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

• S. Jitomirskaya

Atlanta, October 10, 2016

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Almost Mathieu operators

(Hλ,α,θΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn v(θ) = 2 cos 2π(θ), α irrational,

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Almost Mathieu operators

(Hλ,α,θΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn v(θ) = 2 cos 2π(θ), α irrational, Tight-binding model of 2D Bloch electrons in magnetic fields

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Almost Mathieu operators

(Hλ,α,θΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn 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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Almost Mathieu operators

(Hλ,α,θΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn 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)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Almost Mathieu operators

(Hλ,α,θΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn 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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Almost Mathieu operators

(Hλ,α,θΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn 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 hθ α is a dimensionless parameter equal to the ratio of flux through a lattice cell to one flux quantum.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Hofstadter butterfly

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Hofstadter butterfly

Gregory Wannier to Lars Onsager: “It looks much more complicated than I ever imagined it to be”

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic spectral transitions

1D Quasiperiodic operators: (hθΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic spectral transitions

1D Quasiperiodic operators: (hθΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn Transitions in the coupling λ

• riginally 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 + iǫ) = 0, ǫ > 0 implies pure ac spectrum for all α, θ

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic spectral transitions

1D Quasiperiodic operators: (hθΨ)n = Ψn+1 + Ψn−1 + λv(θ + nα)Ψn Transitions in the coupling λ

• riginally 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 + iǫ) = 0, ǫ > 0 implies pure ac spectrum for all α, θ

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Lyapunov exponent

Given E ∈ R and θ ∈ T, solve Hλ,α,θψ = Eψ over CZ:

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Lyapunov exponent

Given E ∈ R and θ ∈ T, solve Hλ,α,θψ = Eψ over CZ: transfer matrix: AE(θ) :=

• E − λv(θ)

−1 1

• ψn

ψn−1

• = AE

n (α, θ)

• ψ0

ψ−1

• AE

n (α, θ) := A(θ + α(n − 1)) . . . A(θ)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Lyapunov exponent

Given E ∈ R and θ ∈ T, solve Hλ,α,θψ = Eψ over CZ: transfer matrix: AE(θ) :=

• E − λv(θ)

−1 1

• ψn

ψn−1

• = AE

n (α, θ)

• ψ0

ψ−1

• AE

n (α, θ) := A(θ + α(n − 1)) . . . A(θ)

The Lyapunov exponent (LE): L(α, E) := lim

n→∞

1 n

• T

log ||AE

(n)(x)||dx ,

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Lyapunov exponent

Given E ∈ R and θ ∈ T, solve Hλ,α,θψ = Eψ over CZ: transfer matrix: AE(θ) :=

• E − λv(θ)

−1 1

• ψn

ψn−1

• = AE

n (α, θ)

• ψ0

ψ−1

• AE

n (α, θ) := A(θ + α(n − 1)) . . . A(θ)

The Lyapunov exponent (LE): L(α, E) := lim

n→∞

1 n

• T

log ||AE

(n)(x)||dx ,

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic transitions in the supercritical (L > 0) regime

Small denominators - resonances - (v(θ + kα) − v(θ + ℓα))−1 are in competition with eL(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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Exponential strength of a resonance: β(α) := lim sup

n→∞ −ln ||nα||R/Z

|n| and δ(α, θ) := lim sup

n→∞ −ln ||2θ + nα||R/Z

|n| α is Diophantine if β(α) = 0 θ is α-Diophantine if δ(α) = 0

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Exponential strength of a resonance: β(α) := lim sup

n→∞ −ln ||nα||R/Z

|n| and δ(α, θ) := lim sup

n→∞ −ln ||2θ + nα||R/Z

|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).

