Analytic quasiperiodic Schr odinger operators at small coupling - PowerPoint PPT Presentation

Analytic quasiperiodic Schr¨ odinger operators at small coupling Milivoje Lukic (Rice University) joint work with Ilia Binder, David Damanik, Michael Goldstein Western States Mathematical Physics Meeting February 12, 2017

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Setup odinger operator H V = − d 2 Quasi-periodic Schr¨ dx 2 + V : potential given by V ( x ) = U ( ω x ) with sampling function U : T ν → R and frequency ω ∈ R ν . Analytic sampling function, with small coupling: � c ( m ) e 2 π im θ U ( θ ) = m ∈ Z ν | c ( m ) | ≤ ε e − κ 0 | m | for some ε > 0, 0 < κ 0 ≤ 1. Diophantine frequency ω = ( ω 1 , . . . , ω ν ) ∈ R ν , m ∈ Z ν \ { 0 } | m ω | ≥ a 0 | m | − b 0 , for some 0 < a 0 < 1 , ν < b 0 < ∞ . All our results will hold for ε < ε 0 ( a 0 , b 0 , κ 0 )

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Direct spectral theory Direct spectral properties (Elliason, Damanik–Goldstein): H V has purely a.c. spectrum, Floquet solutions for a.e. E ∈ σ ( H V ) � ( E − m , E + Spectrum σ ( H V ) = [ E , ∞ ) \ m ) m ∈ Z ν \{ 0 } exponential in m decay of gap sizes E + m − E − m , m ≤ 2 ε e − κ 0 E + m − E − 2 | m | polynomial in m , m ′ decay of distances between gaps, dist ([ E − m , E + m ] , [ E − m ′ , E + m ′ ]) ≥ a | m | − b if m � = m ′ , | m ′ | ≥ | m | m ≤ ε ′ e − κ | m | for some κ ≥ 4 κ 0 , then | c ( m ) | ≤ (2 ε ′ ) 1 / 2 e − κ If E + m − E − 2 | m | What about inverse spectral theory?

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Reflectionless operators Green’s function is formally given by G ( x , y ; z ) = � δ x , ( H V − z ) − 1 δ y � V is reflectionless if Re G (0 , 0; E + i 0) = 0 for Lebesgue-a.e. E ∈ S = σ ( H V ) Periodic and “finite-gap” quasiperiodic operators are reflectionless If V is periodic, the set of periodic potentials Q with spectrum σ ( H Q ) = σ ( H V ) is topologically a torus, parametrized by Dirichlet data Kotani: almost periodic operators with pure a.c. spectrum are reflectionless Craig, 1989: definition of Dirichlet data for reflectionless operators and their evolution in x for spectra obeying some conditions Sodin–Yuditskii, 1995: reflectionless operators on homogeneous spectra are almost periodic

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Isospectral torus of small quasi-periodic Schr¨ odinger operators � c ( m ) e 2 π im ω x , | c ( m ) | ≤ ε e − κ 0 | m | V ( x ) = m ∈ Z ν Theorem (Damanik–Goldstein–Lukic) Let 0 < ε ≤ ε 1 ( a 0 , b 0 , κ 0 ) . Assume Q ∈ L ∞ ( R ) is reflectionless and σ ( H Q ) = σ ( H V ) . Then � d ( m ) e 2 π im ω x , Q ( x ) = m ∈ Z ν with √ − κ 0 � � m ∈ Z ν . | d ( m ) | ≤ 4 ε exp 4 | m | , Quasi-periodicity, the frequency ω , analyticity of the sampling function, are all encoded in the spectrum of H V

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Proof through periodic approximations � c ( m ) e 2 π im ω x , | c ( m ) | ≤ ε e − κ 0 | m | V ( x ) = m ∈ Z ν Use periodic approximants ˜ � c ( m ) e 2 π im ˜ ω x V ( x ) = m ∈ Z ν ω ∈ Q ν for ˜ To compare isospectral tori and evolutions of Dirichlet data, we need uniform estimates on distances between gaps and bands (theory of periodic potentials is insufficient: estimates would grow exponentially with period) Consider the quotient group ω ) := Z ν / { m ∈ Z ν : m ˜ Z (˜ ω = 0 } and adapt the Damanik–Goldstein multiscale analysis to a method on Z (˜ ω ) Construct periodic approximations ˜ Q of Q such that σ ( H ˜ Q ) = σ ( H ˜ V )

Small quasi-periodic operators Isospectral torus KdV equation Comb domains KdV equation with almost periodic initial data Consider the initial value problem for the KdV equation: ∂ t u − 6 u ∂ x u + ∂ 3 x u = 0 u ( x , 0) = V ( x ) By Lax pair representation, solutions give isospectral families H u ( · , t ) McKean–Trubowitz, 1976: If V ∈ H n ( T ), then there is a global solution u ( x , t ) on T × R and this solution is H n ( T )-almost periodic in t (i.e., u ( · , t ) = F ( ζ t ) for some continuous F : T ∞ → H n ( T ) and ζ ∈ R ∞ ) Conjecture (Deift): If V : R → R is almost periodic, then there is a global solution u ( x , t ) that is almost periodic in t .

