Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions and universal hierarchical structure of eigenfunctions

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1 Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions and universal hierarchical structure of eigenfunctions S. Jitomirskaya Atlanta, October 10, 2016

2 Almost Mathieu operators (H λ,α,θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n v(θ) = 2 cos 2π(θ), α irrational,

3 Almost Mathieu operators (H λ,α,θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n v(θ) = 2 cos 2π(θ), α irrational, Tight-binding model of 2D Bloch electrons in magnetic fields

4 Almost Mathieu operators (H λ,α,θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n v(θ) = 2 cos 2π(θ), α irrational, Tight-binding model of 2D Bloch electrons in magnetic fields First introduced by R. Peierls in 1933

5 Almost Mathieu operators (H λ,α,θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n v(θ) = 2 cos 2π(θ), α irrational, Tight-binding model of 2D Bloch electrons in magnetic fields First introduced by R. Peierls in 1933 Further studied by a Ph.D. student of Peierls, P.G. Harper (1955)

6 Almost Mathieu operators (H λ,α,θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n v(θ) = 2 cos 2π(θ), α irrational, Tight-binding model of 2D Bloch electrons in magnetic fields First introduced by R. Peierls in 1933 Further studied by a Ph.D. student of Peierls, P.G. Harper (1955) Is called Harper s model

7 Almost Mathieu operators (H λ,α,θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n v(θ) = 2 cos 2π(θ), α irrational, Tight-binding model of 2D Bloch electrons in magnetic fields First introduced by R. Peierls in 1933 Further studied by a Ph.D. student of Peierls, P.G. Harper (1955) Is called Harper s model With a choice of Landau gauge effectively reduces to h θ α is a dimensionless parameter equal to the ratio of flux through a lattice cell to one flux quantum.

8 Hofstadter butterfly

9 Hofstadter butterfly Gregory Wannier to Lars Onsager: It looks much more complicated than I ever imagined it to be

10 Hofstadter butterfly Gregory Wannier to Lars Onsager: It looks much more complicated than I ever imagined it to be David Jennings described it as a picture of God

12 S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral

13 Hierarchical structure driven by the continued fraction expansion of the magnetic flux: eigenfunctions

14 Hierarchical structure driven by the continued fraction expansion of the magnetic flux Predicted by M. Azbel (1964) Spectrum: only known that the spectrum is a Cantor set (Ten Martini problem)

15 Hierarchical structure driven by the continued fraction expansion of the magnetic flux Predicted by M. Azbel (1964) Spectrum: only known that the spectrum is a Cantor set (Ten Martini problem) Eigenfunctions: History: Bethe Ansatz solutions (Wiegmann, Zabrodin, et al) Sinai, Hellffer-Sjostrand, Buslaev-Fedotov

16 Hierarchical structure driven by the continued fraction expansion of the magnetic flux Predicted by M. Azbel (1964) Spectrum: only known that the spectrum is a Cantor set (Ten Martini problem) Eigenfunctions: History: Bethe Ansatz solutions (Wiegmann, Zabrodin, et al) Sinai, Hellffer-Sjostrand, Buslaev-Fedotov remained a challenge even at the physics level

17 Hierarchical structure driven by the continued fraction expansion of the magnetic flux Predicted by M. Azbel (1964) Spectrum: only known that the spectrum is a Cantor set (Ten Martini problem) Eigenfunctions: History: Bethe Ansatz solutions (Wiegmann, Zabrodin, et al) Sinai, Hellffer-Sjostrand, Buslaev-Fedotov remained a challenge even at the physics level Today: universal self-similar exponential structure of eigenfunctions throughout the entire localization regime.