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Exponential strength of a resonance: β(α) := lim sup

n→∞ −ln ||nα||R/Z

|n| and δ(α, θ) := lim sup

n→∞ −ln ||2θ + nα||R/Z

|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)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Exponential strength of a resonance: β(α) := lim sup

n→∞ −ln ||nα||R/Z

|n| and δ(α, θ) := lim sup

n→∞ −ln ||2θ + nα||R/Z

|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)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Conjecture for the sharp transition (1994): If β(α) = 0, then λ0 = eδ(α,θ) is the transition line:

Hλ,α,θ has purely singular continuous spectrum for |λ| < eδ(α,θ), Hλ,α,θ has Anderson localization (stronger than pure point spectrum ) for |λ| > eδ(α,θ).

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Conjecture for the sharp transition (1994): If β(α) = 0, then λ0 = eδ(α,θ) is the transition line:

Hλ,α,θ has purely singular continuous spectrum for |λ| < eδ(α,θ), Hλ,α,θ has Anderson localization (stronger than pure point spectrum ) for |λ| > eδ(α,θ).

(all θ, Diophantine α )

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Conjecture for the sharp transition (1994): If β(α) = 0, then λ0 = eδ(α,θ) is the transition line:

Hλ,α,θ has purely singular continuous spectrum for |λ| < eδ(α,θ), Hλ,α,θ has Anderson localization (stronger than pure point spectrum ) for |λ| > eδ(α,θ).

(all θ, Diophantine α ) If δ(α, θ) = 0, then L(E) = β(α) is the transition line.

Hλ,α,θ has purely singular continuous spectrum for L(E) < β(α) Hλ,α,θ has Anderson localization for L(E) > β(α).

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Conjecture for the sharp transition (1994): If β(α) = 0, then λ0 = eδ(α,θ) is the transition line:

Hλ,α,θ has purely singular continuous spectrum for |λ| < eδ(α,θ), Hλ,α,θ has Anderson localization (stronger than pure point spectrum ) for |λ| > eδ(α,θ).

(all θ, Diophantine α ) If δ(α, θ) = 0, then L(E) = β(α) is the transition line.

Hλ,α,θ has purely singular continuous spectrum for L(E) < β(α) Hλ,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)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Pure point to singular continuous transition conjecture

Conjecture for the sharp transition (1994): If β(α) = 0, then λ0 = eδ(α,θ) is the transition line:

Hλ,α,θ has purely singular continuous spectrum for |λ| < eδ(α,θ), Hλ,α,θ has Anderson localization (stronger than pure point spectrum ) for |λ| > eδ(α,θ).

(all θ, Diophantine α ) If δ(α, θ) = 0, then L(E) = β(α) is the transition line.

Hλ,α,θ has purely singular continuous spectrum for L(E) < β(α) Hλ,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).

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

We say φ is a generalized eigenfunction if it is a polynomially bounded solution of Hλ,α,θφ = Eφ.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

We say φ is a generalized eigenfunction if it is a polynomially bounded solution of Hλ,α,θφ = Eφ. Let U(k) =

• φ(k)

φ(k − 1)

• .
• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

We say φ is a generalized eigenfunction if it is a polynomially bounded solution of Hλ,α,θφ = Eφ. Let U(k) =

• φ(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 Ak satisfy

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

We say φ is a generalized eigenfunction if it is a polynomially bounded solution of Hλ,α,θφ = Eφ. Let U(k) =

• φ(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 Ak satisfy f (|k|)e−ε|k| ≤ ||U(k)|| ≤ f (|k|)eε|k|, and g(|k|)e−ε|k| ≤ ||Ak|| ≤ g(|k|)eε|k|.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

(all α, Diophantine θ) Let pn

qn be the continued fraction expansion of α. For any qn 2 ≤ k < qn+1 2 , define explicit functions f (k), g(k) as

follows(depend on α through the sequence of qn):

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

(all α, Diophantine θ) Let pn

qn be the continued fraction expansion of α. For any qn 2 ≤ k < qn+1 2 , define explicit functions f (k), g(k) as

follows(depend on α through the sequence of qn): Case 1: q

8 9

n+1 ≥ qn 2 or k ≥ qn.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

(all α, Diophantine θ) Let pn

qn be the continued fraction expansion of α. For any qn 2 ≤ k < qn+1 2 , define explicit functions f (k), g(k) as

follows(depend on α through the sequence of qn): Case 1: q

8 9

n+1 ≥ qn 2 or k ≥ qn.

