On the spectra of separable 2D almost Mathieu operators

We consider separable 2D discrete Schr\"odinger operators generated by 1D almost Mathieu operators. For fixed Diophantine frequencies we prove that for sufficiently small couplings the spectrum must be an interval. This complements a result by J. Bourgain establishing that for fixed couplings the spectrum has gaps for some (positive measure) Diophantine frequencies. Our result generalizes to separable multidimensional discrete Schr\"odinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. The proof is based on the study of the thickness of the spectrum of the almost Mathieu operator, and utilizes the Newhouse Gap Lemma on sums of Cantor sets.

independence of the spectrum for operators with quasiperiodic potentials follows from the fact that the underlying topological dynamical system (T b , Z) is minimal [14].
Let d ≥ 2 be a positive integer. We consider the d-dimensional analog operator H generated by almost Mathieu operators. Specifically, H is the discrete Schrödinger operator on 2 (Z d ) defined by [ Hψ](n) = m∈{e1,...,e d } ψ(n + m) + ψ(n − m) + k∈{1,...,d} 2λ k cos(2π(n k α k + ω k )) ψ(n) for every ψ ∈ 2 (Z d ), n ∈ Z d ; {e 1 , . . . , e d } is the standard basis. The operator H and the spectrum Σ = Σ λ1,α1,ω1 + · · · + Σ λ d ,α d ,ω d are the main objects of study in this paper. The theory of the almost Mathieu operator provides insight into the theory of H. Consider the following theorem established by A. Avila and D. Damanik in 2008 [3]: If λ = 1 and α ∈ T \ Q, then the integrated density of states of the almost Mathieu operator is absolutely continuous. Along with a theorem by Steinhaus, which states that the sum of two sets with positive Lebesgue-measure contains an open interval, we immediately obtain the following proposition. Proposition 1.1. Assume at least two among λ 1 , . . . , λ d are not equal to 1 and α 1 , . . . , α d ∈ T \ Q. Then Σ has a dense interior.
We say a few words on the proof of Theorem 1.2. Observe Σ is the sum of Cantor spectra Σ 1 + · · · + Σ d , where henceforth Σ k := Σ λ k ,α k ,ω k . Indeed, the potential of H is separable. In general, the spectra of separable multidimensional operators are sums of the spectra of 1D operators. This fact can be found within [12]. See also another proof involving the convolution of density of states measures within [13]. The Newhouse Gap Lemma may be utilized to guarantee that Σ 1 + · · · + Σ d is an interval thereby establishing Theorem 1.2. The notion of thickness, which is a quantitative characterization of nonempty compact subsets K of R often denoted τ (K), was utilized by S. Newhouse in the 1970s [31,32,33] to prove the namesake Newhouse Gap Lemma. See also for a short proof section 4.2 on page 63 within [34]. The definition of thickness can be found within subsection 2.6. We abridge the Newhouse Gap Lemma: Let K 1 and K 2 be nonempty compact subsets of R. Assume the maximal-gap-lengths of K 1 and K 2 are sufficiently small relative to the diameters of K 1 and K 2 , and 1 ≤ τ (K 1 ) · τ (K 2 ). Then K 1 + K 2 is an interval. S. Astels in 1999 [1,2] generalized the Newhouse Gap Lemma to obtain the following; see Theorem 2.7 for the unabridged version. Theorem 1.4 (S. Astels [1,2]). Let K 1 , . . . , K d (d ≥ 2) be nonempty compact subsets of R. Assume the maximal-gap-lengths of K 1 , . . . , K d are sufficiently small relative to the diameters of K 1 , . . . , K d , . Then K 1 + · · · + K d is an interval. Because λ k is small, one can think of H λ k ,α k ,ω k as a perturbation of the discrete Laplacian whose spectrum is the interval [−2, 2]. Specifically, Σ k converges to [−2, 2] in the Hausdorff metric as λ k → 0. Also, the diameter of Σ k converges to the diameter of [−2, 2] as λ k → 0. Indeed, |diam σ(A) − diam σ(B)| ≤ 2 dist Haus (σ(A), σ(B)) ≤ 2 A − B for bounded self-adjoint operators A and B. Therefore Σ k has a small maximal-gap-length relative to the diameters of Σ 1 , . . . , Σ d . As a result, to utilize the Newhouse Gap Lemma it is enough to establish that the thickness of the spectrum of the almost Mathieu operator approaches infinity as the coupling approaches zero.
Theorem 1.5. The spectrum of the almost Mathieu operator H λ,α,ω with irrational frequency fixed and assumed to satisfy a Diophantine condition has thickness approaching infinity as the coupling λ approaches zero i.e.
