The spectrum of dynamically defined The main claim Pictures - PowerPoint PPT Presentation

The spectrum of dynamically defined operators Helge Kr¨ uger Introduction The spectrum of dynamically defined The main claim Pictures operators The unitary case Anderson localization Future work Helge Kr¨ uger Caltech September 7, 2011

Schr¨ odinger operators The spectrum of dynamically defined The discrete Laplacian acting on the square summable sequences operators ℓ 2 ( Z ) is given by Helge Kr¨ uger Introduction ∆ ψ ( n ) = ψ ( n + 1) + ψ ( n − 1) . (1) The main claim Pictures For a potential, i.e. a bounded sequence, V : Z → R , we call The unitary case H = ∆ + V a Schr¨ odinger operator. Anderson localization Future work For V ( n ) i.i.d.r.v. with distribution supported in [ a , b ], H = ∆ + V is called the Anderson model. we have that the spectrum is given by σ ( H ) = range ( V ) + σ (∆) = [ a − 2 , b + 2] . (2) For V ( n ) = 2 λ cos(2 π ( n ω + x )) with ω irrational and λ � = 0, we have the Almost–Mathieu operator. Then the spectrum is always a Cantor set.

Schr¨ odinger operators The spectrum of dynamically defined The discrete Laplacian acting on the square summable sequences operators ℓ 2 ( Z ) is given by Helge Kr¨ uger Introduction ∆ ψ ( n ) = ψ ( n + 1) + ψ ( n − 1) . (1) The main claim Pictures For a potential, i.e. a bounded sequence, V : Z → R , we call The unitary case H = ∆ + V a Schr¨ odinger operator. Anderson localization Future work For V ( n ) i.i.d.r.v. with distribution supported in [ a , b ], H = ∆ + V is called the Anderson model. we have that the spectrum is given by σ ( H ) = range ( V ) + σ (∆) = [ a − 2 , b + 2] . (2) For V ( n ) = 2 λ cos(2 π ( n ω + x )) with ω irrational and λ � = 0, we have the Almost–Mathieu operator. Then the spectrum is always a Cantor set.

Schr¨ odinger operators The spectrum of dynamically defined The discrete Laplacian acting on the square summable sequences operators ℓ 2 ( Z ) is given by Helge Kr¨ uger Introduction ∆ ψ ( n ) = ψ ( n + 1) + ψ ( n − 1) . (1) The main claim Pictures For a potential, i.e. a bounded sequence, V : Z → R , we call The unitary case H = ∆ + V a Schr¨ odinger operator. Anderson localization Future work For V ( n ) i.i.d.r.v. with distribution supported in [ a , b ], H = ∆ + V is called the Anderson model. we have that the spectrum is given by σ ( H ) = range ( V ) + σ (∆) = [ a − 2 , b + 2] . (2) For V ( n ) = 2 λ cos(2 π ( n ω + x )) with ω irrational and λ � = 0, we have the Almost–Mathieu operator. Then the spectrum is always a Cantor set.

Number theory of ω n (mod 1) and ω n 2 (mod 1) The spectrum of ω n (mod 1) and ω n 2 (mod 1) are both equidistributed in [0 , 1]. dynamically defined operators Helge Kr¨ uger Let N ≥ 2 and define Introduction The main claim (mod 1) } N { β 1 < β 2 < · · · < β N } = { ω n n =1 Pictures The unitary case and Anderson localization { γ 1 < γ 2 < · · · < γ N } = { ω n 2 (mod 1) } N n =1 . Future work Then the set of lengths { l j = β j +1 − β j , j = 1 , . . . , N − 1 } consists of just three elements, whereas { ℓ j = γ j +1 − γ j , j = 1 , . . . , N − 1 } obey Poisson statistics for generic ω .

Number theory of ω n (mod 1) and ω n 2 (mod 1) The spectrum of ω n (mod 1) and ω n 2 (mod 1) are both equidistributed in [0 , 1]. dynamically defined operators Helge Kr¨ uger Let N ≥ 2 and define Introduction The main claim (mod 1) } N { β 1 < β 2 < · · · < β N } = { ω n n =1 Pictures The unitary case and Anderson localization { γ 1 < γ 2 < · · · < γ N } = { ω n 2 (mod 1) } N n =1 . Future work Then the set of lengths { l j = β j +1 − β j , j = 1 , . . . , N − 1 } consists of just three elements, whereas { ℓ j = γ j +1 − γ j , j = 1 , . . . , N − 1 } obey Poisson statistics for generic ω .

Number theory of ω n (mod 1) and ω n 2 (mod 1) The spectrum of ω n (mod 1) and ω n 2 (mod 1) are both equidistributed in [0 , 1]. dynamically defined operators Helge Kr¨ uger Let N ≥ 2 and define Introduction The main claim (mod 1) } N { β 1 < β 2 < · · · < β N } = { ω n n =1 Pictures The unitary case and Anderson localization { γ 1 < γ 2 < · · · < γ N } = { ω n 2 (mod 1) } N n =1 . Future work Then the set of lengths { l j = β j +1 − β j , j = 1 , . . . , N − 1 } consists of just three elements, whereas { ℓ j = γ j +1 − γ j , j = 1 , . . . , N − 1 } obey Poisson statistics for generic ω .

