Spectral inequality for the Schr¨ odinger equation −△ g + V ( x ) in R d Gilles Lebeau † , IvanMoyano ‡ † D ´ epartementdeMath ´ ematiques , Universit´ e Nice Sophia Antipolis, lebeau@unice.fr ‡ CenterforMathematicalSciences , im 449@ dpmms . cam . ac . uk Marrakech, April 19, 2018 Premier (?) Congr` es Franco Marocain de Math´ ematiques Appliqu´ ees Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Outline Spectral inequality on a compact manifold 1 Spectral inequality for −△ in R d , a short review 2 Main result 3 Sketch of proof 4 Interpolation for holomorphic functions Holomorphic extensions Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Outline Spectral inequality on a compact manifold 1 Spectral inequality for −△ in R d , a short review 2 Main result 3 Sketch of proof 4 Interpolation for holomorphic functions Holomorphic extensions Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Spectral inequality on a compact manifold Let ( M , g ) be a smooth connected compact manifold. Let △ g be the (negative) Laplace operator on M , and let −△ g e j = λ j e j , 0 = λ 0 < λ 1 ≤ λ 2 ≤ ... be its spectral decomposition. The following spectral inequality was proved by Jerison-Lebeau and Lebeau-Zuazua in 1998-1999. Theorem Let ω ⊂ M be a non void open subset of M. There exists constants A = A ( ω ) > 0 , C = C ( ω ) > 0 such that for all λ > 0 and all sequence { z j } j ∈ N of complex numbers, one has | z j | 2 ≤ Ae C λ 1 / 2 � � � z j e j ( x ) | 2 d g x . | (1.1) ω λ j <λ λ j <λ Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Outline Spectral inequality on a compact manifold 1 Spectral inequality for −△ in R d , a short review 2 Main result 3 Sketch of proof 4 Interpolation for holomorphic functions Holomorphic extensions Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
In the rest of the talk, we will work in R d , and ω ⊂ R d will denote a measurable set satisfying the density assumption: ∃ R , δ > 0 , x ∈ R d mes { t ∈ ω, | x − t | < R } ≥ δ. inf (2.1) such that Theorem Let ω ⊂ R d be a measurable set satisfying the geometric condition (2.1). There exists constants A = C ( ω ) , C = C ( ω ) > 0 such that the following hods true. For all µ > 0 and all f ∈ L 2 ( R d ) , such that support (ˆ f ) ⊂ { ξ ∈ R d , | ξ | ≤ µ } , where ˆ f is the Fourier transform of f , one has � � f � 2 L 2 ( R d ) ≤ Ae C µ | f ( x ) | 2 dx . (2.2) ω Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Using techniques from Harmonic Analysis, Logvinenko and Sereda proved ( Tero. Funk. Anal. i Prilozen , 1974) that in 1-d, the condition E ⊂ R measurable s.t. ∃ γ > 0 , a > 0 s.t. mes ( E ∩ I ) ≥ γ (2.3) mes ( I ) whenever I is an interval of length a , is sufficient to ensure that, when support(ˆ f ) ⊂ [ − b , b ]: � | f ( x ) | 2 dx ≥ C � f � 2 ∃ C = C ( γ, a , b ) > 0 s.t. L 2 ( R ) . (2.4) E On the other hand, the authors were not able to quantify the dependence of C with respect to the parameters a , b , γ . This was achieved in the one-dimensional case by Kovrojkine ( The Uncertainty principle for relatively dense sets and lacunary spectra , 2002) where the author proves that � γ � ab +1 ∃ K > γ such that C ( γ, a , b ) = , K Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Outline Spectral inequality on a compact manifold 1 Spectral inequality for −△ in R d , a short review 2 Main result 3 Sketch of proof 4 Interpolation for holomorphic functions Holomorphic extensions Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Let g be a given Riemannian metric on R d . Let ∆ g be the (negative) Laplace operator defined by the metric g . Let V = V ( x ) a real potential function such that lim x →∞ V ( x ) = 0. One defines the Schr¨ odinger operator associated to ( g , V ) by H g , V := − 1 2∆ g + V ( x ) (3.1) With reasonable hypothesis on ( g , V ), H g , V is a (bounded from below) unbounded self adjoint operator in L 2 ( R d ), and its spectrum σ ( g , V ) ⊂ [ E 0 , ∞ [ satisfies the following. σ ( g , V ) ∩ ] − ∞ , 0[ is purely discrete with eigenvalues of finite multiplicity, and its only possible accumulation point is 0. σ ( g , V ) ∩ ]0 , ∞ [ is absolutely continuous. For E ∈ R , we denote by Π E the spectral projector on ] − ∞ , E [ associated to H g , V . . Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
