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 ( H λ,α,θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n v ( θ ) = 2 cos 2 π ( θ ) , α irrational, S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions
Almost Mathieu operators ( H λ,α,θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n v ( θ ) = 2 cos 2 π ( 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 ( H λ,α,θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n v ( θ ) = 2 cos 2 π ( 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 ( H λ,α,θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n v ( θ ) = 2 cos 2 π ( 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 ( H λ,α,θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n v ( θ ) = 2 cos 2 π ( 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 ( H λ,α,θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n v ( θ ) = 2 cos 2 π ( 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. 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: ( h θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )Ψ n S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions
Arithmetic spectral transitions 1D Quasiperiodic operators: ( h θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )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 + i ǫ ) = 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: ( h θ Ψ) n = Ψ n + 1 + Ψ n − 1 + λ v ( θ + n α )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 + i ǫ ) = 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 H λ,α,θ ψ = E ψ over C Z : S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions
Lyapunov exponent Given E ∈ R and θ ∈ T , solve H λ,α,θ ψ = E ψ over C Z : transfer matrix: � � E − λ v ( θ ) − 1 A E ( θ ) := 1 0 � � � � ψ n ψ 0 = A E n ( α, θ ) ψ n − 1 ψ − 1 A E 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 H λ,α,θ ψ = E ψ over C Z : transfer matrix: � � E − λ v ( θ ) − 1 A E ( θ ) := 1 0 � � � � ψ n ψ 0 = A E n ( α, θ ) ψ n − 1 ψ − 1 A E n ( α, θ ) := A ( θ + α ( n − 1 )) . . . A ( θ ) The Lyapunov exponent (LE): 1 � log || A E L ( α, E ) := lim ( n ) ( x ) || dx , n n →∞ T S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions
Lyapunov exponent Given E ∈ R and θ ∈ T , solve H λ,α,θ ψ = E ψ over C Z : transfer matrix: � � E − λ v ( θ ) − 1 A E ( θ ) := 1 0 � � � � ψ n ψ 0 = A E n ( α, θ ) ψ n − 1 ψ − 1 A E n ( α, θ ) := A ( θ + α ( n − 1 )) . . . A ( θ ) The Lyapunov exponent (LE): 1 � log || A E L ( α, E ) := lim ( n ) ( x ) || dx , n n →∞ T S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions
Arithmetic transitions in the supercritical ( L > 0) regime Small denominators - resonances - ( v ( θ + k α ) − 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 S. Jitomirskaya Quasiperiodic Schrodinger operators: sharp arithmetic spectral transitions
Pure point to singular continuous transition conjecture Exponential strength of a resonance: n →∞ − ln || n α || R / Z β ( α ) := lim sup | n | and n →∞ − ln || 2 θ + n α || R / Z δ ( α, θ ) := lim sup | 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: n →∞ − ln || n α || R / Z β ( α ) := lim sup | n | and n →∞ − ln || 2 θ + n α || R / Z δ ( α, θ ) := lim sup | 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
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