A V = − ∆ + V H 1 0 (Ω) ∩ H 2 (Ω) . D ( A V ) = • Purely discrete spectrum: σ ( A V ) = σ d ( A V ) = { λ n ( V ) , n ≥ 1 } , λ 1 ( V ) ≤ λ 2 ( V ) ≤ . . . • { ϕ n ( · , V ) , n ≥ 1 } = L 2 (Ω)-orthonormal basis of eigenfncs of A V : A V ϕ n ( · , V ) = λ n ( V ) ϕ n ( · , V ) , n ≥ 1 . ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions Multi-dimensional Borg-Levinson theorem • Nachmann, Sylvester, Uhlmann ’88 and Novikov ’88: L ∞ (Ω) ∋ V �→ � � λ n ( V ) , ∂ ν ϕ n ( · , V ) | ∂ Ω n ≥ 1 is injective , where ∂ ν φ = ∇ 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. ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions 3D inverse spectral problem From now on ω is a connected bounded open subset of R 2 with C 2 -boundary and we consider the 3D unbounded waveguide : Ω = ω × R = { x = ( x ′ , x 3 ) , x ′ = ( x 1 , x 2 ) ∈ ω, x 3 ∈ R } . The scalar potential V ∈ L ∞ (Ω; R ) is 1-periodic wrt x 3 , V ( x ′ , x 3 + 1) = V ( x ′ , x 3 ) , ( x ′ , x 3 ) ∈ Ω . and we introduce the self-adjoint operator A V = A acting in L 2 (Ω): � A = − ∆ + V H 1 0 (Ω) ∩ H 2 (Ω) . D ( A ) = • Filonov, Kachkovskyi ’09: σ ( A ) = σ ac ( A ). • Question : Uniqueness of V from (partial) spectral data of A ? ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions Floquet-Bloch-Gel’fand transform • For f ∈ C ∞ 0 (Ω) and θ ∈ [0 , 1), + ∞ e − ik θ f ( x ′ , x 3 + k ) , x ′ ∈ ω, x 3 ∈ R . � ( U f ) θ ( x ) = k = −∞ • Quasi-periodicity : ( U f ) θ ( x ′ , x 3 + 1) = e i θ ( U f ) θ ( x ′ , x 3 ). • Elementary cell : ˜ Ω = ω × (0 , 1). • U extends to a unitary transform from L 2 (Ω) onto � ⊕ L 2 (˜ Ω) d θ = L 2 ((0 , 1) d θ ; L 2 (˜ Ω)) , (0 , 1) ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions Fiber decomposition of A • Main properties of U : � U ∂ f � ∂ ( U f ) θ ( i ) ( x ) = ( x ) , z = x 1 , x 2 , x 3 ∂ z ∂ z θ V ( x )( U f ) θ ( x ) since V ( x ′ , x 3 + 1) = V ( x ′ , x 3 ) . ( ii ) ( U ( Vf )) θ ( x ) = • Consequence: ( U Af ) θ = ( − ∆ + V )( U f ) θ , θ ∈ [0 , 1) . • Fiber operators A ( θ ), θ ∈ [0 , 1), acting in L 2 (˜ Ω) as A ( θ ) = − ∆ + V � 0 ( ω )) ∩ H 1 (˜ u ∈ L 2 ((0 , 1) dx 3 ; H 1 Ω) , ∆ u ∈ L 2 ( Y ) , D ( A ( θ )) = ∂ ℓ x 3 u ( · , 1) = e i θ ∂ ℓ � x 3 u ( · , 0) , ℓ = 0 , 1 . ´ Eric Soccorsi Inverse problems in a quantum waveguide
Introduction Inverse spectral problem Spectral analysis Uniqueness result Time dependent case More on band functions Direct integral decomposition This fiber decomposition may be reformulated as � ⊕ U A U ∗ = A ( θ ) d θ, (0 , 1) which is equivalent to: ( i ) ∀ f ∈ D ( A ) , ( U f ) θ ∈ D ( A ( θ )) , a.e. θ ∈ (0 , 1); ( ii ) ( U Af ) θ = A ( θ )( U f ) θ , a.e. θ ∈ (0 , 1); (0 , 1) �A ( θ )( U f ) θ � 2 Ω) d θ = � Af � 2 � ( iii ) L 2 (Ω) . L 2 (˜ ´ Eric Soccorsi Inverse problems in a quantum waveguide
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