SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERATORS WITH NON-ERGODIC RANDOM POTENTIAL Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile Atlanta, 8 October, 2016 Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
The Model On ℓ 2 ( Z d ) define the operator ∆ by u ( n + k ) − 2 d u ( n ) , u ∈ ℓ 2 ( Z d ) . � (∆ u )( n ) = | k | + = 1 The random potential V ω is the multiplication operator on ℓ 2 ( Z d ) given by ( V ω u )( n ) = ( 1 + | n | α ) ω n u ( n ) , α > 0 where ( ω n ) n ∈ Z d are iid real random variables uniformly distributed on [ 0 , 1 ] . Ω = [ 0 , 1 ] Z d , B Ω , P = ⊗ µ Consider the probability space . � � The random operators H ω on ℓ 2 ( Z d ) H ω = − ∆ + V ω , ω = ( ω n ) n ∈ Z d ∈ Ω . Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Gordon-Jakši´ c-Molchanov-Simon Define { a j } j ≥ 1 , ( a 0 = ∞ ) given by | φ ( n ) − φ ( m ) | 2 , φ ∈ ℓ 2 ( Z d ) , � inf a j = � φ � = 1 ( n , m ) φ ∈ C ( A j ) | n − m | + = 1 A j ⊂ Z d with # A j = j and A j are connected. A path between points n , m ∈ Z d is a sequence of sites τ = ( n 1 , n 2 , · · · , n k ) , n 1 = n , n k = m , | n j + 1 − n j | + = 1 . X ⊂ Z d is connected if any two points in X can be connected with a path which lies within X . { a j } j ≥ 1 is a strictly decreasing sequence . a j < a j − 1 , j = 1 , 2 , · · · · · · Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
The Spectrum (Gordon-Jakši´ c-Molchanov-Simon) H ω has discrete spectrum a.e ω if and only if α > d . If N ω ( E ) denotes the number of eigenvalues of H ω which are less than E then for α > d and for a.e ω d N ω ( E ) = O ( E α ) as E → ∞ . Fixed a k ∈ N and d / k ≥ α > d / ( k + 1 ) for a.e ω σ ( H ω ) = σ pp ( H ω ) σ ess ( H ω ) = [ a k , ∞ ) , # σ disc ( H ω ) < ∞ . Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
IDS (Gordon-Jakši´ c-Molchanov-Simon) L = χ L H ω χ L , χ L is the projection onto ℓ 2 (Λ L ) , Λ L is Define H ω a cube with side length ( 2 L + 1 ) centered at origin. N ω E j ≤ E , E j ∈ σ ( H ω L ( E ) = # { j : L ) } If d / k > α > d / ( k + 1 ) , k ∈ N and E ∈ ( a j , a j − 1 ) , 1 ≤ j ≤ k , then N ω L ( E ) lim L d − j α = N j ( E ) ( Non random ) a . e ω, L →∞ If α = d / k and E ∈ ( a j , a j − 1 ) , 1 ≤ j < k , the above is valid. If E ∈ ( a k , a k − 1 ) then N ω L ( E ) lim = N k ( E ) ( Non random ) a . e ω. lnL L →∞ Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
For each ω , H ω L is a matrix (symmetric) of order ( 2 L + 1 ) d . L ) = ( 2 L + 1 ) d . # σ ( H ω In other words average spacing between two consecutive eigenvalue of H ω L inside ( a j , a j − 1 ) is of order L − ( d − j α ) . Now we want study how the eigenvalues of H ω L are accumulating (are they following any rule) insdie ( a j , a j − 1 ) , as L gets large. Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Problem Formulation (Dolai-Mallick) � Define ξ ω δ L d − α ( E j − E ) ( · ) , E ∈ ( a 1 , ∞ ) . L , E ( · ) = j L , E ( I ) = # { j : E j ∈ E + L − ( d − α ) I } I ⊂ R ξ ω { ξ ω L , E } is a sequence of integer valued random measure (i.e sequence of Point processs). We want study the Weak limit of ξ ω L , E Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Our result (Dolai-Mallick) For d ≥ 3, max { 2 , d 2 } < α < d and E ∈ S the sequence of point process { ξ ω L , E } L converges weakly to the Poisson point processs with intensity measure N ′ 1 ( E ) dx . � n � N ′ 1 ( E ) | B | = e − N ′ 1 ( E ) | B | lim � ω : ξ ω � L , E ( B ) = n , n ∈ L →∞ P n ! N ∪ { 0 } , B is bounded Borel set of R . In above S ⊂ ( a 1 , ∞ ) such that N ′ 1 ( E ) > 0 , E ∈ S Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Idea