Spectral estimates and Random Weighted Sobolev Inequalities Spectral estimates and Random Weighted Sobolev Inequalities Didier Robert in collaboration with Aur´ elien Poiret and Laurent Thomann Conference in honor of Johannes Sj¨ ostrand CIRM, September 23, 2013 1
Spectral estimates and Random Weighted Sobolev Inequalities Content 1 Introduction, Random series 2 Probabilities on scales of Hilbert spaces 3 Spectral estimates for polynomial potentials 4 Probabilistic weighted Sobolev estimates 5 Random Quantum Ergodicity 2
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series The simplest model is the 1-D torus T = R / 2 π Z with its Sobolev c n e inx then � f � 2 � � (1 + | n | ) 2 s | c n | 2 . spaces. Let f ( x ) = H s ( T ) = n ∈ Z n ∈ Z By the usual Sobolev embeddings, if f ∈ H 1 / 2 − 1 / p ( T ) with p ≥ 2 then f ∈ L p ( T ). Paley and Zygmund (1930) have improved this result allowing random coefficients. X n ( ω ) c n e inx where { X n } is a sequence of � Let f ω ( x ) = n ∈ Z independent Bernoulli random variables. 3
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series The simplest model is the 1-D torus T = R / 2 π Z with its Sobolev c n e inx then � f � 2 � � (1 + | n | ) 2 s | c n | 2 . spaces. Let f ( x ) = H s ( T ) = n ∈ Z n ∈ Z By the usual Sobolev embeddings, if f ∈ H 1 / 2 − 1 / p ( T ) with p ≥ 2 then f ∈ L p ( T ). Paley and Zygmund (1930) have improved this result allowing random coefficients. X n ( ω ) c n e inx where { X n } is a sequence of � Let f ω ( x ) = n ∈ Z independent Bernoulli random variables. If f ∈ L 2 ( T ) then for all p ≥ 2, a.s f ω ∈ L p ( T ). 4
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series log α (1 + | n | ) | c n | 2 < + ∞ then a.s � Moreover if for some α > 1, n ∈ Z f ω ∈ C ( T ). Many other results concerning random trigonometric series were obtained by Paley and Zygmund, as it is detailed in the beautiful book of J-P. Kahane (Some random series of functions). 5
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series The setting of random trigonometric series was extended to Riemannian compact manifolds for orthonormal basis of eigenfunctions of the Laplace-Beltrami operator, in particular by Burq, Lebeau, Tvzetkov. The main motivations and applications are the following : 6
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series The setting of random trigonometric series was extended to Riemannian compact manifolds for orthonormal basis of eigenfunctions of the Laplace-Beltrami operator, in particular by Burq, Lebeau, Tvzetkov. The main motivations and applications are the following : (1) To get existence and well posedness results for non linear PDE (wave equation or Schr¨ odinger equation) in supercritical cases. (2) For linear self-adjoint PDE with high multiplicity eigenvalues (Laplace on the 2-sphere; harmonic oscillator with D ≥ 2) find basis of eigenfunctions satisfying ”better” L ∞ estimates or satisfying a quantum ergodic property (Zelditch considered the 2-sphere (1992), recently Burq-Lebeau have improved his result) 7
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series In our joint work with A. Poiret and L. Thomann we extend the odinger operators in L 2 ( R d ), d ≥ 2. Burq-Lebeau analysis to Schr¨ Moreover we consider more general probability measures on the spectral subspaces E h satisfying a Gaussian concentration property. 8
Spectral estimates and Random Weighted Sobolev Inequalities Introduction, Random series In our joint work with A. Poiret and L. Thomann we extend the odinger operators in L 2 ( R d ), d ≥ 2. Burq-Lebeau analysis to Schr¨ Moreover we consider more general probability measures on the spectral subspaces E h satisfying a Gaussian concentration property. Here we shall only consider the applications (2). For the details, see the link to our preprint: Random weighted Sobolev inequalities and application to Hermite functions and the soon forthcoming paper : Random weighted Sobolev inequalities for Schr¨ odinger operator with superquadratic potentials. Concerning applications of our results to NLS in supercritical cases with (or without) harmonic potential, see the link to our preprint: Probabilistic global well-posedness for the supercritical nonlinear harmonic oscillator. 9
