On the spectrum of critical almost Mathieu operators in the rational case
aa r X i v : . [ m a t h . SP ] J u l On the spectrum of critical almost Mathieu operators inthe rational case
S. Jitomirskaya, L. Konstantinov, I. Krasovsky
Abstract
We derive a new Chambers-type formula and prove sharper upper bounds on themeasure of the spectrum of critical almost Mathieu operators with rational frequencies.
The Harper operator, a.k.a. the discrete magnetic Laplacian , is a tight-binding model of anelectron confined to a 2D square lattice in a uniform magnetic field orthogonal to the latticeplane and with flux 2 πα through an elementary cell. It acts on ℓ ( Z ) and is usually givenin the Landau gauge representation( H ( α ) ψ ) m,n = ψ m,n − + ψ m,n +1 + e − i παn ψ m − ,n + e i παn ψ m +1 ,n , (1)first considered by Peierls [17], who noticed that it makes the Hamiltonian separable andturns it into the direct integral in θ of operators on ℓ ( Z ) given by:( H α,θ ϕ )( n ) = ϕ ( n −
1) + ϕ ( n + 1) + 2 cos 2 π ( αn + θ ) ϕ ( n ) , α, θ ∈ [0 , . (2)In physics literature, it also appears under the names Harper’s or the Azbel-Hofstadtermodel, with both names used also for the discrete magnetic Laplacian H ( α ). In mathematics,it is universally called the critical almost Mathieu operator. In addition to importance inphysics, this model is of special interest, being at the boundary of two reasonably wellunderstood regimes: (almost) localization and (almost) reducibility, and not being amenableto methods of either side. Recently, there has been some progress in the study of the finestructure of its spectrum [6, 7, 9, 13, 15].Denote the spectrum of an operator H, as a set, by σ ( H ). An important object isthe union of σ ( H α,θ ) over θ, which coincides with the spectrum of H ( α ). We denote it S ( α ) := σ ( H ( α )) = ∪ θ ∈ [0 , σ ( H α,θ ). Note that by the general theory of ergodic operators, if α is irrational, σ ( H α,θ ) is independent of θ . We denote the Lebesgue measure of a set A by | A | .For irrational α , the Lebesgue measure | S ( α ) | = 0, and S ( α ) is a set of Hausdorff di-mension no greater than 1 / The name “discrete magnetic Laplacian” was first introduced by M. Shubin in [18]. This name was originally introduced by Barry Simon [19]. α ∈ Q and a strong continuity. For rational α = p q , where p , q are coprimepositive integers, Last obtained the bounds [14, Lemma 1]:2( √ q < (cid:12)(cid:12)(cid:12)(cid:12) S (cid:18) p q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) < eq , (3)where e = exp(1) = 2 . . . . . While the upper bound in (3) was sufficient for the argumentof [8], the measure of the spectrum is subject to another conjecture of Thouless [20, 21]:that in the limit p n /q n → α , we have q n | S ( p n /q n ) | → c , where c = 32 C c /π = 9 . . . . , C c = P ∞ k =0 ( − k (2 k + 1) − being the Catalan constant. Thouless provided a partly heuristicargument in the case p n = 1, q n → ∞ . A rigorous proof for α = 0 and p n = 1 or p n = 2, q n odd, was given in [5].The purpose of this note is to present a sharper upper bound, for all α ∈ Q : Theorem 1.