18 Arithmetic spectral transitions 1D Quasiperiodic operators: (h θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n

19 Arithmetic spectral transitions 1D Quasiperiodic operators: (h θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n Transitions in the coupling λ originally approached by KAM (Dinaburg, Sinai, Bellissard, Frohlich-Spencer-Wittwer, Eliasson) nonperturbative methods (SJ, Bourgain-Goldstein for L > 0; Last,SJ,Avila for L = 0) reduced the transition to the transition in the Lyapunov exponent (for analytic v): L(E) > 0 implies pp spectrum for a.e. α, θ L(E + iɛ) = 0, ɛ > 0 implies pure ac spectrum for all α, θ

20 Arithmetic spectral transitions 1D Quasiperiodic operators: (h θ Ψ) n = Ψ n+1 + Ψ n 1 + λv(θ + nα)ψ n Transitions in the coupling λ originally approached by KAM (Dinaburg, Sinai, Bellissard, Frohlich-Spencer-Wittwer, Eliasson) nonperturbative methods (SJ, Bourgain-Goldstein for L > 0; Last,SJ,Avila for L = 0) reduced the transition to the transition in the Lyapunov exponent (for analytic v): L(E) > 0 implies pp spectrum for a.e. α, θ L(E + iɛ) = 0, ɛ > 0 implies pure ac spectrum for all α, θ

21 Lyapunov exponent Given E R and θ T, solve H λ,α,θ ψ = Eψ over C Z :

22 Lyapunov exponent Given E R and θ T, solve H λ,α,θ ψ = Eψ over C Z : transfer matrix: A E (θ) := ( ψn ψ n 1 ( ) E λv(θ) ) = A E n (α, θ) ( ) ψ0 ψ 1 A E n (α, θ) := A(θ + α(n 1))... A(θ)

23 Lyapunov exponent Given E R and θ T, solve H λ,α,θ ψ = Eψ over C Z : transfer matrix: A E (θ) := ( ψn ψ n 1 ( ) E λv(θ) ) = A E n (α, θ) ( ) ψ0 ψ 1 A E n (α, θ) := A(θ + α(n 1))... A(θ) The Lyapunov exponent (LE): 1 L(α, E) := lim log A E n (n) n (x) dx, T

24 Lyapunov exponent Given E R and θ T, solve H λ,α,θ ψ = Eψ over C Z : transfer matrix: A E (θ) := ( ψn ψ n 1 ( ) E λv(θ) ) = A E n (α, θ) ( ) ψ0 ψ 1 A E n (α, θ) := A(θ + α(n 1))... A(θ) The Lyapunov exponent (LE): 1 L(α, E) := lim log A E n (n) n (x) dx, T

25 Arithmetic transitions in the supercritical (L > 0) regime Small denominators - resonances - (v(θ + kα) v(θ + lα)) 1 are in competition with e L(E) l k. L very large compared to the resonance strength leads to more localization L small compared to the resonance strength leads to delocalization

26 Pure point to singular continuous transition conjecture Exponential strength of a resonance: β(α) := lim sup ln nα R/Z n n and δ(α, θ) := lim sup ln 2θ + nα R/Z n n α is Diophantine if β(α) = 0 θ is α-diophantine if δ(α) = 0

27 Pure point to singular continuous transition conjecture Exponential strength of a resonance: β(α) := lim sup ln nα R/Z n n and δ(α, θ) := lim sup ln 2θ + nα R/Z n n α is Diophantine if β(α) = 0 θ is α-diophantine if δ(α) = 0 For the almost Mathieu, on the spectrum L(E) = max(0, ln λ) (Bourgain-SJ, 2001).

28 Pure point to singular continuous transition conjecture Exponential strength of a resonance: β(α) := lim sup ln nα R/Z n n and δ(α, θ) := lim sup ln 2θ + nα R/Z n n α is Diophantine if β(α) = 0 θ is α-diophantine if δ(α) = 0 For the almost Mathieu, on the spectrum L(E) = max(0, ln λ) (Bourgain-SJ, 2001). λ < 1 pure ac spectrum (Dinaburg-Sinai 76, Aubry-Andre 80, Bellissard-Lima-Testard, Eliasson,..., Avila 2008)

29 Pure point to singular continuous transition conjecture Exponential strength of a resonance: β(α) := lim sup ln nα R/Z n n and δ(α, θ) := lim sup ln 2θ + nα R/Z n n α is Diophantine if β(α) = 0 θ is α-diophantine if δ(α) = 0 For the almost Mathieu, on the spectrum L(E) = max(0, ln λ) (Bourgain-SJ, 2001). λ < 1 pure ac spectrum (Dinaburg-Sinai 76, Aubry-Andre 80, Bellissard-Lima-Testard, Eliasson,..., Avila 2008) λ > 1 no ac spectrum (Ishii-Kotani-Pastur)

30 Pure point to singular continuous transition conjecture Conjecture for the sharp transition (1994): If β(α) = 0, then λ 0 = e δ(α,θ) is the transition line: H λ,α,θ has purely singular continuous spectrum for λ < e δ(α,θ), H λ,α,θ has Anderson localization (stronger than pure point spectrum ) for λ > e δ(α,θ).

31 Pure point to singular continuous transition conjecture Conjecture for the sharp transition (1994): If β(α) = 0, then λ 0 = e δ(α,θ) is the transition line: H λ,α,θ has purely singular continuous spectrum for λ < e δ(α,θ), H λ,α,θ has Anderson localization (stronger than pure point spectrum ) for λ > e δ(α,θ). (all θ, Diophantine α )

32 Pure point to singular continuous transition conjecture Conjecture for the sharp transition (1994): If β(α) = 0, then λ 0 = e δ(α,θ) is the transition line: H λ,α,θ has purely singular continuous spectrum for λ < e δ(α,θ), H λ,α,θ has Anderson localization (stronger than pure point spectrum ) for λ > e δ(α,θ). (all θ, Diophantine α ) If δ(α, θ) = 0, then L(E) = β(α) is the transition line. H λ,α,θ has purely singular continuous spectrum for L(E) < β(α) H λ,α,θ has Anderson localization for L(E) > β(α).

33 Pure point to singular continuous transition conjecture Conjecture for the sharp transition (1994): If β(α) = 0, then λ 0 = e δ(α,θ) is the transition line: H λ,α,θ has purely singular continuous spectrum for λ < e δ(α,θ), H λ,α,θ has Anderson localization (stronger than pure point spectrum ) for λ > e δ(α,θ). (all θ, Diophantine α ) If δ(α, θ) = 0, then L(E) = β(α) is the transition line. H λ,α,θ has purely singular continuous spectrum for L(E) < β(α) H λ,α,θ has Anderson localization for L(E) > β(α). (all α, Diophantine θ ) sc spectrum for β = proved in Gordon, Avron-Simon (82), and for δ = in SJ-Simon (94)

34 Pure point to singular continuous transition conjecture Conjecture for the sharp transition (1994): If β(α) = 0, then λ 0 = e δ(α,θ) is the transition line: H λ,α,θ has purely singular continuous spectrum for λ < e δ(α,θ), H λ,α,θ has Anderson localization (stronger than pure point spectrum ) for λ > e δ(α,θ). (all θ, Diophantine α ) If δ(α, θ) = 0, then L(E) = β(α) is the transition line. H λ,α,θ has purely singular continuous spectrum for L(E) < β(α) H λ,α,θ has Anderson localization for L(E) > β(α). (all α, Diophantine θ ) sc spectrum for β = proved in Gordon, Avron-Simon (82), and for δ = in SJ-Simon (94) pp spectrum for β = δ = 0 proved in SJ (99).

35 Asymptotics in the pp regime We say φ is a generalized eigenfunction if it is a polynomially bounded solution of H λ,α,θ φ = Eφ.

