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

Analytic quasiperiodic Schr¨

• dinger 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

Quasi-periodic Schr¨

• dinger operator HV = − d2

dx2 + V : potential given by

V (x) = U(ωx) with sampling function U : Tν → R and frequency ω ∈ Rν. Analytic sampling function, with small coupling: U(θ) =

• m∈Zν

c(m)e2πimθ |c(m)| ≤ εe−κ0|m| for some ε > 0, 0 < κ0 ≤ 1. Diophantine frequency ω = (ω1, . . . , ων) ∈ Rν, |mω| ≥ a0|m|−b0, m ∈ Zν \ {0} for some 0 < a0 < 1, ν < b0 < ∞. All our results will hold for ε < ε0(a0, b0, κ0)

Small quasi-periodic operators Isospectral torus KdV equation Comb domains

Direct spectral theory

Direct spectral properties (Elliason, Damanik–Goldstein): HV has purely a.c. spectrum, Floquet solutions for a.e. E ∈ σ(HV ) Spectrum σ(HV ) = [E, ∞) \

• m∈Zν\{0}

(E −

m , E + m )

exponential in m decay of gap sizes E +

m − E − m ,

E +

m − E − m ≤ 2εe− κ0

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| If E +

m − E − m ≤ ε′e−κ|m| for some κ ≥ 4κ0, then |c(m)| ≤ (2ε′)1/2e− κ

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) = δx, (HV − z)−1δy V is reflectionless if Re G(0, 0; E + i0) = 0 for Lebesgue-a.e. E ∈ S = σ(HV ) Periodic and “finite-gap” quasiperiodic operators are reflectionless If V is periodic, the set of periodic potentials Q with spectrum σ(HQ) = σ(HV ) 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¨

• dinger operators

V (x) =

• m∈Zν

c(m)e2πimωx, |c(m)| ≤ εe−κ0|m| Theorem (Damanik–Goldstein–Lukic) Let 0 < ε ≤ ε1(a0, b0, κ0). Assume Q ∈ L∞(R) is reflectionless and σ(HQ) = σ(HV ). Then Q(x) =

• m∈Zν

d(m)e2πimωx, with |d(m)| ≤ √ 4ε exp

• − κ0

4 |m|

• ,

m ∈ Zν. Quasi-periodicity, the frequency ω, analyticity of the sampling function, are all encoded in the spectrum of HV

Small quasi-periodic operators Isospectral torus KdV equation Comb domains

Proof through periodic approximations

V (x) =

• m∈Zν

c(m)e2πimωx, |c(m)| ≤ εe−κ0|m| Use periodic approximants ˜ V (x) =

• m∈Zν

c(m)e2πim ˜

ωx

for ˜ ω ∈ Qν 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 Z(˜ ω) := Zν/{m ∈ Zν : m˜ 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: ∂tu − 6u∂xu + ∂3

xu = 0

u(x, 0) = V (x) By Lax pair representation, solutions give isospectral families Hu(·,t) McKean–Trubowitz, 1976: If V ∈ Hn(T), then there is a global solution u(x, t) on T × R and this solution is Hn(T)-almost periodic in t (i.e., u(·, t) = F(ζt) for some continuous F : T∞ → Hn(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

V (x) =

• m∈Zν

c(m)e2πimωx, |c(m)| ≤ εe−κ0|m| Theorem (Binder–Damanik–Goldstein–Lukic) If ε < ε0(a0, b0, κ0), then

1

(existence) there exists a global solution u(x, t);

2

(uniqueness) if ˜ u is another solution on R × [−T, T], and ˜ u, ∂3

x ˜

u ∈ L∞(R × [−T, T]), then ˜ u = u;

3

(x-dependence) for each t, u(·, t) is quasi-periodic in x, u(x, t) =

• m∈Zν

c(m, t)e2πimωx |c(m, t)| ≤ √ 4ε e− κ0

4 |m| 4

(t-dependence) t → u(·, t) is W k,∞(R)-almost periodic in t, for any 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, σac(HV ) = σ(HV ) = S, and S obeys some Craig-type and homogeneity conditions, then

1

(existence) there exists a global solution u(x, t);

2

(uniqueness) if ˜ u is another solution on R × [−T, T], and ˜ u, ∂3

x ˜

u ∈ L∞(R × [−T, T]), then ˜ u = u;

3

(x-dependence) for each t, x → u(x, t) is almost periodic in x;

4

(t-dependence) t → u(·, t) is W 4,∞(R)-almost periodic in t.

Small quasi-periodic operators Isospectral torus KdV equation Comb domains

Trajectory on isospectral torus

Rybkin, 2008: Assume that V is reflectionless and σac(HV ) = σ(HV ) = S. Assume that u(x, t) is a solution such that u, ∂3

xu ∈ 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

V (x) =

• m∈Zν

c(m)e2πimωx, |c(m)| ≤ εe−κ0|m| Almost-periodicity of V implies almost-periodicity of m-functions, so following Johnson–Moser, consider w(z) = lim

L→∞

1 L L m+(x; z)dx Theorem (Binder–Damanik–Goldstein–Lukic) If ε < ε0(a0, b0, κ0), then w(z) is a conformal map from C \ [E, ∞) to a comb domain of the form C+ \

• m∈Zν

{mω + iy | 0 < y < hm} where hm < ε1/2e− κ0

5 |m|

Small quasi-periodic operators Isospectral torus KdV equation Comb domains

Prescribing the heights hm

Theorem (Binder–Damanik–Goldstein–Lukic) If hm < ε′e−κ|m| with ε′ < ε0(a0, b0, κ0) and κ ≥ 5κ0, then there exists V (x) =

• m∈Zν

c(m)e2πimωx, |c(m)| ≤ (ε′)1/4e− κ

3 |m|

which corresponds to the comb domain C+ \

• m∈Zν

{mω + iy | 0 < y < hm} Corollary Within the class of analytic quasi-periodic Schr¨

• dinger operators at small

coupling, we can prescribe an arbitrary set M ⊂ Zν 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!