For all positive coprime integers p and q , (cid:12)(cid:12)(cid:12)(cid:12) S (cid:18) p q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πq . Thus, the upper bound is reduced from 8 e = 21 . . . . to 4 π = 12 . . . . . The way weprove Theorem 1 is very different from that of [14]; we use the chiral gauge representation [8]and Lidskii’s inequalities. The chiral gauge representation of the almost Mathieu operatoralso leads to a new type of Chambers’ relation (equations (14), (15) below). Consider the following operator on ℓ ( Z ):( e H α,θ ϕ )( n ) = 2 sin 2 π ( α ( n − θ ) ϕ ( n − π ( αn + θ ) ϕ ( n +1) , α, θ ∈ [0 , , (4)and define e S ( α ) := ∪ θ ∈ [0 , σ ( e H α,θ ). It was shown in [8, Theorem 3.1] that the operators M α := ⊕ θ ∈ [0 , H α,θ and f M α := ⊕ θ ∈ [0 , e H α,θ are unitarily equivalent, so that S ( α ) = e S ( α/ σ ( H α,θ ) = σ ( e H α,θ ) , in general.) See also related partly non-rigorous consider-ations in [16, 10, 22, 11, 12], and an application of the rational case in [13]. Operator (4)corresponds to the chiral gauge representation of the Harper operator.From now on, we always consider the case of rational α . Furthemore, the analysis belowfor q = 1, q = 2 becomes especially elementary, and gives | S (1) | = 8, | S (1 / | = 4 √
2, sothat Theorem 1 obviously holds in these cases. From now on, we assume q ≥ p is even, define p := p and q := q (note that q is necessarily odd in this case). Thiscorresponds to case I below. If p is odd, define p := p and q := 2 q . This corresponds tocase II below. We note that in either case p and q are coprime and S ( p /q ) = e S ( p/q ).Let b ( x ) := 2 sin(2 πx ), and further identify b n ( θ ) := b (( p/q ) n + θ ). For the operator e H pq ,θ ,Floquet theory states that E ∈ σ ( e H pq ,θ ) if and only if the equation ( e H pq ,θ ϕ )( n ) = Eϕ ( n ) has2 solution { ϕ ( n ) } n ∈ Z satisfying ϕ ( n + q ) = e ikq ϕ ( n ) for all n , and for some real k . Therefore,for a fixed k , there exist q values of E satisfying the eigenvalue equation B θ,k,ℓ ϕ ( ℓ )... ϕ ( ℓ + q − = E ϕ ( ℓ )... ϕ ( ℓ + q − (5)for any ℓ , where B θ,k,ℓ := b ℓ · · · e − ikq b ℓ + q − b ℓ b ℓ +1 · · · b ℓ +1 b ℓ +2 · · · · · · b ℓ + q − b ℓ + q − e ikq b ℓ + q − · · · b ℓ + q − . (6)Thus, the eigenvalues of B θ,k,ℓ are independent of ℓ . The celebrated Chambers’ formula presents the dependence of the determinant of the al-most Mathieu operator with α = p /q restricted to the period q with Floquet boundaryconditions, on the phase θ and quasimomentum k . In the critical case it is given by (see,e.g., [14])det( A θ,k,ℓ − E ) = ∆( E ) − − q (cos(2 πq θ ) + cos( kq )) , (7)where A θ,k,ℓ := a ℓ · · · e − ikq a ℓ +1 · · · a ℓ +2 · · · · · · a ℓ + q − e ikq · · · a ℓ + q − , ℓ ∈ Z , (8) a ( x ) := 2 cos(2 πx ) , a n ( θ ) := a (( p /q ) n + θ ) , (9)and ∆, the discriminant , is independent of θ and k . An immediate corollary of this formulais that S (cid:16) p q (cid:17) = ∆ − ([ − , B θ,k,ℓ − E ) . Indeed, as usual, separatingthe terms containing k in the determinant, we obtain, for the characteristic polynomial D θ,k ( E ) := det( B θ,k,ℓ − E ) : D θ,k ( E ) = D (0) θ ( E ) − ( − q b · · · b q − · kq ) , (10)where D (0) θ ( E ) is independent of k and equal therefore to D θ,k = π q ( E ).For the product of b j ’s we have: In [14], the discriminant differs from ∆( E ) by the factor ( − q . emma 1. b · · · b q − = q − Y j =0 π (cid:18) pq j + θ (cid:19) = 4 sin( πqθ ) sin πq ( θ + 1 /
2) = 2(cos( πq/ − cos πq (2 θ + 1 / . (11) Proof.