We will assume that ( g , V ) satisfy the following hypothesis ( H ) The metric g and the potential V are real analytic and there exists a > 0 such that they extend holomorphicaly in the complex domain U a = {| Im ( z ) | < a } . One has g = Id + ˜ g , where ˜ g is a symbol of degree < 0 in U a . V is a symbol of degree < 0 in U a . – Observe that even in the case g = Id , the assumption on the potential V allows long range perturbation. Short range perturbations are associated to potentials V which are symbols of degree < − 1 in U a . For the analysis of scattering theory for long range perturbation, we refer to H¨ ormander, in The analysis of linear pde’s vol 4, ch. XXX. – Observe also that the metric g may have trapped trajectories. Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Main result For E ∈ R , we define E 1 / 2 by ± √ E 1 / 2 E 1 / 2 � = E for E ≥ 0 , = ± i | E | for E < 0 . ± ± Theorem Let ( g , V ) satisfying hypothesis ( H ). There exists constants A = A ( ω, g , V ) , C = C ( ω, g , V ) > 0 such that for all E ∈ R and for all f ∈ L 2 ( R d ) , one has � Π E f � L 2 ( R d ) ≤ A | e CE 1 / 2 ± | � Π E f � L 2 ( ω ) . (3.2) Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Main result Observe that under the hypothesis ( H ), which allows long range perturbation, we may have dim ( range (Π 0 )) = ∞ . In particular, inequality 3.2 implies that any function f ∈ range (Π 0 ) satisfies: f ( x ) = 0 for all x ∈ ω ⇒ f = 0 . In fact, in the course of the proof, we will show that any f ∈ range (Π 0 ) extends holomorphicaly in U a for a > 0 small enough. Thus uniqueness holds true for any measurable set ω of positive measure. Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Outline Spectral inequality on a compact manifold 1 Spectral inequality for −△ in R d , a short review 2 Main result 3 Sketch of proof 4 Interpolation for holomorphic functions Holomorphic extensions Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
The first main ingredient is to use the following ” classical” interpolation Let ω ⊂ R d satisfying the inequality for holomorphic functions. hypothesis ( H ). There exists constants C int = C int ( ω, a ) > 0 and δ = δ ( ω, a ) ∈ (0 , 1) such that � δ �� � 1 − δ �� � R d | f | 2 dx ≤ C int | f | 2 dx | f | 2 | dz | , (4.1) ω U a for any f ∈ L 2 ( U a ) ∩ H ( U a ) satisfying � | f ( x + iy ) | 2 dx < ∞ sup (4.2) 0 ≤ b < a | y | = b Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
Poisson kernel We denote by d π E ( x , y ) the kernel of the spectral measure of H g , V , i.e � f , g ∈ L 2 ( R d ) . d π E ( f , g ) = R d × R d f ( x ) g ( y ) d π E ( x , y ) , Recall that d π E ( f , f ) is the positive measure on the line E ∈ R equal to the derivative of the left continuous and non decreasing function E �→ � Π E ( f ) � 2 L 2 . The Poisson kernel P s , ± ( x , y ) is the smooth function on ]0 , ∞ [ × R d × R d given by the formula � e − sE 1 / 2 ± d π E ( x , y ) . P s , ± ( x , y ) = (4.3) R For any f ∈ L 2 ( R d ), the smooth function on ]0 , ∞ [ × R d defined by � u ( s , x ) = R d P s , ± ( x , y ) f ( y ) dy satisfies the elliptic boundary problem ( − ∂ 2 s → 0 + u ( s , x ) = f ( x ) in L 2 ( R d ) . s + H g , V ) u = 0 , lim (4.4) Marrakech, April 19, 2018 Premier (?) Congr` es Lebeau-Moyano (Universit´ e Nice-Sophia Antipolis and University of Cambridge.) Spectral Inequality / 20
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