Schr odinger operator with Sturm potentials Fractal dimensions Liu - PowerPoint PPT Presentation

Schr¨ odinger operator with Sturm potential Cookie-Cutter-like sets Sketch of proof Schr¨ odinger operator with Sturm potentials —Fractal dimensions Liu Qinghui Beijing Institute of Technology Joint work with Qu Yanhui and Wen Zhiying Hong Kong, Dec. 12, 2012 1 / 19

Schr¨ odinger operator with Sturm potential Cookie-Cutter-like sets Sketch of proof Outline Schr¨ odinger operator with Sturm potential 1 Spectrum study history recent result Cookie-Cutter-like sets 2 Cantor set Cookie-Cutter set and Cookie-Cutter like set Sketch of proof 3 Bounded variation and bounded covariation Deal with different types Homogeneous Moran set 2 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Schr¨ odinger operator with Sturm potential odinger operator on l 2 ( Z ) : Schr¨ ( H α,V ψ ) n = ψ n − 1 + ψ n +1 + v n ψ n , ∀ n ∈ Z , ∀ ψ ∈ l 2 ( Z ) . ( v n ) n ∈ Z : potential. Sturm potential: v n = V χ [1 − α, 1) ( nα + φ mod 1) , ∀ n ∈ Z , α = [0; a 1 , a 2 , · · · ] : frequency V > 0 : coupling; φ ∈ [0 , 1) : phase (take φ = 0 ) Spectrum σ ( H α,V ) = { x ∈ R : xI − H α,V no bounded inverse } := σ . 1989, Bellissard and et. al.(BIST), Commun. Math. Phys. ∀ V > 0 , α irrational, L [ σ ] = 0 . 3 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Schr¨ odinger operator with Sturm potential odinger operator on l 2 ( Z ) : Schr¨ ( H α,V ψ ) n = ψ n − 1 + ψ n +1 + v n ψ n , ∀ n ∈ Z , ∀ ψ ∈ l 2 ( Z ) . ( v n ) n ∈ Z : potential. Sturm potential: v n = V χ [1 − α, 1) ( nα + φ mod 1) , ∀ n ∈ Z , α = [0; a 1 , a 2 , · · · ] : frequency V > 0 : coupling; φ ∈ [0 , 1) : phase (take φ = 0 ) Spectrum σ ( H α,V ) = { x ∈ R : xI − H α,V no bounded inverse } := σ . 1989, Bellissard and et. al.(BIST), Commun. Math. Phys. ∀ V > 0 , α irrational, L [ σ ] = 0 . 3 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Schr¨ odinger operator with Sturm potential odinger operator on l 2 ( Z ) : Schr¨ ( H α,V ψ ) n = ψ n − 1 + ψ n +1 + v n ψ n , ∀ n ∈ Z , ∀ ψ ∈ l 2 ( Z ) . ( v n ) n ∈ Z : potential. Sturm potential: v n = V χ [1 − α, 1) ( nα + φ mod 1) , ∀ n ∈ Z , α = [0; a 1 , a 2 , · · · ] : frequency V > 0 : coupling; φ ∈ [0 , 1) : phase (take φ = 0 ) Spectrum σ ( H α,V ) = { x ∈ R : xI − H α,V no bounded inverse } := σ . 1989, Bellissard and et. al.(BIST), Commun. Math. Phys. ∀ V > 0 , α irrational, L [ σ ] = 0 . 3 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Schr¨ odinger operator with Sturm potential odinger operator on l 2 ( Z ) : Schr¨ ( H α,V ψ ) n = ψ n − 1 + ψ n +1 + v n ψ n , ∀ n ∈ Z , ∀ ψ ∈ l 2 ( Z ) . ( v n ) n ∈ Z : potential. Sturm potential: v n = V χ [1 − α, 1) ( nα + φ mod 1) , ∀ n ∈ Z , α = [0; a 1 , a 2 , · · · ] : frequency V > 0 : coupling; φ ∈ [0 , 1) : phase (take φ = 0 ) Spectrum σ ( H α,V ) = { x ∈ R : xI − H α,V no bounded inverse } := σ . 1989, Bellissard and et. al.(BIST), Commun. Math. Phys. ∀ V > 0 , α irrational, L [ σ ] = 0 . 3 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Fractal dimensions Let α = [0; a 1 , a 2 , · · · ] , K ∗ = lim sup ( a 1 · · · a n ) 1 /n , ( a 1 · · · a n ) 1 /n K ∗ = lim inf n n 2004, L., Wen, Potential Analysis , V > 20 , • if K ∗ < ∞ , then 0 < dim H σ < 1 • if K ∗ = ∞ , then dim H σ = 1 . L., Qu, Wen, preprint, V > 25 , • if K ∗ < ∞ , then 0 < dim B σ < 1 • if K ∗ = ∞ , then dim B σ = 1 . 4 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Asymptotic property of Fractal dimension 2008, Damanik et. al., CMP , α = [0; a 1 , a 2 , · · · ] , a n ≡ 1 , √ lim V →∞ (log V ) dim B σ = − log( 2 − 1) . 2007, L., Peyri` ere, Wen, Comptes Randus Mathematique , sup n a n < ∞ , V > 20 , s ∗ , s ∗ pre-dim, dim H σ ≤ s ∗ ≤ s ∗ ≤ dim B σ , V →∞ s ∗ log V = − log f ∗ ( α ) . V →∞ s ∗ log V = − log f ∗ ( α ) , lim lim 2011, Fan, L., Wen, Ergodic Theory and Dynamical Systems , sup n a n < ∞ , then dim H σ = s ∗ ≤ s ∗ = dim B σ L., Qu, Wen, preprint, V > 25 , no restriction on { a n } , V →∞ (log V )dim H σ = − log f ∗ ( α ) , lim V →∞ (log V )dim B σ = − log f ∗ ( α ) . lim 5 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Case of bounded quotient 2011, Fan, L., Wen, Ergodic Theory and Dynamical Systems . Theorem Let α = [0; a 1 , a 2 , · · · ] , sup n a n < ∞ , V > 20 , dim B σ = s ∗ . dim H σ = s ∗ , Theorem If α = [0; a 1 , a 2 , a 3 , · · · ] with ( a n ) n ≥ 1 ultimate periodic, V > 20 s ∗ = s ∗ . For ( a n ) n ≥ 1 ultimately periodic, we give an algorithm so that one can estimation s ∗ in any accuracy. 6 / 19