To evaluate the product of b j ’s, we expand sine in terms of exponentials and use theformula 1 − z − q = Q q − j =0 (1 − z − e πi pq j ). An alternative derivation can go along the lines ofthe proof of Lemma 9.6 in [1]. (cid:4) Substituting (11) into (10), we have D θ,k ( E ) = D (0) θ ( E ) − − q sin( πqθ ) sin πq ( θ + 1 /
2) cos( kq ) . (12)We can further obtain the dependence of D (0) θ ( E ) on θ : Lemma 2. D (0) θ ( E ) = e ∆( E ) + (cid:26) , q odd πqθ ) − , q even , where the discriminant e ∆( E ) := D (0) θ =0 ( E ) is independent of θ .Proof. Since D θ,k ( E ) is independent of ℓ , it is 1 /q periodic in θ , i.e., D θ,k ( E ) = D θ +1 /q,k ( E ),and by (10) so is D (0) θ ( E ). Therefore, since, clearly, D (0) θ ( E ) = P qn = − q c n ( E ) e πiθn , the terms c k other than k = mq vanish, and D (0) θ ( E ) has the following Fourier expansion: D (0) θ ( E ) = c ( E ) + c q e πiqθ + c − q e − πiqθ . It is easily seen that the c q and c − q can be obtained from the expansion of the determinantand that, moreover, they do not depend on E. Expanding D (0) θ ( E ) with E = 0 in rows andcolumns (cf. [13]), we obtain D (0) θ (0) = D θ,k = π q (0) = (cid:26) , q odd( − q/ ( b b · · · b q − + b b · · · b q − ) , q even . (13)This gives c q = c − q = 0 for q odd, and c q = Q q − j =0 e πi pq j + Q q − j =0 e πi pq (2 j +1) = 2 = c − q , for q even. It remains to denote e ∆( E ) = c ( E ) for q odd, and e ∆( E ) = c ( E ) + 4 for q even, andthe proof is complete. (cid:4) We therefore have, by (12) and Lemma 2:
Lemma 3 (Chambers-type formula) . D θ,k ( E ) = e ∆( E ) + 4( − ( q − / sin(2 πqθ ) cos( kq ) , q odd . (14) D θ,k ( E ) = e ∆( E ) − − cos(2 πqθ ))(1 + ( − q/ cos( kq )) , q even . (15)4ote that e ∆( E ) is a polynomial of degree q independent of k ∈ R and θ ∈ [0 , σ ( e H pq ,θ ) is the union of the eigenvalues of B θ,k,ℓ over k , acollection of q intervals.We make the following observations.Case I: q is odd.By (14), D θ,k ( E ) ≡ det( B θ,k,ℓ − E ) = 0 if and only if e ∆( E ) = 4( − ( q +1) / sin(2 πqθ ) cos( kq ).Thus, σ ( e H pq ,θ ) is the preimage of [ − | sin(2 πqθ ) | , | sin(2 πqθ ) | ] under the mapping e ∆( E ). If θ = m/ (2 q ), m ∈ Z , σ ( e H pq , m q ) is a collection of q points where e ∆( E ) = 0. (In this case, b ( m/ (2 q )) = 0, so that e H splits into the direct sum of an infinite number of copies of a q -dimensional matrix.) We note that the spectra σ ( e H pq ,θ ) for different θ are nested in oneanother as θ grows from 0 to 1 / (4 q ); in particular, for each θ ∈ [0 , σ ( e H pq ,θ ) = e ∆ − ([ − | sin(2 πqθ ) | , | sin(2 πqθ ) | ]) ⊆ σ ( e H pq ,θ = q ) = e ∆ − ([ − , . (16)This implies that all the maxima of e ∆( E ) are no less than 4, and all the minima are nogreater than −
4. Moreover, taking the union over all θ ∈ [0 ,
1) gives: e S (cid:18) pq (cid:19) = σ ( e H pq ,θ = q ) = e ∆ − ([ − , . (17)Clearly, it is sufficient to consider only θ ∈ [0 , / (4 q )].Case II: q is even. This case is similar to case I, so we omit some details for brevity. By(15), D θ,k ( E ) = 0 if and only if e ∆( E ) = 4(1 − cos(2 πqθ ))(1 + ( − q/ cos( kq )). Consideringthe cases k = 0 , πq , it is easy to see that σ ( e H pq ,θ ) is the preimage of [0 , − πqθ )] underthe mapping e ∆( E ). If θ = m/q , m ∈ Z , σ ( e H pq , mq ) is a collection of q points where e ∆( E ) = 0.We note that the spectra σ ( e H pq ,θ ) for different θ are nested in one another as θ grows from0 to 1 / (2 q ); in particular, for each θ ∈ [0 , σ ( e H pq ,θ ) = e ∆ − ([0 , − πqθ )]) ⊆ σ ( e H pq ,θ = q ) = e ∆ − ([0 , . (18)This implies that all the maxima of e ∆( E ) are no less than 16, and all the minima are nogreater than 0. Moreover, taking the union over all θ ∈ [0 ,