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Global existence, uniqueness, and almost periodicity � c ( m ) e 2 π im ω x , | c ( m ) | ≤ ε e − κ 0 | m | V ( x ) = m ∈ Z ν Theorem (Binder–Damanik–Goldstein–Lukic) If ε < ε 0 ( a 0 , b 0 , κ 0 ) , then (existence) there exists a global solution u ( x , t ) ; 1 (uniqueness) if ˜ u is another solution on R × [ − T , T ] , and 2 u , ∂ 3 u ∈ L ∞ ( R × [ − T , T ]) , ˜ x ˜ then ˜ u = u; (x-dependence) for each t, u ( · , t ) is quasi-periodic in x, 3 � c ( m , t ) e 2 π im ω x u ( x , t ) = m ∈ Z ν √ 4 ε e − κ 0 4 | m | | c ( m , t ) | ≤ (t-dependence) t �→ u ( · , t ) is W k , ∞ ( R ) -almost periodic in t, for any 4 integer k ≥ 0 .

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Global existence, uniqueness, and almost periodicity The previous result is a corollary of a more general conditional statement: Theorem (Binder–Damanik–Goldstein–Lukic) If V : R → R is almost periodic, σ ac ( H V ) = σ ( H V ) = S, and S obeys some Craig-type and homogeneity conditions, then (existence) there exists a global solution u ( x , t ) ; 1 (uniqueness) if ˜ u is another solution on R × [ − T , T ] , and 2 u , ∂ 3 u ∈ L ∞ ( R × [ − T , T ]) , ˜ x ˜ then ˜ u = u; (x-dependence) for each t, x �→ u ( x , t ) is almost periodic in x; 3 (t-dependence) t �→ u ( · , t ) is W 4 , ∞ ( R ) -almost periodic in t. 4

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Trajectory on isospectral torus Rybkin, 2008: Assume that V is reflectionless and σ ac ( H V ) = σ ( H V ) = S . Assume that u ( x , t ) is a solution such that u , ∂ 3 x u ∈ L ∞ ( R × [ − T , T ]) for some T > 0. Then, u ( · , t ) is reflectionless for all t Prove that the t -evolution of Dirichlet data is given by a Lipshitz vector field, conclude uniqueness Use finite-gap approximants to prove existence Use the Sodin–Yuditskii map to prove almost periodicity in t

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Comb domains � c ( m ) e 2 π im ω x , | c ( m ) | ≤ ε e − κ 0 | m | V ( x ) = m ∈ Z ν Almost-periodicity of V implies almost-periodicity of m -functions, so following Johnson–Moser, consider � L 1 w ( z ) = lim m + ( x ; z ) dx L L →∞ 0 Theorem (Binder–Damanik–Goldstein–Lukic) If ε < ε 0 ( a 0 , b 0 , κ 0 ) , then w ( z ) is a conformal map from C \ [ E , ∞ ) to a comb domain of the form � C + \ { m ω + iy | 0 < y < h m } m ∈ Z ν where h m < ε 1 / 2 e − κ 0 5 | m |

Small quasi-periodic operators Isospectral torus KdV equation Comb domains Prescribing the heights h m Theorem (Binder–Damanik–Goldstein–Lukic) If h m < ε ′ e − κ | m | with ε ′ < ε 0 ( a 0 , b 0 , κ 0 ) and κ ≥ 5 κ 0 , then there exists � c ( m ) e 2 π im ω x , | c ( m ) | ≤ ( ε ′ ) 1 / 4 e − κ 3 | m | V ( x ) = m ∈ Z ν which corresponds to the comb domain � C + \ { m ω + iy | 0 < y < h m } m ∈ Z ν Corollary Within the class of analytic quasi-periodic Schr¨ odinger operators at small coupling, we can prescribe an arbitrary set M ⊂ Z ν and there exists V such that E + m − E − m > 0 ⇐ ⇒ m ∈ M In particular, we have, within this class, approximation by finite gap potentials just like in the periodic case.

Thank you!