Schr¨ odinger operator with Sturm potential Spectrum Cookie-Cutter-like sets study history Sketch of proof recent result Case of unbounded quotient L., Qu, Wen, preprint. Theorem Let α = [0; a 1 , a 2 , · · · ] , V > 25 , dim B σ = s ∗ . dim H σ = s ∗ , V →∞ s ∗ · log V = − log f ∗ ( α ) . V →∞ s ∗ · log V = − log f ∗ ( α ) , lim lim s ∗ , s ∗ are continuous on V . Key techniques Cookie-Cutter-like structure trace formula Homogeneous Moran set 7 / 19

Schr¨ odinger operator with Sturm potential Cantor set Cookie-Cutter-like sets Cookie-Cutter set and Cookie-Cutter like set Sketch of proof Cantor set 0 ≤ x ≤ 1 � 3 x, 2 Let I = [0 , 1] , f : I → R , f ( x ) = 2 < x ≤ 1 . 1 3(1 − x ) , Then E = { x ∈ I : ∀ n ≥ 0 , f n ( x ) ∈ I } = Cantor set, and dim H E = dim P E = dim B E = log 2 h µ ( f ) log 3 = sup log | Df | dµ. � µ : f − inv Cookie-Cutter: f non-linear. Cookie-Cutter-like: change f n to f n ◦ f n − 1 ◦ · · · ◦ f 1 8 / 19