If ℓqn ≤ k < (ℓ + 1)qn with ℓ ≥ 1, set f (k) = e−|k−ℓqn| ln |λ|¯ r n

ℓ + e−|k−(ℓ+1)qn| ln |λ|¯

r n

ℓ+1,

and g(k) = e−|k−ℓqn| ln |λ| qn+1 ¯ r n

ℓ

+ e−|k−(ℓ+1)qn| ln |λ| qn+1 ¯ r n

ℓ+1

, where for ℓ ≥ 1, ¯ r n

ℓ = e−(ln |λ|−

ln qn+1 qn

+ ln ℓ

qn )ℓqn.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Asymptotics in the pp regime

(all α, Diophantine θ) Let pn

qn be the continued fraction expansion of α. For any qn 2 ≤ k < qn+1 2 , define explicit functions f (k), g(k) as

follows(depend on α through the sequence of qn): Case 1: q

8 9

n+1 ≥ qn 2 or k ≥ qn.

If ℓqn ≤ k < (ℓ + 1)qn with ℓ ≥ 1, set f (k) = e−|k−ℓqn| ln |λ|¯ r n

ℓ + e−|k−(ℓ+1)qn| ln |λ|¯

r n

ℓ+1,

and g(k) = e−|k−ℓqn| ln |λ| qn+1 ¯ r n

ℓ

+ e−|k−(ℓ+1)qn| ln |λ| qn+1 ¯ r n

ℓ+1

, where for ℓ ≥ 1, ¯ r n

ℓ = e−(ln |λ|−

ln qn+1 qn

+ ln ℓ

qn )ℓqn.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

If qn

2 ≤ k < qn, set

f (k) = e−k ln |λ| + e−|k−qn| ln |λ|¯ r n

1 ,

and g(k) = ek ln |λ|.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

If qn

2 ≤ k < qn, set

f (k) = e−k ln |λ| + e−|k−qn| ln |λ|¯ r n

1 ,

and g(k) = ek ln |λ|. Case 2. q

8 9

n+1 < qn 2 and qn 2 ≤ k ≤ min{qn, qn+1 2 }.

Set f (k) = e−k ln |λ|, and g(k) = ek ln |λ|.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

If qn

2 ≤ k < qn, set

f (k) = e−k ln |λ| + e−|k−qn| ln |λ|¯ r n

1 ,

and g(k) = ek ln |λ|. Case 2. q

8 9

n+1 < qn 2 and qn 2 ≤ k ≤ min{qn, qn+1 2 }.

Set f (k) = e−k ln |λ|, and g(k) = ek ln |λ|. Note:f (k) decays exponentially and g(k) grows exponentially. However the decay rate and growth rate are not always the same.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

The behavior of f (k)

¯ r n

ℓ

¯ r n

ℓ+2

¯ r n

ℓ+4

ℓqn (ℓ + 1)qn(ℓ + 2)qn(ℓ + 3)qn(ℓ + 4)qn k

qn+1 2 qn 2

f (k)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

The behavior of g(k)

qn+1 ¯ rn

ℓ

qn+1 ¯ rn

ℓ+2

qn+1 ¯ rn

ℓ+4

ℓqn (ℓ + 1)qn(ℓ + 2)qn(ℓ + 3)qn(ℓ + 4)qn k

qn+1 2 qn 2

g(k)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic spectral transition

Corollary Anderson localization holds throughout the entire conjectured pure point regime.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic spectral transition

Corollary Anderson localization holds throughout the entire conjectured pure point regime. Singular continuous spectrum holds for

• I. λ > eβ(α) (Avila-You-Zhou, 15)
• II. λ > eδ(α,θ) (SJ-Liu, 16)
• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Arithmetic spectral transition

Corollary Anderson localization holds throughout the entire conjectured pure point regime. Singular continuous spectrum holds for

• I. λ > eβ(α) (Avila-You-Zhou, 15)
• II. λ > eδ(α,θ) (SJ-Liu, 16)

Corollary The arithmetic spectral transition conjecture holds as stated.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

History

Localization Method: Avila-SJ: if |λ| > e

16 9 β(α) and δ(α, θ) = 0, then Hλ,α,θ satisfies

AL (Ten Martini Problem)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

History

Localization Method: Avila-SJ: if |λ| > e

16 9 β(α) and δ(α, θ) = 0, then Hλ,α,θ satisfies

AL (Ten Martini Problem) Liu-Yuan extended to the regime |λ| > e

3 2 β(α).