(∀α ∈ DC)[lim λ→0 τ (Σ λ,α,ω ) = +∞]. Theorem 1.5 extends to 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine; combine Theorem 2.4 and Theorem 2.5 and Lemma 4.1. As a result, Theorem 1.2 extends to separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic operators whose potential is analytic and whose frequency is Diophantine. Theorem 1.6. For fixed Diophantine frequencies α 1 , . . . , α d , the spectrum is an interval for separable d-dimensional discrete Schrödinger operators generated by 1D α k -quasiperiodic analytic operators H k = ∆ + V k with α k -dependent sufficiently small norms V k . Theorem 1.6 is necessarily perturbative due to Theorem 1.3. Specifically, the smallness of the norm depends on the Diophantine frequency. We conclude this section with a few words on related mathematical results and questions. The definitions of limit-periodic and almost-periodic potential can be found within [15] and [36], respectively. It suffices to say that the collection of all almostperiodic potentials is a broad class of potentials which contains the periodic and limit-periodic and quasiperiodic potentials. The definitions of box-counting and Hausdorff dimension can be found within [16]. A Cantorval is a nonempty compact subset C of R such that C has no isolated connected components and C has a dense interior; we mention for comparison that a Cantor set is a nonempty compact subset C of R such that C has no isolated points and C has no interior points. (a) We establish a single-interval-characterization for the spectra of separable multidimensional discrete Schrödinger operators generated by 1D quasiperiodic analytic operators H k = ∆ + V k with Diophantine-frequency-dependent sufficiently small norms V k . The same characterization can be said about periodic V k [18] and cannot be said about limit-periodic V k [15]. (b) R. Han and S. Jitomirskaya [18] proved that periodic (not necessarily separable) multidimensional discrete Schrödinger operators have interval spectra when the norm of the potential is small and at least one period is odd. This is the discrete analog of the L. Parnovski [35] resolution of the continuous Bethe-Sommerfeld Conjecture. Moreover, the resolution was extended to quasiperiodic (not necessarily separable) multidimensional continuous Schrödinger operators for almost-all frequencies [26]. This paragraph is the only place where continuous Schrödinger operators are mentioned, and it is done so to motivate the following question. Can one extend the [18,35,26] results to quasiperiodic (not necessarily separable) multidimensional discrete Schrödinger operators for almost-all frequencies? Specifically, can one remove the separability condition from Theorem 1.6? Moreover, can one remove the Diophantine condition from Theorem 1.6? For example, can the frequencies satisfy a Liouvillian condition? In a sense, this means that the irrational frequencies are well approximated by rational numbers.
(c) D. Damanik, J. Fillman, and A. Gorodetski [15] proved that there exists a dense subset B of 1D limit-periodic potentials such that B-type operators have Cantor spectra with zero (lower) boxcounting dimension. Furthermore, separable multidimensional discrete Schrödinger operators generated by B-type operators have Cantor spectra with zero (lower) box-counting dimension. Is it true that the spectra of separable multidimensional discrete Schrödinger operators generated by 1D almost-periodic operators is either a finite union of disjoint intervals or a Cantor set or a Cantorval? In view of Proposition 1.1, does there exist λ 1 , . . . , λ d with at least two not equal to 1 and α 1 , . . . , α d ∈ T \ Q such that Σ is a Cantorval? (d) D. Damanik and A. Gorodetski [12] proved that 1D Fibonacci Hamiltonians have Cantor spectra Σ λ with Hausdorff dimension strictly between zero and one and with thickness strictly greater than zero, and lim λ→0 dim Haus (Σ λ ) = 1 and lim λ→0 τ (Σ λ ) = +∞. Consequently, they establish a single-interval-characterization for the spectra of separable multidimensional discrete Schrödinger operators generated by 1D Fibonacci Hamiltonians with sufficiently small couplings. (e) For fixed s ∈ R \ {−1, 0, 1} and for fixed Diophantine frequency in the 2-torus M. Goldstein, W. Schlag, and M. Voda [17] proved that for sufficiently large couplings the 1D operators with 2-frequency quasiperiodic potential V λ,s,α : Z → R : n → λ(cos(2πnα 1 ) + s cos(2πnα 2 )) must have an interval spectrum. 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]. The [10,25] results pertain to the spectral type but not the topological structure of the spectrum as a set. In section 2 we state the preliminaries. In section 3 we prove the main theorem. In section 4 we prove a lemma used in the proof of the main theorem.

