KdV hierarchy via Abelian coverings and operator identities

We establish precise spectral criteria for potential functions $V$ of reflectionless Schr\"odinger operators $L_V = -\partial_x^2 + V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods extend the classical study of algebro-geometric solutions for the KdV hierarchy to noncompact Riemann surfaces by defining generalized Abelian integrals and analogues of the Baker-Akhiezer function on infinitely connected domains with a uniformly thick boundary satisfying a fractional moment condition.


Introduction
We study the Cauchy problem for the Korteweg de-Vries (KdV) equation and the associated hierarchy of higher-order differential equations for k ∈ N, which we will define precisely below. For the sake of introduction, we simply note that KdV 1 (V ) = 1 4 ∂ 3 x V − 3 2 V ∂ x V is the typical KdV equation, and that KdV k is generally an order 2k + 1 polynomial differential operator. We establish conditions on V in terms of the spectral properties of the associated Schrödinger operator L V = −∂ 2 x + V which guarantee global existence and temporal and spatial almost-periodicity of classical solutions: Theorem 1.1. Assume that the family of operators {L Vx 0 : V x 0 (x) = V (x+x 0 ), x 0 ∈ R} is ergodic. Denote by E their common spectral set, E ac their almost-sure a.c. spectrum, and by ρ(ξ)/ √ ξ the associated integral density of states (i.d.s). Suppose that E is homogeneous, E = E ac , and E satisfies the following moment condition: and that the i.d.s. satisfies the entropy condition Then the Cauchy problem (1.1), (1.2) admits a global solution V (x, t k ) which is uniformly almost-periodic in the time and space coordinates.
We present this result first because of its broad appeal. Recent work [1] of Binder, Damanik, Goldstein, and Lukic has, under stronger assumptions, proven something similar in the case k = 1, and Kotani has announced a related result under an integer moment condition [16]. In our case, Theorem 1.1 is only a facet of our main theorem, Theorem 1.14. In fact, the full content of this paper is the development of a spectrally-dependent Fourier transform, with respect to which the study of reflectionless Schrödinger operators becomes greatly simplified.
Naturally, our result is based on Lax pair representation in the theory of integrable systems, see e.g. [6,11], the spectral theory of ergodic 1-D Schrödinger operators, see e.g. [24], and the functional model approach to the spectral theory see e.g. [22]. The relationship between the KdV equation and the Schrödinger operator via the Lax pair formalism was noted in the 1960s by Lax [18]. At the same time, use of inverse scattering techniques to solve the KdV equation was pioneered by Gardner, Greene, Kruskal, and Miura [8]. The 1970s saw significant further development expounding on these ideas to solve the KdV equation for periodic initial data [4,20,23]. Shortly thereafter, algebro-geometric extensions of the techniques from the periodic setting were developed to address almost-periodic and ergodic initial conditions having finite-gap spectra; see [11] for a textbook treatment on this approach. Extensions of these techniques to the infinite-gap setting have been partially developed in important work by Egorova [7], and more recently in work of Binder, Damanik, Goldstein, and Lukic [1]. Some of our methods, including the ideas of generalized Abelian integrals and conformal mappings onto comb domains, were established by Marchenko in [19]. Other important methods in integrable systems were developed by Deift and Zhao [2].
The sequel will be structured as follows: the remainder of Section 1 establishes definitions and notation en route to our main theorem, Theorem 1.14. Section 2 recalls some basic preliminaries from spectral theory, and establishes the spectral infemum as a normalization point. Section 3 describes in detail the functional models and proves an important identity regarding the reproducing kernels. Finally, Section 4 discusses the generalized Abelian integrals and their relationship to the finite-gap case, and proves the asymptotic expansion of the m-functions up to a certain order under our conditions.