of the proof (Exponential decay of Green’s function) We divide Λ L into N d L numbers of disjoint cubes C p with side length 2 L + 1 N L � d � 2 L + 1 , N L = O ( 2 L + 1 ) ǫ , 0 < ǫ < 1 . Λ L = ∪ C p , | C p | = N L p be the restriction of H ω to C p . Define Let H ω η ω � p , L , E ( · ) = δ L d − α ( x − E ) x ∈ σ ( H ω p ) Using Aizenman-Molcanov method we showed that � s � �� � G ω � ≤ Ce − r | n − m | , z ∈ C + E Λ ( n , m ; z ) Λ − z ) − 1 δ m � , Λ ⊂ Z d G ω Λ ( n , m ; z ) = � δ n , ( H ω Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
N d L � � � � L →∞ � → 0 , f ∈ C + E ω � fd ξ ω fd η ω � L , E − − − − c ( R ) � � p , L , E � � p = 1 x − z , z ∈ C + is 1 Since collection of functions of the form dense in C + c so, to show the above convergence it is enough to verify the following N d 1 L 1 � � � � x − z d ξ ω � x − z d η ω � � → 0 as L → ∞ . � L , E ( x ) − p , L , E ( x ) � � � p = 1 Now the above will follows from exponential decay of Green’s functions. Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Convergence of uniformly asymptotically negligible triangural array N d L � Now we will show η ω p , E , L converges weakly to the Poisson p = 1 point process with intensity measure N ′ 1 dx . To show this it is enough to verify the following three conditions, for any bounded interval I (Con 1) sup ω : η ω = 0 ∀ ǫ > 0 . � � p , L , E ( I ) > ǫ as L → ∞ P 1 ≤ p ≤ N d L N d L (Con 2) � ω : η ω p , L , E ( I ) ≥ 2 = 0 as L → ∞ � � P p = 1 N d L (Con 3) � ω : η ω p , L , E ( I ) ≥ 1 = N ′ � � 1 ( E ) | I | as L → ∞ . P p = 1 Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Minami and Wegner Estimate (Combes-Germinet-Klein) Wegner Estimate 1 + | n | α � − 1 | I | � � � � E TrE H ω L ( I ) ≤ n ∈ Λ L Minami Estimate � 2 � �� � � 1 + | n | α � − 1 | I | L ( I ) − 1 � � TrE H ω L ( I ) TrE H ω ≤ E n ∈ Λ L 1 L d − α E ω � � �� , ν L ( · ) = Tr E H ω L ( · ) N 1 ( x ) = ν ( a 1 , x ) , x > a 1 L →∞ ν L − weakly ν − − − → on ( a 1 , ∞ ) . From Wegner estimate it will follow ν L ( I ) ≤ C | I | and ν ( I ) ≤ | I | . Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Existence of Intensity and it’s positivity Convergence of densities inside ( a 1 , ∞ ) f L ( E ) = d ν L 1 ( E ) = d ν uniformly N ′ dx ( E ) on [ E − δ, E + δ ] , δ > 0 . dx ( E ) − − − − − → L →∞ To show the above convergence we first showed that 1 is analytic and n ∈ Λ L E ω � G ω � ψ L ( z ) = � Λ L ( n , n ; z ) L d − α uniformly bounded on a region G ( ⊂ C ) which contain S ⊂ ( a 1 , ∞ ) . dx ( E ) = 1 The density f L is given by f L ( E ) = d ν L π Im ψ L ( E + i 0 ) . Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
We showed that ν ( a , b ) = N 1 ( b ) − N 1 ( a ) > 0 , | b − a | > 4 d , a , b ∈ ( 4 d , ∞ ) ⊂ ( a 1 , ∞ ) The above can be shown using the min-max principal and the operator inequality A ω L , 4 d . L , 0 ≤ H ω L ≤ A ω A ω n ∈ Λ L b n ω n , A ω L , 4 d = 4 d + � n ∈ Λ L b n ω n , b n = 1 + | n | α . L , 0 = � Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
(Con 1) will follow from Wegner estimate. η ω L , p , E ( I ) ≥ 1 ≤ E ω � η ω � � � P L , p , E ( I ) E + L − ( d − α ) I = E ω � � �� TrE H ω Cp = O ( N − ( d − α ) ) L Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
(Con 2) will follow from Minami estimate. � η ω ≤ E ω � η ω η ω L , p , E ( I ) − 1 � � � �� L , p , E ( I ) ≥ j L , p , E ( I ) P j ≥ 2 � = E ω E + L − ( d − α ) I � � TrE H ω × Cp � �� E + L − ( d − α ) I � � − 1 TrE H ω Cp � � N − 2 ( d − α ) = O . L Dhriti Ranjan Dolai. Pontificia Universidad Católica de Chile Santiago, Chile SPECTRAL STATISTICS OF RANDOM SCHRÖDINGER OPERAT
Recommend
More recommend
