Conformal spectral theory for the monodromy matrix

For any $N\ts N$ monodromy matrix we define the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the Hill operator. The Lyapunov function has (real or complex) branch points, which we call resonances. We determine the asymptotics of the periodic, anti-periodic spectrum and of the resonances at high energy. We show that the endpoints of each gap are periodic (anti-periodic) eigenvalues or resonances (real branch points). Moreover, the following results are obtained: 1) we define the quasimomentum as an analytic function on the Riemann surface of the Lyapunov function; various properties and estimates of the quasimomentum are obtained, 2) we construct the conformal mapping with imaginary part given by the Lyapunov exponent and we obtain various properties of this conformal mapping, which are similar to the case of the Hill operator, 3) we determine various new trace formulae for potentials and the Lyapunov exponent, 4) we obtain a priori estimates of gap lengths in terms of the Dirichlet integral. We apply these results to the Schr\"odinger operators and to first order periodic systems on the real line with a matrix valued complex self-adjoint periodic potential.

Using the quasimomentum as conformal mapping a priori estimates for various parameters of the Hill operator and for the Dirac operator (gap lengths, effective masses,.. in terms of potentials) were obtained in [GT], [KK1], [KK2], [K2]- [K10], [M], [MO]. Conversly (it is significantly more complicated), a priori estimates of potential in terms spectral data (gap lengths, effective masses,..) were obtained [K2], [K5]- [K10], [M] for example, see [K2], where G 2 = n 1 |γ n | 2 and |γ n | 0 is the gap length. Such a priori estimates simplify the proof in the inverse spectral theory for scalar Hill operator [GT], [KK], [K1] and for the periodic Zakharov-Shabat systems [K4], [K5], see also [K6], where the author solved the inverse problem for the operator −y ′′ + u ′ y on L 2 (R), where periodic u ∈ L 2 loc (R). The corresponding theory for the vector case is still modest. It is well known that the spectrum of the Schrödinger operator on the real line with a N × N matrix valued real periodic potential, N > 1 is absolutely continuous and consists of intervals separated by gaps [DS]. We recall results from [CK] about this operator: the Lyapunov function, which is analytic on an associated N-sheeted Riemann surface is determined. Moreover, the conformal mapping with imaginary part given by the Lyapunov exponent is constructed and a priori estimates of gap lengths in terms of potentials are obtained. Some properties of the monodromy matrices and the corresponding the Lyapunov functions for periodic nanotubes were obtained in [KL1], [KL2]. M(z) has the eigenvalues τ j (z), j ∈ 1, N = {1, .., N} such that: ii) If |τ j (z)| = 1 for some (j, z) ∈ 1, N × C, then z ∈ R.
., C r the following asymptotics hold as z = iy, y → ∞. (1.5) Note that monodromy matrices for Schrödinger operators (or canonical systems) with periodic matrix-valued potentials and for Schrödinger operators on periodic nanotubes belong to this class, see [A], [CK], [KL1].
The main goal of our paper is to obtain new results about the Lyapunov functions, the quasimomentum and a priori estimates of gap lengths in terms of potentials for a class of monodromy matrices M N . Firstly, we construct Lyapunov functions and the conformal mapping (averaged quasimomentum) k(·), with imaginary part given by the Lyapunov exponent. In fact, we reformulate some spectral problem for the differential operator with periodic matrix coefficients as problems of conformal mapping theory. Secondly, we obtain various results from the conformal mapping theory. For solving these "new" problems we use some techniques from , [K2], [K6-8] and [CK]. In particular, we use the Poisson integral for the domain C + ∪ (−1, 1) ∪ C − . We apply these results to the Schrödinger operator and to first order periodic systems on the real line with a N × N matrix valued complex selfadjoint periodic potential for any N > 1. We plan to apply these results to study integrable systems [CD1], [CD2], [Ma] and integrated density of states for periodic nanotubes.
An eigenvalue τ (z) of M(z) is called a multiplier. Note that (1.3) yields that if some τ (z), z ∈ C is a multiplier of multiplicity d 1, then 1/τ (z) is a multiplier of multiplicity d.