Generalized Abelian integrals and Hardy spaces on infinitely-connected domains
Algebro-geometric solutions for the KdV hierarchy equations are given by means of the Baker-Akhiezer function, see e.g. [17,28], and books [6,21], see also [11]. This function is associated to a hyperelliptic Riemann surface (with the standard compactification) and a specified point ∞ ∈ S on it. The Baker-Akhiezer function contains two factors: the first one is given by an Abelian integral with a certain singularity at ∞, and the second one is represented as a specific ratio of theta-functions closely related to the so-called prime form [21,IIIb,§1].
Our Baker-Akhiezer function is (1.19). It allows for the much more general case when N = ∞, although we are restricted by the assumption that a j , b j are real. In our normalization 0 < a j < b j and E := R + \ ∪ j≥1 (a j , b j ) does not contain any isolated points. So, we consider the domain which would play a role of the "upper sheet" for a compact hyperelliptic surface S, N < ∞. A classical construction of Abelian integrals is given by means of potential theory [14,Part III,Ch. 9]. Our first assumption is that our domain is standard in this regard: (R) Let G λ 0 (λ) = G(λ, λ 0 ) be the Green function in the domain S + with logarithmic singularity at λ 0 ∈ S + . We assume that G λ 0 is continuous up to the boundary, that is, S + is regular in terms of the potential theory, [10,Theorem 6.3.,p. 95].
We define This space possesses a reproducing kernel, which we denote by k α Since the elements of H 2 S + (α) are multivalued functions, (1.7) creates a state of small uncertainty. To avoid this uncertainty, we provide an alternative definition using the universal covering for S + .
Recall that for the given domain there exists a function λ(z) analytic in the upper half-plane C + and a Fuchsian group Γ π 1 (S + ) such that λ(z) induces a one to one correspondence between the points λ ∈ S + and orbits {γ(z)} γ∈Γ , z ∈ C + . We will normalize this function by the conditions λ(iy) ∈ R − , y > 0, λ(iy) → −∞ as y → ∞.
Due to this normalization, λ(z) is defined up to a positive multiplier. By Γ * we denote the group of unitary characters of the group Γ.
The following definition is basically parallel to Definition 1.2.
is the subspace of the standard H 2 in C + (with respect to the harmonic measure 1 π dx 1+x 2 ) consisting of character-automorphic functions, i.e., The spaces H 2 S + (α) and H 2 Γ (α) correspond by way of the uniformization λ(z): We denote the reproducing kernel in H 2 Γ (α) by k α Γ (z, z 0 ) and remark that the correspondence in Proposition 1.4 completely removes the above-mentioned ambiguity in (1.7). For a textbook discussion of character automorphic Hardy spaces, see [12].
Our second assumption on S + is the Widom condition, which is necessary and sufficient for non-triviality of the spaces H 2 Γ (α) for all α ∈ Γ * .
(PW) We assume that for some (and hence all) λ 0 ∈ S + , we have As soon as (1.8) holds, S + is called a domain of Widom type, and Γ is respectively called a group of Widom type. For λ 0 ∈ R − all critical values ξ = ξ(λ 0 ) are real; moreover, there is exactly one critical point in each gap, ξ j (λ 0 ) ∈ (a j , b j ). The Widom condition (PW) along with our third assumption allows us to extend the notion of Abelian integrals of the second kind: (k-GLC) By the order k gap length condition we mean Under the assumptions (PW) and (k-GLC), we can define the following (see Proposition 4.1 below): Definition 1.5. By the (normalized) order k generalized Abelian integral we mean the multivalued in S + analytic function Θ (k) (λ), whose (single-valued) imaginary part is given by (1.10) As before, the analytic continuation Θ (k) (γ(λ)) along the path γ ∈ π 1 (S + ) is related to the original value Θ (k) (λ) by (1.11) We say that η (k) is an additive character on π 1 (S + ), and that the function Θ (k) (λ) is (additive) character automorphic.
Lemma 1.6. The generalized Abelian differentials dΘ (k) (λ) are of the form where π (k) (λ) is a certain monic polynomial of degree k and c (k) j ∈ (a j , b j ). In the special case k = 0 we will drop the index. M (λ) is called the Martin function of the given domain (with respect to infinity) or Phragmén-Lindelöf function (particularly, with respect to the representation (1.10), see [15,Theorem,p. 407]).