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

History

Localization Method: Avila-SJ: if |λ| > e

16 9 β(α) and δ(α, θ) = 0, then Hλ,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, Hλ,α,θ satisfies AL if |λ| > eβ(α), thus proving the transition line at |λ| > eβ(α) for a.e. θ. However, S can not be described in their proof.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

History

Localization Method: Avila-SJ: if |λ| > e

16 9 β(α) and δ(α, θ) = 0, then Hλ,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, Hλ,α,θ satisfies AL if |λ| > eβ(α), thus proving the transition line at |λ| > eβ(α) for a.e. θ. However, S can not be described in their proof. SJ-Kachkovskiy: alternative argument, still without an arithmetic condition

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

History

Localization Method: Avila-SJ: if |λ| > e

16 9 β(α) and δ(α, θ) = 0, then Hλ,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, Hλ,α,θ satisfies AL if |λ| > eβ(α), thus proving the transition line at |λ| > eβ(α) for a.e. θ. However, S can not be described in their proof. SJ-Kachkovskiy: alternative argument, still without an arithmetic condition

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Local j-maxima

Local j-maximum is a local maximum on a segment |I| ∼ qj. A local j-maximum k0 is nonresonant if ||2θ + (2k0 + k)α||R/Z > κ qj−1ν , for all |k| ≤ 2qj−1 and ||2θ + (2k0 + k)α||R/Z > κ |k|ν , (0.1) for all 2qj−1 < |k| ≤ 2qj. A local j-maximum is strongly nonresonant if ||2θ + (2k0 + k)α||R/Z > κ |k|ν , (0.2) for all 0 < |k| ≤ 2qj.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Universality of behavior at all (strongly) nonresonant local maxima:

Theorem (SJ-W.Liu, 16) Suppose k0 is a local j-maximum. If k0 is nonresonant, then f (|s|)e−ε|s| ≤ ||U(k0 + s)|| ||U(k0)|| ≤ f (|s|)eε|s|, (0.3) for all 2s ∈ I, |s| > qj−1

2 .

If k0 is strongly nonresonant, then f (|s|)e−ε|s| ≤ ||U(k0 + s)|| ||U(k0)|| ≤ f (|s|)eε|s|, (0.4) for all 2s ∈ I.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Universal hierarchical structure

All α, Diophantine θ, pp regime. Let k0 be the global maximum Theorem (SJ-W. Liu, 16) There exists ˆ n0(α, λ, ς, ǫ) < ∞ such that for any k ≥ ˆ n0, nj−k ≥ ˆ n0 + k, and 0 < ani < eς ln |λ|qni , i = j − k, . . . , j, for all 0 ≤ s ≤ k there exists a local nj−s-maximum banj ,anj−1,...,anj−s such that the following holds:

• I. |banj − (k0 + anjqnj)| ≤ qˆ

n0+1,

II.For s ≤ k, |banj ,...,anj−s − (banj ,...,anj−s+1 + anj−sqnj−s)| ≤ qˆ

n0+s+1.

