Anderson localization for two interacting quasiperiodic particles

Abstract

with the operator H(θ 1 , θ 2 ) satisfying Anderson localization in the region of energies E (2.4) |E − U j | λ e (log λ) µ , j = 1, . . . , N int , for all ω ∈ Ω(U, λ, θ 1 , θ 2 ) and for all possible translations of U.Remark 2.4.(1) The part of the spectrum removed by (2.4) is contained in N int intervals of size o(|λ|).(2) The condition on separability of v is irrelevant in Theorem 2.2. The potential v(θ 1 ) + v(θ 2 ) can be replaced by an analytic function w(•, •) ∈ C ω (T 2 ) of two variables that is not constant on any straight line segment, with the same proof. (3) An analogue of Theorem 2.3 can also be obtained for non-separable case, assuming |E − (λw i + U j )| λε for every w i such that w(θ 1 , θ 2 ) ≡ w i on some straight line segment, and every value U j of U, also with the same proof. (4) As discussed above, the inclusion of a background potential U of low complexity could have been done already in [9], as well as in the other papers that establish perturbative results by semi-algebraic techniques and do not involve Lyapunov exponents/cocycles (for example, in [10]). ( 5) Suppose that v satisfies Type II symmetry, U = 0, θ 1 = θ 2 = 1/4. Then one can easily check that ψ(n 1 , n 2 ) = (−1) n 1 δ n 1 n 2 solves the eigenvalue equation H(1/4, 1/4)ψ = 0. While this does not contradict purely point spectrum, all known proofs of Anderson localization show that any solution of the eigenvalue equation decays exponentially, which does not allow the existence of states like ψ. This example suggests that some stronger versions of localization can break down at zero energy, but only in symmetric cases (because otherwise Theorem 2.2 holds). Possible scenarios of delocalization at zero energy in different models are described in [20,15]. (6) The condition in Theorem 2.3 of analyticity of v in the strip of size 20 is technical and can possibly be removed with some extra work. (7) The operator family H(θ 1 , θ 2 ) is not ergodic because U is not assumed to have any translational invariance. However, one can prove localization simultaneously for all translations of U. (8) Our results are perturbative, in the sense that one always has to remove a positive measure set of frequencies. However, our requirements on the frequency are more explicit. The bound on λ 0 can be expressed, in principle, through the Diophantine constant C dio of ω (see (5.1)). The parameter ε freq in Theorems 2.2 and 2.3 is, essentially, the measure of frequencies for which (5.1) does not hold with this C dio . Afterwards, as usually happens in localization proofs, one has to remove an extra set of frequencies of measure zero, for which we do not have any arithmetic description, and which depends on θ 1 , θ 2 , λ, and other parameters. (9) Theorems 2.2 and 2.3 are formulated for the case of a single phase (θ 1 , θ 2 ).However, one can extend them for a full measure set of phases, see Remark 9.1. We do not believe this argument is new, however, in the case of perturbative results, it has not been explicitly stated in the literature.

Anderson localization for multi-frequency quasi-periodic operators on $$\pmb {\mathbb {Z}}^{d}$$

“…in the study of interacting particles, and our proof does not apply in this setting. A localization result for a model with b = 1, d = 2 was recently obtained by Bourgain-Kachkovskiy [10].…”

On the existence of full dimensional KAM torus for nonlinear Schrödinger equation

“…the amplitude I = (I n ) n∈N * of E restricted as 1 4 ǫ 2 e −2rn θ ≤ |I n | ≤ 4ǫ 2 e −2rn θ ; the existence of KAM tori as well as the linear stability of such tori. Also see [4,5,10,26] for the related problem.…”

On the Spectra of Separable 2D Almost Mathieu Operators

“…Similar but distinct quasiperiodic (not necessarily separable) operators have been studied by S. Jitomirskaya, W. Liu, and Y. Shi [25]. The [10,25] results pertain to the spectral type but not the topological structure of the spectrum as a set. In Sect.…”

“…This complements a result by J. Bourgain [8] establishing that for fixed small coupling the 1D operator with potential V λ,1,α has gaps in its spectrum for some (positive measure) Diophantine frequencies. (f) The separable operator considered in Theorem 1.6 but with added background potential (so the resulting operator is not necessarily separable) has been studied by J. Bourgain and I. Kachkovskiy [10]. Similar but distinct quasiperiodic (not necessarily separable) operators have been studied by S. Jitomirskaya, W. Liu, and Y. Shi [25].…”