Let L = M +M −1 2 and Φ(ν, z) = det(L(z) − νI N ). Let ∆ j (z), j ∈ 1, N be the zeros of Φ(ν, z) = 0, where m, n = {m, m + 1, .., n}. This is an algebraic equation in ν of degree N. The coefficients of Φ(ν, z) are entire in z ∈ C. It is well known (see e.g. [Fo]) that the roots ∆ j (z), j ∈ 1, N constitute one or several branches of one or several analytic functions that have only algebraic singularities in C. Thus the number of zeros of Φ(ν, z) = 0 is a constant N e with the exception of some special values of z (see below the definition of a resonance). In general, there is an infinite number of such points on the plane. If all functions ∆ j (z), j ∈ 1, N are distinct, then N e = N. If some of them are identical, then N e < N and Φ(z, ν) = 0 is permanently degenerate.
By definition, the number z 0 is a periodic eigenvalue if z 0 is a zero of the function det (M(z) − I N ). The number z 1 is an anti-periodic eigenvalue if z 1 is a zero of the function det(M(z) + I N ). We need the following preliminary results Theorem 1.1. Let M ∈ M N . Then there exist analytic functions ∆ s , s = 1, .., N 0 N on some N s -sheeted Riemann surface R s , N s 1 having the following properties: i) There exist disjoint subsets ω s ⊂ 1, N, s ∈ 1, N 0 , ω s = 1, N such that all branches of ∆ s , s ∈ 1, N 0 are given by ∆ j (z) = 1 2 (τ j (z) + τ −1 j (z)), j ∈ ω s and satisfy iii) Each function ρ s , s = 1, .., N 0 , given by (1.7) is entire and real on the real line, (1.7) iv) The following identity holds true where z ± n are either periodic (anti-periodic) eigenvalues or real branch points of ∆ j (for some j ∈ 1, N), which are zeros of ρ (below we call such points resonances).
N , then ρ is not a polynomial, since ρ is bounded on R. 2) Let the surface R = ∪ N 0 1 R s be the union of the disjoint Riemann surfaces R s and let ∆ = { ∆ s , s = 1, ., N 0 } be the corresponding analytic function on R. Let ζ → z be the standard projection from the surface R into the complex plane C. We set ζ ∈ R and z = φ(ζ) ∈ C. The surface R is a N-sheeted branch covering of the complex plane, equipped with the natural projection ζ → z. Below we will identify (locally) the point ζ ∈ R and the point z = φ(ζ) ∈ C (see [Fo], Chapter 4). In this case we set Imζ = Imφ(ζ).
Remark. 1) This result is important to describe the spectrum of Schrödinger operators with periodic potentials on armchair nanotubes [BBKL]. 2) It is very difficult to determine the positions of resonances. We can use only the Levinson Theorem (see Sect. 2) and Theorem 1.2. It is similar to another case of poles (other resonances) for scattering for Schrödinger operator with compactly supported potentials on the real line see [K11], [Z]. We consider the conformal mapping associated with M ∈ M N . We need functions from the subharmonic counterpart of the Cartwright class of the entire functions given by We recall the class of functions from [KK1] given by We note that SK + m+1 ⊂ SK + m , m 0. Introduce the simple conformal mapping η : 1], and η(z) = 2z + o(1) as |z| → ∞. (1.11) Note that η(z) = η(z), z ∈ C \ [−1, 1], since η(z) > 1 for any z > 1. The properties of the function ∆ j imply |η( ∆ s (ζ))| > 1, ζ ∈ R + s = {ζ ∈ R s : Im ζ > 0}. Thus we can define the quasimomentum k j (we fix some branch of arccos and ∆ j (z)) and the function q j by k j (z) = arccos ∆ j (z) = i log η(∆ j (z)), q j (z) = Im k j (z) = log |η(∆ j (z))|, k ∈ 1, N (1.12) and z ∈ R + 0 = C + \β + , β + = β∈B( e ∆)∩C + [β, β +i∞), where B(f ) is the set of all branch points of the function f . The branch points of k j in C + belong to B( ∆). Define the averaged quasimomentum k, the density p and the Lyapunov exponent q by Define the sets σ (N ) = {z ∈ R : ∆ 1 (z), .., ∆ N (z) ∈ [−1, 1]}, and For the function k(z) = p(z) + iq(z), z = x + iy ∈ C + we formally introduce the integrals (1.14) Let C us denote the class of all real upper semi-continuous functions h : R → R. With any h ∈ C us we associate the "upper" domain K(h) = {k = p + iq ∈ C : q > h(p), p ∈ R}. We formulate our first main result.
for some absolute constant C 0 > 0.