• III. if qˆ

n0+k ≤ |(x − banj ,anj−1,...,anj−k | ≤ cqnj−k, then for

s = 0, 1, ..., k, f (xs)e−ε|xs| ≤ ||U(x)|| ||U(banj ,anj−1,...,anj−s )|| ≤ f (xs)eε|xs|, Moreover, every local nj−s-maximum on the interval banj ,anj−1,...,anj−s+1 + [−eǫ ln λqnj−s , eǫ ln λqnj−s ] is of the form banj ,anj−1,...,anj−s for some anj−s.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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+1xs)e−ε|xs| ≤ ||U(xs)|| ||U(bKj,Kj−1,...,Kj−s)|| ≤ f ((−1)s+1xs)eε|xs|, where xs = x − bKj,Kj−1,...,Kj−s.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Further corollaries

Corollary Let ψ(k) be any solution to Hλ,α,θψ = Eψ that is linearly independent with respect to φ(k). Let ¯ U(k) =

• ψ(k)

ψ(k − 1)

• ,

then g(|k|)e−ε|k| ≤ ||¯ U(k)|| ≤ g(|k|)eε|k|. Let 0 ≤ δk ≤ π

2 be the angle between vectors U(k) and ¯

U(k). Corollary We have lim sup

k→∞

ln δk k = 0, and lim inf

k→∞

ln δk k = −β.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Corollary We have i) lim sup

k→∞

ln ||Ak|| k = lim sup

k→∞

ln ||¯ U(k)|| k = ln |λ|, ii) lim inf

k→∞

ln ||Ak|| k = lim inf

k→∞

ln ||¯ U(k)|| k = ln |λ| − β. iii) Outside an explicit sequence of lower density zero, lim

k→∞

ln ||Ak|| k = lim

k→∞

ln ||¯ U(k)|| k = ln |λ|.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Corollary We have i) lim supk→∞

− ln ||U(k)|| k

= ln |λ|, ii) lim infk→∞

− ln ||U(k)|| k

= ln |λ| − β. iii) There is an explicit sequence of upper density 1 − 1

2 β ln |λ|,

along which lim

k→∞

− ln ||U(k)|| k = ln |λ|. iv) There is an explicit sequence of upper density 1

2 β ln |λ|,along

which lim sup

k→∞

− ln ||U(k)|| k < ln |λ|.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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.Kr¨ uger-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).

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Key ideas of the proof

Resonant points (small divisors): k : ||kα||R/Z or ||2θ + kα||R/Z is small.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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)

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• f transfer matrices to obtain lower bounds on solutions

Gordon potential (periodicity): |V (j + qn) − V (j)| is small (control by ||qnα|| ≃ 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 aqj−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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Assume E is a generalized eigenvalue and φ is the associated generalized eigenfunction (|φ(n)| < 1 + |n|). Let ϕ be another solution of Hu = Eu. Let U(k) =

• φ(k)

φ(k − 1)

• and

¯ U(k) =

• ϕ(k)

ϕ(k − 1)

• .

Step 1:Sharp estimates for the non-resonant points. ||U(k)|| ≃ e− ln λ|k−ki|||U(ki)|| + e− ln λ|k−ki+1|||U(ki+1)|| ||¯ U(k)|| ≃ e− ln λ|k−ki|||¯ U(ki)|| + e− ln λ|k−ki+1|||¯ U(ki+1)|| where ki is the resonant point and k ∈ [ki, ki+1].

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

Assume E is a generalized eigenvalue and φ is the associated generalized eigenfunction (|φ(n)| < 1 + |n|). Let ϕ be another solution of Hu = Eu. Let U(k) =

• φ(k)

φ(k − 1)

• and

¯ U(k) =

• ϕ(k)

ϕ(k − 1)

• .

Step 1:Sharp estimates for the non-resonant points. ||U(k)|| ≃ e− ln λ|k−ki|||U(ki)|| + e− ln λ|k−ki+1|||U(ki+1)|| ||¯ U(k)|| ≃ e− ln λ|k−ki|||¯ U(ki)|| + e− ln λ|k−ki+1|||¯ U(ki+1)|| where ki is the resonant point and k ∈ [ki, ki+1]. Step 2:Sharp estimates for the resonant points. ||U(ki+1)|| ≃ e−c(ki,ki+1)|ki+1−ki|||U(ki)|| ||¯ U(ki+1)|| ≃ ec′(ki,ki+1)|ki+1−ki|||¯ U(ki)|| where c(ki, ki+1), c′(ki, ki+1) can be given explicitly.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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 = eβ 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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions

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.

• S. Jitomirskaya

Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions