Anderson Hamiltonian with white noise potential Chouk Khalil Hu - PowerPoint PPT Presentation

Review of the one-dimensional case Anderson Hamiltonian with white noise potential Chouk Khalil Hu Berlin Joint work with R.Allez February 9, 2016 Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Construction and Statistical properties odinger operator on [0 , 1] d for d = 1 , 2 , 3 Goal 1: Define the stochastic random Schr¨ H := − ∆ + η where η is a Gaussian white noise; for x, y ∈ R d , E [ η ( x ) η ( y )] = δ ( x − y ) . Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Construction and Statistical properties odinger operator on [0 , 1] d for d = 1 , 2 , 3 Goal 1: Define the stochastic random Schr¨ H := − ∆ + η where η is a Gaussian white noise; for x, y ∈ R d , E [ η ( x ) η ( y )] = δ ( x − y ) . With possibly Dirichlet, Periodic or Neumann boundary conditions. Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Construction and Statistical properties odinger operator on [0 , 1] d for d = 1 , 2 , 3 Goal 1: Define the stochastic random Schr¨ H := − ∆ + η where η is a Gaussian white noise; for x, y ∈ R d , E [ η ( x ) η ( y )] = δ ( x − y ) . With possibly Dirichlet, Periodic or Neumann boundary conditions. Goal 2: It is a self-adjoint operator; What about its statistical spectral properties? Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Motivation and related question Describe the long time behavior (stationary) of the PAM solution with the corresponding ∂ t u = ∆ u + ηu u (0 , x ) = 1 Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Motivation and related question Describe the long time behavior (stationary) of the PAM solution with the corresponding ∂ t u = ∆ u + ηu u (0 , x ) = 1 � t � � � � ∼ t → + ∞ exp − t Λ 1 ( η, [ − t, t ] d ) u ( t, x ) = E x exp( η ( B s )d s ) 0 Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Motivation and related question Describe the long time behavior (stationary) of the PAM solution with the corresponding ∂ t u = ∆ u + ηu u (0 , x ) = 1 � t � � � � ∼ t → + ∞ exp − t Λ 1 ( η, [ − t, t ] d ) u ( t, x ) = E x exp( η ( B s )d s ) 0 If d = 1 , connection with random matrix theory. The stochastic Airy operator arises as the scaling limit of Hermitian Gaussian matrices at the edge of the spectrum. Can we expect same result in more large dimension. Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Motivation and related question Describe the long time behavior (stationary) of the PAM solution with the corresponding ∂ t u = ∆ u + ηu u (0 , x ) = 1 � t � � � � ∼ t → + ∞ exp − t Λ 1 ( η, [ − t, t ] d ) u ( t, x ) = E x exp( η ( B s )d s ) 0 If d = 1 , connection with random matrix theory. The stochastic Airy operator arises as the scaling limit of Hermitian Gaussian matrices at the edge of the spectrum. Can we expect same result in more large dimension. Anderson-Localization? Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Schr¨ odinger Operator with smooth potential Smooth potential Let V a L ∞ ( T d L ) function and we define the operator H V f = − ∆ f + fV L ) . H V is a self-adjoint unbounded operator of L 2 ( T d for all f ∈ H 2 ( T d L ) with domain H 2 ( T d L ) . Spectral analysis of H V The spectrum of H V coincide with the punctual spectrum and 1 S p ( H V ) = { Λ c 1 ( V ) ≤ Λ c 2 ( V ) ≤ ... ≤ Λ c n ( V ) ... } without accumulation point and such that Λ n ( V ) → + ∞ L ) = � n ∈ N Ker (Λ n − H V ) L 2 ( T 2 2 Λ n ( V ) = min F ⊂ H 2 ( T 2 ) max f ∈ F ; � f � L 2 =1 � H V f, f � 3 | Λ n ( V ) − Λ n ( ˜ V ) | ≤ � V − ˜ V � ∞ 4 Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case One dimensional case With Periodic boundary conditions, − d 2 dx 