Localization revisited Fr ed eric Klopp Jeffrey Schenker - PDF document

Localization revisited Fr´ ed´ eric Klopp Jeffrey Schenker Sorbonne Universit´ e Michigan State University Quantissima in the Serenissima III Venice, 20/08/2019 F. Klopp (Sorbonne Universit´ e) Localization Localization in I On Z d or R d , consider H ω = − ∆ + V ω , a random Z d -ergodic Schr¨ odinger operator. Localization holds in I ⊂ R if ∃ µ > 0 s.t. with P ≥ 1 − L − p , ∃ q > 0 s.t. for L large, any e.v. E ∈ I of H ω , L assoc. to eigenfcts s.t. ∃ x E ∈ [ − L , L ] d , one has � L p in mathematical papers (M) | x |≤ L | ϕ E ( x ) | e µ | x − x E | � max 1 in physics papers (P) Known: the bound (P) cannot hold for all eigenfunctions with good probability (Lifshits tails states). the bound (M) is optimal (again Lifshits tails states). Questions: in the localized region, where does the truth lie between (M) and (P)? More precisely, how many states satisfy (P)? how many states satisfy no estimate “better than (M)”? how many states satisfy an estimate “in between (P) and (M)”? F. Klopp (Sorbonne Universit´ e) Localization

On Z d or R d , consider H ω = − ∆ + V ω , a Z d -ergodic R.S.O. s.t. if d ( Λ , ˜ Λ ) > r , ( H ω ) | Λ and ( H ω ) | ˜ (IAD): Λ are independent. Well known: ∃ Σ ⊂ R s.t. σ ( H ω ) = Σ a.s.; the integrated density of states exists i.e. for any E ∈ R , ω -a.s. 1 N ( E ) : = lim | Λ L | # { eigenvalues of ( H ω ) | Λ L ≤ E } L → + ∞ ∃ I ⊆ Σ , µ , p , q , L 0 > 0 s.t., Assume (Localization): for L ≥ L 0 , with P ≥ 1 − L − p , for E ∈ I ∩ σ (( H ω ) | Λ L ) , for ϕ E norm. eigenfct of ( H ω ) | Λ L ass. to E , ∃ x E loc. center s.t. | x |≤ L � ϕ E � 2 ( x ) e µ | x − x E | ≤ L q . max Here � ϕ � 2 ( x ) = � ϕ � L 2 ( x +] − 1 / 2 , 1 / 2 ] d ) and � ϕ � 2 , ∞ = max � ϕ � 2 ( x ) . x ∈ Λ L Theorem µ < µ , ∃ L 0 > 0 s.t., for L ≥ L 0 , with P ≥ 1 − L − p , for For any 0 < ˜ E ∈ I ∩ σ (( H ω ) | Λ L ) , for ϕ E norm. eigenfct of ( H ω ) | Λ L ass. to E, ∃ x E loc. cent. s.t. µ � ϕ E � − 2 / d µ | x − x E | ≤ e 2 ( d + 1 ) / d ˜ | x |≤ L � ϕ E � 2 ( x ) e ˜ max 2 , ∞ (1) F. Klopp (Sorbonne Universit´ e) Localization For normalized eigenfunctions, the inverse of the ( 2 , ∞ ) -norm serves as a measure of extension of eigenfunction. The maximal distance between localization centers of an eigenfct is controlled by its ( 2 , ∞ ) -norm; more precisely, for 0 < α < 1, � � x E , 2 ( d + 1 ) / d � ϕ E � − 2 / d � � x ; α � ϕ E � 2 , ∞ ≤ � ϕ E � 2 ( x ) ⊂ B 2 , ∞ − log � ϕ E � 2 , ∞ − log α . Let us now count the number of eigenfunctions with a given extension. Theorem For µ , I , p as above, ∃ C , t 0 , L 0 > 0 s.t. for L ≥ L 0 , for 0 < t ≤ t 0 , with P ≥ 1 − L − p , one has # { ev E of ( H ω ) | Λ L in I ass. to ϕ E s.t. � ϕ E � 2 , ∞ ≤ t } ≤ e − t − 2 / d / C . (2) | Λ L | So eigenfcts with a large extension are rare. Estimate (P) holds for most eigenfcts while (M) is needed only for a small number of eigenfcts. F. Klopp (Sorbonne Universit´ e) Localization

Optimality of the upper bound on the counting function: a lower bound Let us assume: (HL) there exists an increasing function t > 0 �→ ν ( t ) > 0 such that, ∀ p > 0, for L large, with P ω ≥ 1 − L − p , one has P ω ′ ( � ( V ω ) | Λ L − ( V ω ′ ) | Λ L � ∞ ≤ t ) ≥ e −| Λ L | ν ( t ) . Example: for the Anderson model with single site distribution density that is compactly supported, bounded and lower bound by a constant on its support, one can pick ν ( t ) = C | log t | for some constant C > 0. More generally, Proposition For H ω the (discrete or continuous) Anderson model with an a.c. single site distribution with compactly supported density. Then, there exists C > 0 such that (HL) holds for ν ( t ) = C | log t | . F. Klopp (Sorbonne Universit´ e) Localization One proves Theorem Assume (HL). For µ , p and I as above, for δ ∈ ( 0 , 1 ) ∃ C , t 0 , L 0 > 0 s.t. for L ≥ L 0 , with P ≥ 1 − L − p , for ( log L / C ) − d / 2 ≤ t ≤ t 0 , one has # { ev E of ( H ω ) | Λ L in I ass. to ϕ s.t. � ϕ � 2 , ∞ ≤ t } ≥ e − Ct − 2 / d ν ( t 1 + δ ) . (3) | Λ L | In the case of Proposition 0.3, for the discrete or continuous Anderson model, the bound (3) becomes # { ev E of ( H ω ) | Λ L in I ass. to ϕ s.t. � ϕ � ∞ ≤ t } ≥ e Ct 2 / d log t . | Λ L | Expected: upper bound of the same form as lower bound (3) where 1 + δ is replaced by 1 − δ (similar to Lifshits tails). F. Klopp (Sorbonne Universit´ e) Localization

Consequence of the above theorems Theorem For a < b and t > 0 , the limit � � e.v. in [ a , b ] ∩ I of ( H ω ) | Λ L ass. to ϕ s.t. � ϕ � 2 , ∞ ≤ t # N ∞ ([ a , b ] , t ) : = lim (4) N ( I ) ·| Λ L | L → + ∞ exists almost surely and is a.s. independent of ω . Moreover, ( E , t ) �→ N ( E , t ) : = lim a ↓− ∞ N ([ a , E ] , t ) exists and is the distribution function of a probability measure on R × R + supported in I × R + . Proposition Assume (HL). There exists C > 0 such that, for δ ∈ ( 0 , 1 ) t ≥ C and a < b, one has N ([ a , b ] ∩ I ) e − Ct 2 / d ν ( t 1 + δ ) ≤ N ∞ ([ a , b ] , t ) ≤ N ([ a , b ] ∩ I ) e − t 2 / d / C N ( I ) N ( I ) F. Klopp (Sorbonne Universit´ e) Localization