Generic continuous spectrum for multi-dimensional quasi-periodic - PowerPoint PPT Presentation

Generic continuous spectrum for multi-dimensional quasi-periodic Schr¨ odinger operators with rough potentials Joint work with Rui Han Fan Yang University of California, Irvine 35th Annual Western States Mathematical Physics Meeting February 13, 2017 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 1 / 19

Main object Multi-dimensional discrete quasi-periodic Schr¨ odinger operators ∑ u m + f ( T n x ) u n , ( H α, x u ) n = | m − n | =1 f : T d → R is the potential function, T is the shift on T d with frequency vector α = ( α 1 , α 2 , ..., α d ) ∈ T d , x = ( x 1 , x 2 , ..., x d ) ∈ T d is the phase vector, m , n ∈ Z d and | m | = ∑ d i =1 m i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 2 / 19

Main object Multi-dimensional discrete quasi-periodic Schr¨ odinger operators ∑ u m + f ( T n x ) u n , ( H α, x u ) n = | m − n | =1 f : T d → R is the potential function, T is the shift on T d with frequency vector α = ( α 1 , α 2 , ..., α d ) ∈ T d , x = ( x 1 , x 2 , ..., x d ) ∈ T d is the phase vector, m , n ∈ Z d and | m | = ∑ d i =1 m i . d = 1 ⇒ many (sharp) results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 2 / 19

Main object Multi-dimensional discrete quasi-periodic Schr¨ odinger operators ∑ u m + f ( T n x ) u n , ( H α, x u ) n = | m − n | =1 f : T d → R is the potential function, T is the shift on T d with frequency vector α = ( α 1 , α 2 , ..., α d ) ∈ T d , x = ( x 1 , x 2 , ..., x d ) ∈ T d is the phase vector, m , n ∈ Z d and | m | = ∑ d i =1 m i . d = 1 ⇒ many (sharp) results d > 1 ⇒ very few results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 2 / 19

Excluding point spectrum for one-dimensional operators ( H α, x u ) n = u n +1 + u n − 1 + f ( x + n α ) u n . Repetitions in potential leads to empty point spectrum: Developed by Gordon in 1976 H¨ older continuous potential: Avila-You-Zhou (sharp results for Almost Mathieu operator) Generic continuous potential: Boshernitzan-Damanik (one-dimensional operator with multi-frequency) Sturmian Hamiltonian: many authors Singular potential: Simon (Maryland model), Jitomirskaya-Liu (sharp results for Maryland model), Jitomirskaya-Y (meromorphic potential) Measurable potential: Gordon (one-dimensional operator with multi-frequency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 3 / 19

Excluding point spectrum for multi-dimensional operators General continuous potential: Simon’s Wonderland theorem implies empty point spectrum for generic frequencies. For a long time, there is no arithmetically quantitative result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 4 / 19

Excluding point spectrum for multi-dimensional operators General continuous potential: Simon’s Wonderland theorem implies empty point spectrum for generic frequencies. For a long time, there is no arithmetically quantitative result. H¨ older continuous potential: Damanik: one frequency is sufficiently Liouville, the other frequencies are rational, for any x , H α, x has no point spectrum Breakthrough by Gordon: when all the frequencies are sufficiently Liouville, for any x , H α, x has no ℓ 1 ( Z d ) solution. Gordon-Nemirovski: when all the frequencies are sufficiently Liouville, for any x , H α, x has no point spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 4 / 19

Theorem (Boshernitzan-Damanik, 2007) Let ( H α, x u ) n = u n +1 + u n − 1 + f ( x + n α ) u n , be a one-dimensional quasi-periodic operator with multi-dimensional frequency α ∈ T d . Then for any frequency, for generic continuous potential f , H α, x has no point spectrum for a.e. x ∈ T d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 5 / 19

Our main result for continuous potentials Theorem Assume there exists an infinite sequence Q = { τ ( n ) = ( τ ( n ) 1 , τ ( n ) 2 , ..., τ ( n ) d ) } such that τ ( n ) · · · τ ( n ) ∥ τ ( n ) 1 d lim α i ∥ T = 0 (1) i τ ( n ) n →∞ i for any i = 1 , 2 , ..., d . Then for generic continuous potential f , the multi-dimensional operator H α, x has no point spectrum for a.e. x ∈ T d . Remark d = 2, (1) holds for a.e. ( α 1 , α 2 ). Actually (1) is equivalent to ( α 1 , α 2 ) are not both of bounded type. d ≥ 3, using Borel-Cantelli’s Lemma, (1) holds on a zero measure set of α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 6 / 19