1) gives: e S (cid:18) pq (cid:19) = σ ( e H pq ,θ = q ) = e ∆ − ([0 , . (19)Clearly, it is sufficient to consider only θ ∈ [0 , / (2 q )].In this case of even q we can say more about the form of e ∆( E ). Note that b (0) = b q/ (0) =0 and b k (0) = b − k (0). Recall that by Floquet theory, D θ,k ( E ) = det( B θ,k,ℓ − E ) is independentof the choice of ℓ . For convenience, choose ℓ = − q/ B θ =0 ,k,ℓ = − q/ decomposes into a direct sum, and moreover e ∆( E ) = D θ =0 ,k ( E ) = ( − q/ P q/ ( − E ) P q/ ( E ),where P q/ ( E ) is a polynomial of degree q/
2, odd if q/ q/ e ∆( E ) = P q/ ( E ) is a square.The discriminants e ∆( E ) ≡ e ∆ p/q ( E ) and ∆( E ) ≡ ∆ p /q ( E ) are related in the followingway: Lemma 4.
For q odd, e ∆ p/q ( E ) = ∆ p /q ( E ) , p = 2 p, q = q. (20) For q even, e ∆ p/q ( E ) = ∆ p /q ( E ) , p = p, q = q/ . (21) Proof.
Case I: q is odd. Here, by our definitions at the start of the section, p = 2 p and q = q . e ∆ p/q ( E ) and ∆ p /q ( E ) are polynomials in E of degree q with the same coefficient − E q . Since e ∆( E ) = ∆( E ) = ± q ≥ q + 1 distinct edges of the bands (cf. [4,3.3]), these polynomials coincide: e ∆( E ) = ∆( E ) for each E .Case II: q is even. Here, p = p and q = q/ e S (cid:16) pq (cid:17) = S (cid:16) p q (cid:17) is the preimage of [0 , e ∆ p/q and of [ − ,
4] under ∆ p /q , hence also of [0 ,
16] under ∆ p /q . On the other hand,we have seen above that e ∆( E ) = P q/ ( E ) for some polynomial P q/ ( E ) of degree q/ q .Thus, P q/ ( E ) and ∆ ( E ) coincide at the 2 q ≥ q + 1 (for q odd) and 2 q − ≥ q + 1 (for q even) distinct edges of the bands (cf. [4, 3.3]; the central bands merge for q even), sothese polynomials of degree q are equal: e ∆( E ) = ∆ ( E ) for each E . (cid:4) The rest of the proof follows the argument of [3], namely it uses Lidskii’s inequalities tobound | e S ( pq ) | . The key observation is that choosing ℓ appropriately, we can make the cornerelements of the matrix B θ,k,ℓ very small, of order 1 /q when q is large. This is not possible todo in the standard representation for the almost Mathieu operator. Here are the details.Case I: q is odd. Assume without loss that ( − ( q +1) / > θ ∈ (0 , / (4 q )]. (If ( − ( q +1) / <
0, the analysis is similar.) Then the eigenvalues { λ i ( θ ) } qi =1 of B θ,k =0 ,ℓ labelled in decreasingorder are the edges of the spectral bands where e ∆( E ) reaches its maximum 4 sin(2 πqθ ) onthe band; and the eigenvalues { b λ i ( θ ) } qi =1 of B θ,k = π/q,ℓ labelled in decreasing order are theedges of the spectral bands where e ∆( E ) reaches its minimum − πqθ ) on the band.Then | σ ( e H pq ,θ ) | = q X j =1 ( − q − j ( b λ j ( θ ) − λ j ( θ )) = ( q +1) / X j =1 ( b λ j − ( θ ) − λ j − ( θ )) + ( q − / X j =1 ( λ j ( θ ) − b λ j ( θ )); b λ j ( θ ) − λ j ( θ ) > , if j is odd; b λ j ( θ ) − λ j ( θ ) < , if j is even. (22)6ow we view B θ,k = π/q,ℓ as B θ,k =0 ,ℓ with the added perturbation B θ,k = π/q,ℓ − B θ,k =0 ,ℓ = − b ℓ + q − − b ℓ + q − , which has the eigenvalues { E i ( θ ) } qi =1 given by: E q ( θ ) = − | b ℓ + q − ( θ ) | < E q − ( θ ) = · · · = E ( θ ) = 0 < | b ℓ + q − ( θ ) | = E ( θ ) . The Lidskii inequalities (e.g., (2.51) in [3]) are:
Theorem 2.