Using this theorem we deduce that the function h(p) = q(x(p)), p ∈ R is continuous on 1. The Schrödinger operator. We consider the self-adjoint operator It is well known (see p.1486-1494 [DS], [Ge]) that the spectrum σ(S) of S is absolutely continuous and consists of non-degenerate intervals [λ + n−1 , λ − n ], n = 1, .., N G ∞ and let λ + 0 = 0. These intervals are separated by the gaps γ n = (λ − n , λ + n ) with the length |γ n | > 0. Introduce the fundamental N × N-matrix solutions ϕ(t, z), ϑ(t, z) of the equation (1.25) where I N , N 1 is the identity N ×N matrix. Here and below we use the notation ( ′ ) = ∂/∂t. We define the monodromy 2N × 2N-matrix M, the matrix J and the trace T m , m ∈ Z by It is well known that M(·) ∈ M 2N , see [GL], [Kr]. The functions M and T m , m ∈ Z are entire and det M = 1. Let τ j , j ∈ 1, 2N be the eigenvalues of M. It is a root of the algebraic gives the spectral point z 2 ∈ σ(S) of multiplicity 2, see [CK]. For complex V we have ∆ j , j ∈ 1, 2N, where each ∆ j (z) ∈ [−1, 1] gives the spectral point z 2 ∈ σ(S) of multiplicity 1. The zeros of D(1, √ λ) ( and D(−1, √ λ)) (counted with multiplicity) are the periodic (anti-periodic) eigenvalues for the equation −y ′′ + V y = λy with periodic (anti-periodic) boundary conditions. Let g = ∪ n∈Z g n where g n = (z − n , z + n ), z ± n = λ ± n > 0 and g −n = g n , n 1. We formulate our main result.
ii) The averaged quasimomentum k = 1 2N 2N 1 k j is analytic in C + and k : C + → k(C + ) = K(h) is a conformal mapping onto K(h) for some h ∈ C us and q = Im k has an harmonic extension from C + into Ω = C + ∪ C − ∪ g given by q(z) = q(z), z ∈ C − and q(z) > 0 for any z ∈ Ω. Furthermore, q ∈ SK + 2 ∩ C(C) and there exist branches k j , j ∈ 1, 2N such that the following asymptotics, identities and estimates hold: for some absolute constant C 0 .
Remark. 1) Properties of the Lyapunov functions are formulated in Theorems 1.1, 1.2.
2) Various properties of the quasimomnetum k j are formulated in Theorems 1.3, 1.4, 3.1.
3) The existence of real and complex resonances was proved in [BBK] for the Schrödinger operator on the real line with a 2 × 2 matrix real valued periodic potential V ∈ H . 4) If the potential V ∈ H is real and a matrix 1 0 V (t)dt has distinct eigenvalues, then the operator S has only finite number complex resonances [CK]. 5) Let σ(m, A) denote the spectrum of a self-adjoint operator A of multiplicity m, m 0. We have the following simple corollary from Theorem 1.5: Let σ(S) = σ(2N, S) = R + for some V = V * ∈ H . Then V = 0. There are two simple proofs: a) if σ(S) = σ(N, S) = R + , then all gaps are close and the identity (1.29) yields V = 0. b) if σ(S) = σ(N, S) = R + , then all gaps are close and the estimate (1.33) yields V = 0. 6) Recall that the so-called Borg Theorem for periodic systems was proved in [CHGL], [GKM] for general cases.
2. The periodic canonical systems. Consider the operator K given by It is well known (see [DS] p.1486-1494, [Ge]) that the spectrum σ(K) of K is absolutely continuous and consists of non-degenerated intervals σ n . These intervals are separated by the gaps (1.34) It is well known that the monodromy matrixψ(1, ·) ∈ M N , see p.109 [YS], [Kr]. The eigenvalues τ (z) of ψ(1, z) are the multiplier of K: to each of them corresponds a solution f The zeros of D(1, z) (or D(−1, z)) are the eigenvalues of periodic (anti-periodic) problem for the equation −iJy ′ + V y = zy.
Theorem 1.6. Let V = V * ∈ H 0 . Then ψ(1, z) belongs to M 0,0 N , and the averaged quasimomentum k = 1 N N 1 k j is analytic in C + and k : C + → k(C + ) = K(h) is a conformal mapping for some h ∈ C us . Furthermore, and there exist branches k j , j ∈ 1, N such that the following asymptotics, identities and estimates hold: for some absolute constant C 0 .
Remark. 1) In the proof we use arguments from [K4-5] and [CK]. Theorem 1.6 generalizes the result of [CK] to the case of the canonical systems with periodic matrix potential. 2) We have the following simple corollary from Theorem 1.6: Let σ(K) = σ(N, K) = R for some V = V * ∈ H 0 . Then V = 0. The proof is similar to the case of the Schrödinger operator, see Remark 5) after Theorem 1.5. 3) In [K3] for periodic canonical system with with a specific 4×4 matrix real valued symmetric periodic potential the existence of real and complex resonances is proved. 4) Recall the estimates 1 √ 2 G V 2 G (1 + G ) for the case N 1 = N 2 = 1 from [K8]. The plan of our paper is as follows. In Sect. 2 we obtain the basic properties of the Lyapunov functions. In Sect. 3 we obtain the main properties of the quasimomentum and we prove the basic Theorems 1.3 and 1.4, devoted to the conformal mapping theory and Theorems 1.2 about resonances. In Sect. 4 we obtain the results for the Schrödinger operator and the first order systems and Theorem 1.5, 1.6 will be proved.
Proof. i) It is well known that the "polynomial" Φ(ν, z), z, ν ∈ C is given by (2.1), see p. 331-333 [RS]. Using the identity (1.3) we obtain This gives that each T m = Tr L m , m ∈ N is the entire and is real on the real line. ii) Let some some ∆ j (z) ∈ [−1, 1], z ∈ C. Then τ j (z) satifies |τ j (z)| = 1 and Definition M yields z ∈ R.
We recall some well known facts about entire functions, see [Koo]. An entire function f (z) is said to be of exponential type if there is a constant γ such that |f (z)| const. e γ|z| everywhere. The infimum over the set of γ for which such an inequality holds is called the type of f . The function f is said to belong to the Cartwright class if f (z) is entire, of exponential type, and R log + |f (x)|dx 1+x 2 < ∞. The function f is said to belong to the Cartwright class E C (a + , a − ), if f is entire, of exponential type, and the following conditions are fulfilled: Due to the Paley-Wiener Theorem, if f ∈ Cart and f (x) − cos x ∈ L 2 (R), then f = 1 −1φ (t)e −itz dt for someφ ∈ L 2 (−1, 1). Denote by N (r, f ) the total number of zeros of f with modulus r. Recall the following well known result (see p.69, [Koo]). Theorem (Levinson). Let the function f ∈ E C (1, 1). Then N (r, f ) = 2 π r + o(r) as r → ∞. and for each δ > 0 the number of zeros of f with modulus r lying outside both of the two sectors | arg z|, | arg z − π| < δ is o(r) for r → ∞.
We determine asymptotics of ρ s , ∆ m and the zeros of ρ s , s ∈ 1, N 0 .

The conformal mappings
In this section we study properties of the quasimomentum and prove theorems about the conformal mapping. Recall that the Lyapunov function ∆ s (ζ) is analytic on some N s -sheeted Riemann surface R s and R = ∪ N 0 1 R s . Let z = x+iy ∈ C be the natural projection of ζ ∈ R, B( ∆) be the set of all branch points of the Lyapunov function and R ± = {ζ ∈ R : ± Im ζ > 0}. Recall that the simply connected domains R ± 0 = C ± \ β ± ⊂ C ± and define a domain Due to the Definition M, ∆(ζ) / ∈ [−1, 1] for ζ ∈ R + . Recall that q(ζ) = | log η( ∆(ζ))| is the single-valued on R + imaginary part of the (in general, many-valued on R + ) quasimomentum We denote by q j (z), (z, j) ∈ C + × 1, N, the branches of q(ζ) and by p j (z), k j (z), z ∈ R + 0 , the single-valued branches of p(ζ), k(ζ), respectively.
Theorem 3.1. Let M ∈ M N and let s ∈ 1, N 0 (N 0 is defined in Theorem 1.1). Then the function q s (ζ) = log |η( ∆ s (ζ))| is subharmonic on the Riemann surface R s . Moreover, 1) If M ∈ M 0 N , then the following asymptotics hold If ∆ s (z) = cos z + O(1/z) as |z| → ∞, then the following asymptotics hold: 2) Let ∆ j be analytic on some bounded interval Y = (α, β) ⊂ R for some j ∈ ω s . Then i) If ∆ j (z) ∈ R \ [−1, 1] for all z ∈ Y , then k j (·) has an analytic extension from R + 0 into ii) If ∆ j (z) / ∈ R for any z ∈ Y , then there exists a branch ∆ j ′ , j ′ ∈ ω s such that ∆ j ′ (z) = ∆ j (z) for any z ∈ Y . The functions ∆ j (z) and k j ′ + k j have analytic extensions from Proof. The proof repeats the proof of Theorem 4.1 from [CK].
Recall the needed properties of the functions q ∈ SC defined in Sect. 1 and k = p + iq. It is well known, that p ∈ C(C + ) and 1 2π (∂ 2 x + ∂ 2 y )q = µ q (in a sense of distribution) is a so-called Riesz measure of the function v. Moreover, the following identities are fulfilled: Now we recall the well known fact (see [Ah]). Theorem (Nevanlinna). i) Let µ be a Borel measure on R such that R (1+x 2r )dµ(x) < ∞ for some integer r 0. Then for each s > 0 the following asymptotics hold ii) Let f be an analytic function in C + such that Im f (z) 0 for all z ∈ C + and Im f (iy) = c 0 y −1 +. . .+c 2p−1 y −2r +O(y −2r−1 ) as y → ∞ (3.13) for some c 0 , ..c 2r−1 ∈ R and r 0. Then f (z) = C + R dµ(t) t−z , z ∈ C + , for some Borel measure µ on R such that R (1 + x 2r )dµ(x) < ∞ and C ∈ R.
Proof of Theorem 1.3. i) The proof repeats the proof of Theorem 4.2 from [CK].
ii) Below for each real harmonic function q(z), z ∈ C + we introduce an analytic function k = p + iq in C + , where (−p) is some harmonic conjugate of q for C + . If q ∈ SK + 0 , then the function k = p + iq in C + is defined by (3.14) Due to i), the function q = 1 N N 1 q j ∈ SC ∩ C(C) and q is positive in C + . Let z = iy, y → ∞, using q m (iy) = y + o(1) (see (3.1) ) we obtain Thus these asymptotics and Φ(z, 0) = (cos N z) exp i − 2r Then by i) and the Nevanlinna Theorem, the function q ∈ SK + 2r . We need the following result from [KK1]: Let q ∈ SK + m for some m 0 and q = const. Then k : C + → k(C + ) = K(h) is a conformal mapping for some h ∈ C us , h 0. Moreover, the following asymptotics, estimates and identities are fulfilled: where I S n , I D n , Q n , n 0 are given by (1.14). Thus the above results give that k : C + → k(C + ) = K(h) is a conformal mapping for some h ∈ C us . Moreover, asymptotics (1.17) and identities (1.18), (1.19) hold true.
4 Proof of Theorems 1.5 and 1.6 We begin with some notational convention. A vector h = {h n } N 1 ∈ C N has the Euclidean norm |h| 2 = N 1 |h n | 2 , while a N × N matrix A has the operator norm given by |A| = sup |h|=1 |Ah|. Note that |A| 2 Tr A * A.
1 The first order periodic systems. In this case J = I N 1 ⊕ (−I N 2 ). Below we use arguments from [K4], [K5]. We need the identities (4.1) The solution of the equation −iJψ ′ + V ψ = zψ, ψ 0 (z) = I N satisfies the integral equation and ψ is given by Proceeding by induction, We need the following estimates.  since |t − 2t 1 + 2t 2 · · · + (−1) n 2t n | t, which yields (4.8). This shows that for each t > 0 the series (4.3) converges uniformly on bounded subsets of C × H 0 . Each term of this series is an entire function. Hence the sum is an entire function. Summing the majorants we obtain estimates (4.9). The proof of asymptotics in (4.10) is standard (see e.g. [K4] or [K5]). (4.10) implies (4.11). Assume that the sequence V ν → V weakly in H , as ν → ∞. Then each term ψ n (t, z, V ν ) → ψ n (t, z, V ) uniformly on bounded subsets of R × C and fixed n 1. Then (4.9) gives that ψ(t, z, V ν ) → ψ(t, z, V ) uniformly on bounded subsets of R × C.
We formulate the following results from [CK].