Our main result for measurable potentials Theorem Let the potential f be measurable, then for generic α ∈ T d , H α, x has no point spectrum for a.e. x . Remark: This theorem extends a result of Gordon (2016) for d = 1 to the multi-dimensional setting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 7 / 19

Rokhlin-Halmos Lemma Let T be an invertible measure-preserving transformation on a measure space ( X , M , µ ) with µ ( X ) = 1. We assume T is aperiodic. Then for any ϵ > 0 and n ∈ N , there exists an E ∈ M such that: the sets E , TE , ..., T n − 1 E are pairwise disjoint; µ ( ∪ n − 1 j =0 T j E ) > 1 − ϵ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 8 / 19

Sketch of proof for continuous potentials j =1 τ ( n ) Notations: for 1 > δ > 0, let Γ n = (2 d + δ ) ∏ d . Denote j x ⋆ y = ( x 1 y 1 , x 2 y 2 , ..., x d y d ), x , y ∈ R d . Proof: By (1), for any k ∈ N , take n k so that Γ n k α i ∥ < 1 ∥ τ ( n k ) for any i = 1 , 2 , ..., d . i τ ( n k ) k 2 i Step 1 By Rokhlin-Halmos Lemma, there exists O n k so that { O n k + j ⋆ α } ∥ j ∥ ∞ ≤ ( k +1)Γ nk are pairwise disjoint , and 1 − 2 − k − 1 | O n k | > (2( k + 1)Γ n k + 1) d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 9 / 19

Step 2 Cut O n k into { K n k , l } l =1 , 2 ,..., s nk , so that s nk 1 − 2 − k ∑ | K n k , l | > (2( k + 1)Γ n k + 1) d , l =1 and diam ( K n k , l ) < 1 k . Let I n k = { m ∈ Z d : 0 ≤ m i ≤ τ ( n k ) − 1 for any i = 1 , 2 , ..., d } and define i K n k , l + m ⋆ α + j ⋆ τ ( n k ) ⋆ α. ∪ U n k , l , m = ( nk ) | j i |≤ k Γ nk /τ i Step 3 U n k , l , m satisfies √ diam ( U n k , l , m ) < 2 d +1 . k for 1 ≤ l ≤ s n k , m ∈ I n k , U n k , l , m are pairwise disjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 10 / 19

Let F n k = { f ∈ C ( T d ) : f is constant on each U n k , l , m } , and define G n k = { g ∈ C ( T d ) : there exists f ∈ F n k s . t . ∥ g − f ∥ 0 < 1 2 k − 2 d + γ 2 d + δ Γ nk } . Step 4 k ≥ t G n k is a generic set of C ( T d ). G = ∩ ∪ t ≥ 1 If f ∈ G , then there exists subsequence { ˜ n k } of { n k } such that f ∈ G ˜ n k . For any k > 4 d − 1 + 2 ∥ f ∥ 0 , let ∪ T ˜ n k = ( K ˜ n k , l + j ⋆ α ) . 1 ≤ l ≤ s ˜ nk , ∥ j ∥ ∞ ≤ ( k − 1)Γ ˜ nk Step 5 For x ∈ T ˜ n k , we have: n k ) ⋆ α ) − f ( x ) | < (4 d − 1 + 2 ∥ f ∥ 0 ) − 2 d + γ | f ( x + j ⋆ τ (˜ 2 d + δ Γ ˜ nk . max (˜ nk ) j ∈ Z d , | j i |≤ Γ ˜ nk /τ i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 11 / 19

Step 6 T ˜ n k satisfies: ) d ( (2 k − 2)Γ ˜ n k + 1 (1 − 2 − k ) , | T ˜ n k | ≥ (2 k + 2)Γ ˜ n k + 1 thus ∑ | T c ∑ n k | = (1 − | T ˜ n k | ) < ∞ . ˜ k k Therefore, a.e. x ∈ T d belongs to infinitely many T ˜ n k . For such full measure set of x , combining Step 5 and the following Theorem (Gordon-Nemirovski), H α, x has no point spectrum. □ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UC Irvine Generic continuous spectrum February 13, 2017 12 / 19