For any q × q self-adjoint matrix M , we denote is eigenvalues by E ( M ) ≥ E ( M ) ≥ · · · ≥ E q ( M ) . For q × q self-adjoint matrices A and B , we have: E i ( A + B ) + · · · + E i m ( A + B ) ≤ E i ( A ) + · · · + E i m ( A ) + E ( B ) + · · · + E m ( B ); E i ( A + B ) + · · · + E i m ( A + B ) ≥ E i ( A ) + · · · + E i m ( A ) + E q − m +1 ( B ) + · · · + E q ( B ) , for any ≤ i < · · · < i m ≤ q . Applying these inequalities with A = B θ,k =0 ,ℓ , B = B θ,k = π/q,ℓ − B θ,k =0 ,ℓ gives: ( q +1) / X j =1 ( b λ j − ( θ ) − λ j − ( θ )) ≤ ( q +1) / X j =1 E j ( θ ) = E ( θ ); ( q − / X j =1 ( λ j ( θ ) − b λ j ( θ )) ≤ − q X j =( q − / E j ( θ ) = − E q ( θ ) . Substituting these into (22), we obtain: | σ ( e H pq ,θ ) | ≤ E ( θ ) − E q ( θ ) = 4 | b ℓ + q − ( θ ) | . (23)Moreover, by the invariance of D θ,k ( E ) under the mapping b n b n + m , for n = 0 , , . . . , q − m , we can choose any ℓ in (23), so that | σ ( e H pq ,θ ) | ≤ ℓ | b ℓ + q − ( θ ) | . (24)In particular, (cid:12)(cid:12)(cid:12)(cid:12) e S (cid:18) pq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = | σ ( e H pq ,θ = q ) | ≤ ℓ (cid:12)(cid:12)(cid:12)(cid:12) b ℓ + q − (cid:18) q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 4 · (cid:12)(cid:12)(cid:12)(cid:12) sin 2 π (cid:18) q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πq . (25)Therefore, | S ( p q ) | = | e S ( pq ) | ≤ πq = πq , as required.Case II: q is even. This case is similar to case I, so we omit some details for brevity. Thistime, the Lidskii equations of Theorem 2 show that | e S ( pq ) | ≤ πq . Indeed, as in (24), we have(note the doubling of the eigenvalues for e ∆( E ) = 0) | σ ( e H pq ,θ ) | ≤ ℓ | b ℓ + q − ( θ ) | . (26)7n particular, (cid:12)(cid:12)(cid:12)(cid:12) e S (cid:18) pq (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = | σ ( e H pq ,θ = q ) | ≤ ℓ (cid:12)(cid:12)(cid:12)(cid:12) b ℓ + q − (cid:18) q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 4 · (cid:12)(cid:12)(cid:12)(cid:12) sin 2 π (cid:18) q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ πq . (27)Therefore, | S ( p q ) | = | e S ( pq ) | ≤ πq = πq , as required.This completes the proof of Theorem 1. Acknowledgment
The work of S.J. was supported by NSF DMS-1901462. The work of I.K. was supported bythe Leverhulme Trust research programme grant RPG-2018-260.
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