Number theory of ω n (mod 1) and ω n 2 (mod 1) The spectrum of ω n (mod 1) and ω n 2 (mod 1) are both equidistributed in [0 , 1]. dynamically defined operators Helge Kr¨ uger Let N ≥ 2 and define Introduction The main claim (mod 1) } N { β 1 < β 2 < · · · < β N } = { ω n n =1 Pictures The unitary case and Anderson localization { γ 1 < γ 2 < · · · < γ N } = { ω n 2 (mod 1) } N n =1 . Future work Then the set of lengths { l j = β j +1 − β j , j = 1 , . . . , N − 1 } consists of just three elements, whereas { ℓ j = γ j +1 − γ j , j = 1 , . . . , N − 1 } obey Poisson statistics for generic ω .

Number theory of ω n (mod 1) and ω n 2 (mod 1) The spectrum of ω n (mod 1) and ω n 2 (mod 1) are both equidistributed in [0 , 1]. dynamically defined operators Helge Kr¨ uger Let N ≥ 2 and define Introduction The main claim (mod 1) } N { β 1 < β 2 < · · · < β N } = { ω n n =1 Pictures The unitary case and Anderson localization { γ 1 < γ 2 < · · · < γ N } = { ω n 2 (mod 1) } N n =1 . Future work Then the set of lengths { l j = β j +1 − β j , j = 1 , . . . , N − 1 } consists of just three elements, whereas { ℓ j = γ j +1 − γ j , j = 1 , . . . , N − 1 } obey Poisson statistics for generic ω .

The skew-shift Schr¨ odinger operator Define the skew-shift T : T 2 → T 2 , T = R / Z , The spectrum of dynamically defined operators T ( x , y ) = ( x + 2 ω, x + y ) (mod 1) . (3) Helge Kr¨ uger Then ω n 2 = T n ( ω, 0) 2 (mod 1). Introduction The main claim It thus makes sense instead of considering the potential Pictures V ( n ) = f ( n 2 ω (mod 1)), to consider potentials given by The unitary case Anderson localization V ( n ) = λ f ( T n ( x , y )) (4) Future work for f : T 2 → R . These then form an ergodic family of potentials. For sufficiently regular f , the spectrum of ∆ + V Conjecture: consists of finitely many intervals and is Anderson localized. This means it behaves as in the random case. Progress : Large coupling ( λ ≫ 1): Bourgain–Goldstein–Schlag, Bourgain, Bourgain–Jitomirskaya, K. Small coupling (0 < λ ≪ 1): Bourgain. largely open Necessity of regularity: Avila–Bochi–Damanik, Boshernitzan–Damanik.

The skew-shift Schr¨ odinger operator Define the skew-shift T : T 2 → T 2 , T = R / Z , The spectrum of dynamically defined operators T ( x , y ) = ( x + 2 ω, x + y ) (mod 1) . (3) Helge Kr¨ uger Then ω n 2 = T n ( ω, 0) 2 (mod 1). Introduction The main claim It thus makes sense instead of considering the potential Pictures V ( n ) = f ( n 2 ω (mod 1)), to consider potentials given by The unitary case Anderson localization V ( n ) = λ f ( T n ( x , y )) (4) Future work for f : T 2 → R . These then form an ergodic family of potentials. For sufficiently regular f , the spectrum of ∆ + V Conjecture: consists of finitely many intervals and is Anderson localized. This means it behaves as in the random case. Progress : Large coupling ( λ ≫ 1): Bourgain–Goldstein–Schlag, Bourgain, Bourgain–Jitomirskaya, K. Small coupling (0 < λ ≪ 1): Bourgain. largely open Necessity of regularity: Avila–Bochi–Damanik, Boshernitzan–Damanik.

The skew-shift Schr¨ odinger operator Define the skew-shift T : T 2 → T 2 , T = R / Z , The spectrum of dynamically defined operators T ( x , y ) = ( x + 2 ω, x + y ) (mod 1) . (3) Helge Kr¨ uger Then ω n 2 = T n ( ω, 0) 2 (mod 1). Introduction The main claim It thus makes sense instead of considering the potential Pictures V ( n ) = f ( n 2 ω (mod 1)), to consider potentials given by The unitary case Anderson localization V ( n ) = λ f ( T n ( x , y )) (4) Future work for f : T 2 → R . These then form an ergodic family of potentials. For sufficiently regular f , the spectrum of ∆ + V Conjecture: consists of finitely many intervals and is Anderson localized. This means it behaves as in the random case. Progress : Large coupling ( λ ≫ 1): Bourgain–Goldstein–Schlag, Bourgain, Bourgain–Jitomirskaya, K. Small coupling (0 < λ ≪ 1): Bourgain. largely open Necessity of regularity: Avila–Bochi–Damanik, Boshernitzan–Damanik.