36 Asymptotics in the pp regime We say φ is a generalized eigenfunction if it is a( polynomially) φ(k) bounded solution of H λ,α,θ φ = Eφ. Let U(k) =. φ(k 1)

37 Asymptotics in the pp regime We say φ is a generalized eigenfunction if it is a( polynomially) φ(k) bounded solution of H λ,α,θ φ = Eφ. Let U(k) =. φ(k 1) Theorem (SJ-W.Liu, 16) There exist explicit universal functions f, g s.t. throughout the entire predicted pure point regime, for any generalized eigenfunction φ and any ε > 0, there exists K such that for any k K, U(k) and A k satisfy

38 Asymptotics in the pp regime We say φ is a generalized eigenfunction if it is a( polynomially) φ(k) bounded solution of H λ,α,θ φ = Eφ. Let U(k) =. φ(k 1) Theorem (SJ-W.Liu, 16) There exist explicit universal functions f, g s.t. throughout the entire predicted pure point regime, for any generalized eigenfunction φ and any ε > 0, there exists K such that for any k K, U(k) and A k satisfy f ( k )e ε k U(k) f ( k )e ε k, and g( k )e ε k A k g( k )e ε k.

39 Asymptotics in the pp regime (all α, Diophantine θ) Let pn q n be the continued fraction expansion of α. For any q n 2 k < q n+1 2, define explicit functions f (k), g(k) as follows(depend on α through the sequence of q n ):

40 Asymptotics in the pp regime (all α, Diophantine θ) Let pn q n be the continued fraction expansion of α. For any q n 2 k < q n+1 2, define explicit functions f (k), g(k) as follows(depend on α through the sequence of q n ): Case 1: q 8 9 n+1 qn 2 or k q n.

41 Asymptotics in the pp regime (all α, Diophantine θ) Let pn q n be the continued fraction expansion of α. For any q n 2 k < q n+1 2, define explicit functions f (k), g(k) as follows(depend on α through the sequence of q n ): Case 1: q 8 9 n+1 qn 2 or k q n. If lq n k < (l + 1)q n with l 1, set and f (k) = e k lqn ln λ r n l + e k (l+1)qn ln λ r n l+1, g(k) = e k lqn ln λ q n+1 r n l + e k (l+1)qn ln λ q n+1 r l+1 n, where for l 1, r l n = e (ln λ ln q n+1 + ln l qn qn )lqn.

42 Asymptotics in the pp regime (all α, Diophantine θ) Let pn q n be the continued fraction expansion of α. For any q n 2 k < q n+1 2, define explicit functions f (k), g(k) as follows(depend on α through the sequence of q n ): Case 1: q 8 9 n+1 qn 2 or k q n. If lq n k < (l + 1)q n with l 1, set and f (k) = e k lqn ln λ r n l + e k (l+1)qn ln λ r n l+1, g(k) = e k lqn ln λ q n+1 r n l + e k (l+1)qn ln λ q n+1 r l+1 n, where for l 1, r l n = e (ln λ ln q n+1 + ln l qn qn )lqn.

43 If qn 2 k < q n, set f (k) = e k ln λ + e k qn ln λ r n 1, and g(k) = e k ln λ.

44 If qn 2 k < q n, set f (k) = e k ln λ + e k qn ln λ r n 1, and Case 2. q 8 9 n+1 < qn 2 Set g(k) = e k ln λ. and qn 2 k min{q n, q n+1 2 }. f (k) = e k ln λ, and g(k) = e k ln λ.

45 If qn 2 k < q n, set f (k) = e k ln λ + e k qn ln λ r n 1, and Case 2. q 8 9 n+1 < qn 2 Set g(k) = e k ln λ. and qn 2 k min{q n, q n+1 2 }. f (k) = e k ln λ, and g(k) = e k ln λ. Note:f (k) decays exponentially and g(k) grows exponentially. However the decay rate and growth rate are not always the same.

46 The behavior of f (k) f (k) r n l r n l+2 r n l+4 q n 2 q lq n (l + 1)q n (l + 2)q n (l + 3)q n (l + 4)q n+1 n k 2

47 The behavior of g(k) g(k) q n+1 r n l+4 q n+1 r n l+2 q n+1 r n l q n 2 q lq n (l + 1)q n (l + 2)q n (l + 3)q n (l + 4)q n+1 n k 2

48 Arithmetic spectral transition Corollary Anderson localization holds throughout the entire conjectured pure point regime.

49 Arithmetic spectral transition Corollary Anderson localization holds throughout the entire conjectured pure point regime. Singular continuous spectrum holds for I. λ > e β(α) (Avila-You-Zhou, 15) II. λ > e δ(α,θ) (SJ-Liu, 16)

50 Arithmetic spectral transition Corollary Anderson localization holds throughout the entire conjectured pure point regime. Singular continuous spectrum holds for I. λ > e β(α) (Avila-You-Zhou, 15) II. λ > e δ(α,θ) (SJ-Liu, 16) Corollary The arithmetic spectral transition conjecture holds as stated.

51 History Localization Method: Avila-SJ: if λ > e 16 9 β(α) and δ(α, θ) = 0, then H λ,α,θ satisfies AL (Ten Martini Problem)

52 History Localization Method: Avila-SJ: if λ > e 16 9 β(α) and δ(α, θ) = 0, then H λ,α,θ satisfies AL (Ten Martini Problem) Liu-Yuan extended to the regime λ > e 3 2 β(α).

53 History Localization Method: Avila-SJ: if λ > e 16 9 β(α) and δ(α, θ) = 0, then H λ,α,θ satisfies AL (Ten Martini Problem) Liu-Yuan extended to the regime λ > e 3 2 β(α). Reducibility Method: Avila-You-Zhou proved that there exists a full Lebesgue measure set S such that for θ S, H λ,α,θ satisfies AL if λ > e β(α), thus proving the transition line at λ > e β(α) for a.e. θ. However, S can not be described in their proof.

54 History Localization Method: Avila-SJ: if λ > e 16 9 β(α) and δ(α, θ) = 0, then H λ,α,θ satisfies AL (Ten Martini Problem) Liu-Yuan extended to the regime λ > e 3 2 β(α). Reducibility Method: Avila-You-Zhou proved that there exists a full Lebesgue measure set S such that for θ S, H λ,α,θ satisfies AL if λ > e β(α), thus proving the transition line at λ > e β(α) for a.e. θ. However, S can not be described in their proof. SJ-Kachkovskiy: alternative argument, still without an arithmetic condition

55 History Localization Method: Avila-SJ: if λ > e 16 9 β(α) and δ(α, θ) = 0, then H λ,α,θ satisfies AL (Ten Martini Problem) Liu-Yuan extended to the regime λ > e 3 2 β(α). Reducibility Method: Avila-You-Zhou proved that there exists a full Lebesgue measure set S such that for θ S, H λ,α,θ satisfies AL if λ > e β(α), thus proving the transition line at λ > e β(α) for a.e. θ. However, S can not be described in their proof. SJ-Kachkovskiy: alternative argument, still without an arithmetic condition

56 Local j-maxima Local j-maximum is a local maximum on a segment I q j. A local j-maximum k 0 is nonresonant if for all k 2q j 1 and 2θ + (2k 0 + k)α R/Z > 2θ + (2k 0 + k)α R/Z > for all 2q j 1 < k 2q j. A local j-maximum is strongly nonresonant if for all 0 < k 2q j. 2θ + (2k 0 + k)α R/Z > κ q j 1 ν, κ k ν, (0.1) κ k ν, (0.2)

57 Universality of behavior at all (strongly) nonresonant local maxima: Theorem (SJ-W.Liu, 16) Suppose k 0 is a local j-maximum. If k 0 is nonresonant, then f ( s )e ε s U(k 0 + s) U(k 0 ) f ( s )e ε s, (0.3) for all 2s I, s > q j 1 2. If k 0 is strongly nonresonant, then f ( s )e ε s U(k 0 + s) U(k 0 ) f ( s )e ε s, (0.4) for all 2s I.

58 Universal hierarchical structure All α, Diophantine θ, pp regime. Let k 0 be the global maximum Theorem (SJ-W. Liu, 16) There exists ˆn 0 (α, λ, ς, ɛ) < such that for any k ˆn 0, n j k ˆn 0 + k, and 0 < a ni < e ς ln λ qn i, i = j k,..., j, for all 0 s k there exists a local n j s -maximum b anj,a nj 1,...,a nj s such that the following holds: I. b anj (k 0 + a nj q nj ) qˆn0 +1, II.For s k, b anj,...,a nj s (b anj,...,a nj s+1 + a nj s q nj s ) qˆn0 +s+1. III. if qˆn0 +k (x b anj,a nj 1,...,a nj k cq nj k, then for s = 0, 1,..., k, f (x s )e ε xs U(x) U(b anj,a nj 1,...,a nj s ) f (x s)e ε xs, Moreover, every local n j s -maximum on the interval b anj,a nj 1,...,a nj s+1 + [ e ɛ ln λqn j s, e ɛ ln λqn j s ] is of the form b anj,a nj 1,...,a nj s for some a nj s.

59 Universal hierarchical structure of the eigenfunctions Global maximum Local maximum Local maximum

60 Universal reflexive-hierarchical structure Theorem (SJ-W. Liu,16) For Diophantine α and all θ in the pure point regime there exists a hierarchical structure of local maxima as above, such that f (( 1) s+1 x s )e ε xs where x s = x b Kj,K j 1,...,K j s. U(x s ) U(b Kj,K j 1,...,K j s ) f (( 1)s+1 x s )e ε xs,

61 Further corollaries Corollary Let ψ(k) be any solution to H λ,α,θ ψ = Eψ that ( is linearly ψ(k) independent with respect to φ(k). Let Ū(k) = ψ(k 1) then g( k )e ε k Ū(k) g( k )e ε k. ), Let 0 δ k π 2 Corollary We have and be the angle between vectors U(k) and Ū(k). lim sup k lim inf k ln δ k k ln δ k k = 0, = β.

62 Corollary We have i) lim sup k ln A k k = lim sup k ln Ū(k) k = ln λ, ii) lim inf k ln A k k = lim inf k ln Ū(k) k = ln λ β. iii) Outside an explicit sequence of lower density zero, ln A k ln lim = lim Ū(k) = ln λ. k k k k

63 Corollary We have ln U(k) i) lim sup k k = ln λ, ln U(k) ii) lim inf k k = ln λ β. iii) There is an explicit sequence of upper density along which ln U(k) lim = ln λ. k k β β ln λ, iv) There is an explicit sequence of upper density 1 2 ln λ,along which ln U(k) lim sup < ln λ. k k

64 Further applications Upper bounds on fractal dimensions of spectral measures and quantum dynamics for trigonometric polynomials (SJ-W.Liu-S.Tcheremchantzev, SJ-W.Liu). The exact rate for exponential dynamical localization in expectation for the Diophantine case (SJ-H.Krüger-W.Liu). The first result of its kind, for any model. The same universal asymptotics of eigenfunctions for the Maryland Model (R. Han-SJ-F.Yang).

65 Key ideas of the proof Resonant points (small divisors): k : kα R/Z or 2θ + kα R/Z is small.

66 Key ideas of the proof Resonant points (small divisors): k : kα R/Z or 2θ + kα R/Z is small. New way to deal with resonant points in the positive Lyapunov regime (supercritical regime)

67 Key ideas of the proof Resonant points (small divisors): k : kα R/Z or 2θ + kα R/Z is small. New way to deal with resonant points in the positive Lyapunov regime (supercritical regime) Develop Gordon and palindromic methods to study the trace of transfer matrices to obtain lower bounds on solutions Gordon potential (periodicity): V (j + q n ) V (j) is small (control by q n α e β(α)qn ) palindromic potential (symmetry): V (k j) V (j) is small (control by 2θ + kα e δ(α,θ) k ) Bootstrap starting around the (local) maxima leads to effective estimates Reverse induction proof that local j 1-maxima are close to aq j 1 shifts of the local j-maxima, up to a constant scale Deduce that all the local maxima are (strongly) non-resonant and apply reverse induction

68 Assume E is a generalized eigenvalue and φ is the associated generalized eigenfunction ( φ(n) < ( 1 + n ). Let ) ϕ be another φ(k) solution of Hu = Eu. Let U(k) = and φ(k 1) ( ) ϕ(k) Ū(k) =. ϕ(k 1) Step 1:Sharp estimates for the non-resonant points. U(k) e ln λ k k i U(k i ) + e ln λ k k i+1 U(k i+1 ) Ū(k) e ln λ k k i Ū(k i ) + e ln λ k k i+1 Ū(k i+1 ) where k i is the resonant point and k [k i, k i+1 ].

69 Assume E is a generalized eigenvalue and φ is the associated generalized eigenfunction ( φ(n) < ( 1 + n ). Let ) ϕ be another φ(k) solution of Hu = Eu. Let U(k) = and φ(k 1) ( ) ϕ(k) Ū(k) =. ϕ(k 1) Step 1:Sharp estimates for the non-resonant points. U(k) e ln λ k k i U(k i ) + e ln λ k k i+1 U(k i+1 ) Ū(k) e ln λ k k i Ū(k i ) + e ln λ k k i+1 Ū(k i+1 ) where k i is the resonant point and k [k i, k i+1 ]. Step 2:Sharp estimates for the resonant points. U(k i+1 ) e c(k i,k i+1 ) k i+1 k i U(k i ) Ū(k i+1 ) e c (k i,k i+1 ) k i+1 k i Ū(k i ) where c(k i, k i+1 ), c (k i, k i+1 ) can be given explicitly.

70 Current quasiperiodic preprints Almost Mathieu operator: Avila-You-Zhou: sharp transition in α between pp and sc Avila-You-Zhou: dry Ten Martini, non-critical, all α Shamis-Last, Krasovsky, SJ- S. Zhang: gap size/dimension results for the critical case Avila-SJ-Zhou: critical line λ = e β Damanik-Goldstein-Schlag-Voda: homogeneous spectrum, Diophantine α W. Liu-SJ: sharp transitions in α and θ and universal (reflective) hierarchical structure Unitary almost Mathieu: Fillman-Ong-Z. Zhang: complete a.e. spectral description

71 Current quasiperiodic preprints Extended Harper s model: Avila-SJ-Marx: complete spectral description in the coupling phase space (+Erdos-Szekeres conjecture!) R. Han: an alternative argument R. Han-J: sharp transition in α between pp and sc spectrum in the positive Lyapunov exponent regime R. Han: dry Ten Martini (non-critical Diophantine) General 1-frequency quasiperiodic: analytic: SJ- S. Zhang: sharp arithmetic criterion for full spectral dimensionality (quasiballistic motion) R. Han-SJ: sharp topological criterion for dual reducibility to imply localization Damanik-Goldstein-Schlag-Voda: homogeneous spectrum, supercritical monotone: SJ-Kachkovskiy: all coupling localization meromorphic: SJ-Yang: sharp criterion for sc spectrum

72 Current quasiperiodic preprints Maryland model: W. Liu-SJ: complete arithmetic spectral transitions for all λ, α, θ W. Liu: surface Maryland model SJ-Yang: a constructive proof of localization General Multi-frequency: R. Han-SJ: localization-type results with arithmetic conditions (general zero entropy dynamics; including the skew shift) R. Han-Yang: generic continuous spectrum Hou-Wang-Zhou: ac spectrum for Liouville (presence) Avila-SJ: ac spectrum for Liouville (absence) Deift s problem (almost periodicity of KdV solutions with almost periodic initial data) : Binder-Damanik-Goldstein-Lukic: a solution under certain conditions.

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