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces Let { X n } a sequence of complex i.i.d random variables, of common law ν satisfying the following concentration property: there exist constants c , C > 0 independent of N ∈ N such that for all Lipschitz and convex function F : C N − → R Cr 2 − ν ⊗ N � X ∈ C N : � � F � 2 � ≥ r Lip , � � � F ( X ) − E ( F ( X )) ≤ c e ∀ r > 0 , (1) where � F � Lip is the best constant so that | F ( X ) − F ( Y ) | ≤ � F � Lip � X − Y � ℓ 2 . Examples: Gauss law, Bernoulli law and more generally measures with compact support (Talagrand theorem). 10
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces Let K a separable complex Hilbert space and K is a self-adjoint, positive operator on K with a compact resolvent. We denote by { ϕ j , j ≥ 1 } an orthonormal basis of eigenvectors of K , K ϕ j = λ j ϕ j , and { λ j , j ≥ 1 } is the non decreasing sequence of eigenvalues of K (each is repeated according to its multiplicity). Then we get a natural scale of Sobolev spaces associated with K , defined for s ≥ 0 by K s = Dom ( K s / 2 ). 11
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces Let γ = { γ j } j ≥ 1 a sequence of complex numbers such that j | γ j | 2 < + ∞ . � λ s j ≥ 1 γ j ϕ j ∈ K s and v ω � � We denote by v γ = γ = γ j X j ( ω ) ϕ j . We j ≥ 1 j ≥ 1 have E ( � v ω γ � 2 K ) < + ∞ , therefore v ω γ ∈ K s , a.s. We define the measure µ γ on K s as the probability law of the random vector v ω γ . These measures were introduced by Burq-Tzvetkov (2008). They are much more flexible than Gibbs measures known before in some particular cases (Lebowitz, Bourgain). 12
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces Some properties (i) If the support of ν is C and if γ j � = 0 for all j ≥ 1 then the support of µ γ is K s . ∈ K s + ǫ then µ γ ( K s + ǫ ) = 0. (ii) If for some ǫ > 0 we have v γ / (iii) Assume that we are in the particular case where d ν ( x ) = c α e −| x | α d x with α ≥ 2. Let γ = { γ j } and β = { β j } be two complex sequences and assume that � 2 �� a / 2 � γ j � � � − 1 = + ∞ . � � β j � � j ≥ 1 Then the measures µ γ and µ β are mutually singular, i.e there exists a measurable set A ⊂ K s such that µ γ ( A ) = 1 and µ β ( A ) = 0. 13
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces Consider finite dimensional subspaces E h of K defined by spectral localizations depending on a small parameter 0 < h ≤ 1 ( h − 1 is a measure of energy for the quantum Hamiltonian K ). Let I h = [ a h h , b h h [, Λ h = { j , λ j ∈ I h } , N h = #Λ h and E h the spectral subspace of K in the interval I h . Our goal is to find uniform estimates in h ∈ ]0 , h 0 [ for h 0 > 0 small enough. Consider the random vector in E h : � v γ ( ω ) := v γ, h ( ω ) = γ j X j ( ω ) ϕ j . (2) j ∈ Λ h Introduce the squeezing condition: | γ n | 2 ≤ K 0 � | γ j | 2 , ∀ n ∈ Λ h , ∀ h ∈ ]0 , h 0 ] . (3) N h j ∈ Λ h 14
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces To get estimates from below we also need: K 1 | γ j | 2 ≤ | γ n | 2 ≤ K 0 � � | γ j | 2 , ∀ n ∈ Λ h , ∀ h ∈ ]0 , 1] . (4) N h N h j ∈ Λ h j ∈ Λ h 15
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces To get estimates from below we also need: K 1 | γ j | 2 ≤ | γ n | 2 ≤ K 0 � � | γ j | 2 , ∀ n ∈ Λ h , ∀ h ∈ ]0 , 1] . (4) N h N h j ∈ Λ h j ∈ Λ h Why this assumption? For 1-D Hamiltonians the eigenvalues are non degenerate and it is possible to get accurate L ∞ estimates on eigenfunctions. But for D ≥ 2 eigenvalues may have high multiplicities and it is much more difficult to get accurate L ∞ estimates. With condition (3) or (4) we shall see that it is enough to know good estimates for the spectral functions in small energy windows instead of individual eigenfunctions. 16
Spectral estimates and Random Weighted Sobolev Inequalities Probabilities on scales of Hilbert spaces Now consider probabilities on the unit sphere S h of the subspaces E h . The random vector v γ defines a probability measure ν γ, h on E h . We define a probability measure P γ, h on S h as the image of v ν γ, h by v �→ � v � . Examples: 1 • If | γ n | = N for all j ∈ Λ and if X n follows the complex normal √ law N C (0 , 1) then P γ, h is the uniform probability on S h considered in Burq-Lebeau. • Assume that for all n ∈ N , P ( X n = 1) = P ( X n = − 1) = 1 / 2, then P γ, h is a convex sum of 2 N Dirac measures. In the first example P γ, h is invariant by e − itK . 17
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