Preliminaries
Let A : H → H be a bounded operator. The spectrum of A is denoted σ(A). Note σ(A) is a nonempty compact subset of C. Also note, if A is self-adjoint, then σ(A) ⊆ R.

Separable Potentials and The Laplacian. Let
for every n ∈ Z d . The proof of the following theorem can be found within [12].
By the spectral mapping theorem, By Theorem 2.1, Here U m is the unitary operator from 2 (Z d ) to 2 (Z d ) defined by [U m ψ](n) = ψ(n − m) for every ψ ∈ 2 (Z d ), n ∈ Z d . Also here, Φ A is the Borel functional calculus with respect to a bounded operator A.

Quasiperiodic Potentials and The Almost Mathieu Operator. Let
and there exist α, ω ∈ T b such that v is nonconstant and {1, α 1 , . . . , α b } is independent over the rationals and V (n) = v(nα + ω) for every n ∈ Z. The proof of the following theorem can be found within [36].
be a quasiperiodic potential with parameter ω. Let H ω be the Schrödinger operator. Then σ(H ω ) = Σ ω =: Σ is independent of ω. Furthermore, Σ is a nonempty compact subset of R and Σ has no isolated points.
The potential of the almost Mathieu operator V λ,α,ω : Z → R : n → 2λ cos(2π(nα+ω)) is 1-frequency quasiperiodic when α is irrational. By the resolution of the Ten Martini Problem within [4], the spectrum Σ λ,α,ω of the almost Mathieu operator is a Cantor set when α is irrational.

Integrated Density of States. Let
be a quasiperiodic potential with parameter ω. Let H ω be the Schrödinger operator. The integrated density of states (IDS) is Here µ is the normalized Haar measure on T b . Note N (−∞, inf Σ] ≡ 0 and N [sup Σ, +∞) ≡ 1. The proof of Theorem 2.3 can be found within [36] and of Theorem 2.4 can be found within [11]. (i) N is monotone and continuous.
(i) α satisfies a Diophantine condition: for some c > 0, t > b. (ii) v satisfies a regularity condition: v is r-differentiable on T b for some r ≥ 550t.
be a quasiperiodic potential. Let H be the Schrödinger operator. Let N be the IDS. Define Σ := σ(H). Let Gap b (Σ) be the collection of all bounded gaps of Σ. Fix n ∈ Z b \ {0}. The n-th spectral gap of Σ is either the element G (n) in Gap b (Σ) such that N G (n) ≡ nα − nα or the empty set when such an element G (n) does not exist. Moreover, (−∞, inf Σ) ∪ (sup Σ, +∞) is called the 0-th spectral gap of Σ. The proof of the following theorem can be found within [29].
be a quasiperiodic potential with parameters v, α, ω. Let H v,α,ω be the Schrödinger operator. Fix α. Fix r 0 > r > 0. Define be the n-th spectral gap of Σ v . Assume the following.
Note τ (K) = 0 when K has an isolated point, τ (K) = 1 when K is the middle-thirds Cantor set, and τ (K) = +∞ when K is an interval. The proof of Theorem 2.6 can be found within [12] and of Theorem 2.7 can be found within [1,2]. The former theorem is the classical Gap Lemma, but we require the latter theorem which is a generalized version.
Theorem 2.6 (Gap Lemma). Let K 1 and K 2 be nonempty compact subsets of R. For each nonempty compact subset K of R, let Gap b (K) be the collection of all bounded gaps of K. For each nonempty compact subset K of R, define Γ(K) := sup{length(U ) : Theorem 2.7 (Gap Lemma). Let K 1 , . . . , K d (d ≥ 2) be nonempty compact subsets of R. For each nonempty compact subset K of R, let Gap b (K) be the collection of all bounded gaps of K. For each nonempty compact subset K of R, define Γ(K) := sup{length(U ) : U ∈ Gap b (K)}. Assume .
Because Σ λ has no isolated points, Temporarily, assume F λ (n) = sup Σ λ . Therefore By (i-iii), for sufficiently small λ, Because the frequency is fixed and because the spectrum is the same for any phase, we define Σ λ := Σ λ,α,ω . The spectral-thickness τ (Σ λ ) is the infimum-length-ratio of planks over gaps; more details can be found within subsection 2.6. Therefore it can be similarly shown that where κ λ (n) is a positive real-valued function of n. Taking the limit as λ approaches zero concludes the proof of Lemma 4.1. Then lim λ→0 τ (σ(H λ,α,ω )) = +∞.