Schr¨ odinger operator with Sturm potential Cantor set Cookie-Cutter-like sets Cookie-Cutter set and Cookie-Cutter like set Sketch of proof Definition for Cookie-Cutter set Let I = [0 , 1] , I 1 , I 2 ⊂ I , and f : I 1 ∪ I 2 → I satisfy: (i) f | I 1 , f | I 2 is an 1 − 1 mapping to I . (ii) f is c 1+ γ H¨ older: | Df ( x ) − Df ( y ) | ≤ c | x − y | γ . (iii) f is Expansive, 1 < b ≤ | Df ( x ) | ≤ B < ∞ . E = { x ∈ I : ∀ n ≥ 0 , f n ( x ) ∈ I } Cookie-Cutter set of f . 9 / 19

Schr¨ odinger operator with Sturm potential Cantor set Cookie-Cutter-like sets Cookie-Cutter set and Cookie-Cutter like set Sketch of proof Definition of Cookie-Cutter like set Given { ( f k , � q k j =1 I k j , c k , γ k , b k , B k ) } k ≥ 1 satisfy: (i’) f k | I k j is an 1 − 1 mapping to I . (ii’) f k is c 1+ γ k H¨ older (iii’) f k is Expansive. Cookie-Cutter-like set (CC-like set) = { x ∈ I : f k ◦ · · · ◦ f 1 ( x ) ∈ I, ∀ k ≥ 0 } . E 10 / 19

Schr¨ odinger operator with Sturm potential Cantor set Cookie-Cutter-like sets Cookie-Cutter set and Cookie-Cutter like set Sketch of proof Symbol system and pre-dimension Let Ω n = � n k =1 { 1 , · · · , q k } , F n = f n ◦ · · · ◦ f 1 , ∀ ω ∈ Ω n , F n is monotone on I ω , F n ( I ω ) = I. ∀ n > 0 , { I ω } ω ∈ Ω n is a covering of E . ω ∈ Ω k | I ω | s k = 1 , and ∀ k ≥ 1 , let s k satisfies ( ∃ . 1 . ) � s ∗ = lim sup s ∗ = lim inf s k , s k . k k 11 / 19

Schr¨ odinger operator with Sturm potential Cantor set Cookie-Cutter-like sets Cookie-Cutter set and Cookie-Cutter like set Sketch of proof Ma, Rao, Wen, Sci. China A, 2001 Let E be CC-like set for { ( f k , � q k j =1 I k j , c k , γ k , b k , B k ) } k ≥ 1 . Theorem dim H E = s ∗ , dim P E = dim B E = s ∗ . Theorem s ∗ , s ∗ depend continuously on { ( f k , � q k j =1 I k j , c k , γ k , b k , B k ) } k ≥ 1 . σ ( H α,V ) has a kind of CC-like structure (multi-type). Let α = [0; a 1 , a 2 , · · · ] , a k partly determines f k . ( a k ) k ≥ 1 bounded implies bounded expansive. 12 / 19

Schr¨ odinger operator with Sturm potential Cantor set Cookie-Cutter-like sets Cookie-Cutter set and Cookie-Cutter like set Sketch of proof key properties [MRW01] Recall F n = f n ◦ · · · ◦ f 1 , ∀ ω ∈ Ω n , F n ( I ω ) = I . Bounded variation . ∃ ξ ≥ 1 , ∀ n ≥ 1 , ω ∈ Ω n , x, y ∈ I ω , ξ − 1 ≤ | DF n ( x ) | | I ω | ∼ | DF n ( x ) | − 1 . | DF n ( y ) | < ξ, Bounded covariation . ∀ m > k ≥ 1 , ω 1 , ω 2 ∈ Ω k , τ ∈ Ω k +1 ,m , ξ − 2 | I ω 2 ∗ τ | | I ω 2 | ≤ | I ω 1 ∗ τ | | I ω 1 | ≤ ξ 2 | I ω 2 ∗ τ | | I ω 2 | . Existence of Gibbs-like measure . Given β > 0 , there exist η > 0 and a probability measure µ β supported by E such that for any n ≥ 1 and ω 0 ∈ Ω n , we have | I ω 0 | β | I ω 0 | β η − 1 | I ω | β ≤ µ β ( I ω 0 ) ≤ η | I ω | β . � � ω ∈ Ω n ω ∈ Ω n 13 / 19