2 + b ′ ( b ∈ C 1 / 2 − ) t , on the space H 1 ([ − L, L ] , R ) , L > 0 such that Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case One dimensional case With Periodic boundary conditions, − d 2 dx 2 + b ′ ( b ∈ C 1 / 2 − ) t , on the space H 1 ([ − L, L ] , R ) , L > 0 such that If f ∈ H 1 ([ − L, L ] , R ) , f ( t ) b ′ t ∈ H − 1 / 2 − is a well defined as a distribution. Why H 1 ? Let ( f, λ ) a solution of the eigenvalue equation then − d 2 dx 2 f = λf + fb ′ then we can expect that the regularity of f is at least the regularity of b ′ +2 then f ∈ C 3 / 2 − ⊆ H 3 / 2 − Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case The random spectrum Theorem (Fukushima, Nakao (1977)) the following inequality hold C 1 ( b ) � f � H 1 ≤ � Hf, f � L 2 + γ � f � 2 2 ≤ C 2 ( b ) � f � H 1 Then H admit a unique self-adjoint extension. The random spectrum of H is almost surly pure point formed by a sequence Λ k . Furthermore Λ 1 < Λ 2 < Λ 3 < . . . Moreover L 2 = � k ker(Λ k − H ) Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case The random spectrum Theorem (Fukushima, Nakao (1977)) the following inequality hold C 1 ( b ) � f � H 1 ≤ � Hf, f � L 2 + γ � f � 2 2 ≤ C 2 ( b ) � f � H 1 Then H admit a unique self-adjoint extension. The random spectrum of H is almost surly pure point formed by a sequence Λ k . Furthermore Λ 1 < Λ 2 < Λ 3 < . . . Moreover L 2 = � k ker(Λ k − H ) Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Schr¨ odinger operator with white noise potential Goal We want to define and study the follwoing random Schr¨ odinger operator : H = − ∆ + η, d = 2 as an unbounded operator of L 2 ( T 2 L ) . Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Schr¨ odinger operator with white noise potential Goal We want to define and study the follwoing random Schr¨ odinger operator : H = − ∆ + η, d = 2 as an unbounded operator of L 2 ( T 2 L ) . Problem η is a Schwartz distribution and not a function. η ∈ C − 1 − ε ( T 2 L ) for all ε > 0 and not 1 bettre If ϕ a smooth function then the product ϕη is well defined as a distribution ! 2 If ( f, Λ) is an eigenfunction/eigenvalue of H : 3 − ∆ f = fη + Λ f regularity of f = ( regularity of η ) + 2 = 1 − ε = ⇒ the product fη is ill-defined. Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Non homogenous Besov space for j > 0 , ∆ j the projection on the Fourier mode of size 2 j , ∆ − 1 the projection 1 on the Fourier mode less than 2 . � � f ∈ S ′ ( T d ) , we say that f ∈ B α 2 jα � ∆ j f � L p ( T d ) p,q if j ≥− 1 ∈ l q ([ − 1 , + ∞ )) , 2 moreover when p = q = 2 then it coincide with the Sobolev space H α If α ∈ (0 , 1) then C α = B α ∞ , ∞ coincide with the space of α -H¨ older functions. 3 f ∈ B γ p,p , g ∈ B α ∞ , ∞ then the Paraproduct term 4 � f ≺ g = ∆ i f ∆ j g − 1 ≤ i<j − 1 is always well defined and satisfy that f ≺ g ∈ B min( α,α + γ − ) (with good p,p continuity bound). fg = f ≺ g + f ◦ g + f ≻ g with f ≻ g = g ≺ f and f ◦ g = � | i − j |≤ 1 ∆ i f ∆ j g is well defined if α + γ > 0 and in this case it lie in B α + γ p,p . The white noise η satisfy that almost surely η ∈ C − d 2 − ε ( T d ) for all ε > 0 . 5 Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential

Review of the one-dimensional case Paracontrolled distribution and Domain of the operator Decomposition of the eigenfunction If ( f, λ ) is a formal solution of the eigenvalue problem associated to H then − ∆ f = − fη + λf = f ≺ η − f ≻ η + f ◦ η − Λ f � �� � � �� � H − 1 − ε H − ε ⇒ − (1 − ∆) − 1 ( f ≻ η + f ◦ η − Λ f + f ) f = − (1 − ∆) − 1 ( f ≺ η ) � �� � � �� � 1 − ε 2 − ε Chouk KhalilHu BerlinJoint work with R.Allez Anderson Hamiltonian with white noise potential