Functional models on the universal Abelian cover in application to spectral theory of 1-D Schrödinger operators
We can now trace the path towards our Main Theorem. The universal Abelian covering of S + is defined as follows: denote by Γ the commutator subgroup of Γ The quotient Γ/Γ is canonically dual to Γ * by Pontryagin duality. Thus, any of the multi-valued analytic continuations defined above become functions on the surface R := C + /Γ . In particular, we have the covering map λ R : (1.14) Similarly, we consider generalized Abelian integrals as functions on R and denote them by θ (k) ). As one would expect, the group Γ/Γ acts on this surface by where γ ∈ Γ/Γ denotes the equivalence class in Γ/Γ of an element γ ∈ Γ. According to this notation (1.11) has the form The main theorem claims that the KdV hierarchy equation of order k is simply the relation that the two multiplication operators for the functions λ R (p) and θ (k) R (p) commute as actions on a Hardy space H 2 R associated to the Abelian cover R; we define this space below.
We require such notions as inner and outer functions, functions of bounded characteristic (of class N ) and of its Smirnov subclass (or Nevanlinna class N + ) on the Riemann surfaces. These objects are well known in the theory of Hardy spaces in the disc D or half-plane C + [9], particularly the class N + is defined in Ch. II, Sect. 5. We say that the function is outer (inner) on S + or R if its lift to the universal cover C + is outer (inner). We say that F (p), p ∈ R, is of bounded characteristic if it can be represented as a ratio two bounded analytic functions. It is of Smirnov class if in addition the denominator is an outer function.
The integral density of states is a fundamental measure in the spectral theory of ergodic operators. In our case it coincides with dΘ restricted on E = ∂S + . Above we paid tribute to its importance in our definition of H 2 R as a subspace of L 2 -space with respect to dθ R on the boundary ∂R.
The action of the group Γ/Γ on ∂R is dissipative. This means that there exists a fundamental measurable set E ⊂ ∂R such that γE ∩ E = ∅ for γ = 1 Γ/Γ , and, for an arbitrary L 1 -function F (p), Note that essentially E = ∂S + . We define the Hardy spaces H 2 (α) of character automorphic functions (with respect to the action of the group Γ/Γ on R) in the following way.
The space H 2 (α) is a small modification of the spaces H 2 S + (α) ∼ = H 2 Γ (α) defined above -for a precise relationship between these spaces, see Lemma 3.3. The advantage of considering these spaces in this new way is that we can collect all character automorphic Hardy spaces H 2 (α) within H 2 R in a sense we now describe. Consider the collection of functions f (p, α) such that f (p, α) ∈ H 2 (α) for a.e. α ∈ Γ * with respect to the Haar measure dα on Γ * and We denote this space by H = Γ * H 2 (α)dα. The proof of the main theorem concerns two natural Fourier transforms which unify the perspectives of the global functional model on H 2 R and the individual functional models on the character automorphic Hardy spaces H 2 (α). The first such transform maps from where e R (p, α) is our special function (or generalized eigenfunction), i.e., the reproducing kernel related component of the Baker-Akhiezer function For the precise statement of the main theorem -including an explicit formula for e R (p, α) -we need some additional notation and definitions. We wish to distinguish the relationship between the functional model for Jacobi operators with that developed in the main theorem below. In fact, the Fourier transforms above can be considered as a continuous version of the discrete Fourier transform from [30], in which the Green function has to be substituted by the Martin function. Moreover, in this paper we get it as a limit case, see subsection 3.1. We will also substitute the condition (PW) related to the Green function (see (1.8)) by a similar condition related to the Martin function: The condition (PW M ) is equivalent to the entropy condition (1.3) and implies the (PW) condition (1.8); see (4.18) and Corollary 4.14. We will see in subsection 4.3 that it plays an important role in the asymptotics of our special functions e R (p, α).
The functions e R (p, α) are defined by means of canonical products, see (1.21). First, we define the set of divisors with the identification (a j , −1) = (a j , +1) and (b j , −1) = (b j , +1), endowed with the product topology of circles.
Recall that the Blaschke factor in C + is of the form The function Φ(λ(z), λ 0 ) is represented by the Blaschke product along the orbit {γ(z 0 )}, λ(z 0 ) = λ 0 , i.e., Convergence of Blaschke products in (1.21) for all D ∈ D(E) corresponds exactly to the Widom condition (1.8), and convergence of the whole product is guaranteed additionally by the finite gap length condition (1.9) (for k = 1/2). Therefore e(λ, D) is defined as a multi-valued character-automorphic function in S + . Its character is denoted by α(D). This map from divisors D(E) to Γ * was introduced in [31] and called the generalized Abel map; we define it precisely below in Definition 2.5. It is always surjective. If the generalized Abel map is injective, one can define e R (p, α(D)) = e(λ R (p), D). This property of Widom domains is not yet completely understood. One of several equivalent conditions to the DCT property holding is: An equivalent property is given in Remark 1.16 and it is described as the Direct Cauchy Theorem (DCT), which holds in the domain, in subsection 2.1. There is a nice sufficient condition guaranteeing the DCT property holds; specifically, if E is homogeneous, then S + is of Widom type and (DCT) holds. Recall that E is homogenous (in the sense of Carleson) if there exists κ > 0 such that |E ∩ (ξ − δ, ξ + δ)| ≥ κδ, ∀ξ ∈ E and ∀δ > 0.
An example of Widom domain S + such that (DCT) holds but the boundary is not homogeneous is given in [35].
Under our assumptions, we can relate our special functions e R (p, α) to the reproducing kernels of H 2 (α). Define √ λ in the domain C \ R + and extend it as characterautomorphic function on S + . We denote by µ R (p) the corresponding function on R with corresponding character by j ∈ Γ * , Since µ R (p) 2 = λ R (p), j is an order 2 element of the group Γ * , i.e., 2j = 0 Γ * . With this notation, we have the following important identities: Theorem 1.11. Assume that S + is of Widom type with DCT and (1.9) holds for k = 1/2. Then the reproducing kernel of the space H 2 (α) is of the form Respectively, the reproducing kernel K(p, p 0 ) of the space H 2 R has the forms (1.26) Remark 1.12. As a consequence of (1.23) and (1.24), e R (p, α) could instead be defined via the reproducing kernels in the following way While definitions (1.27) and (1.28) demonstrate the relationship between the special function e R and the reproducing kernels k α , we prefer our definition (1.22) because it demonstrates immediately the importance of the DCT condition and is constructive in nature.

The KdV hierarchy via the functional model
The description of the reproducing kernels of H 2 (α) from Theorem 1.11 yields, in fact, the representation for the Weyl-Titchmarsh m-function: Corollary 1.13. In the setting of Theorem 1.11, together with the asymptotics defines a single-valued function m α + (λ) in S + with positive imaginary part in C + . This function possesses the reflectionless property (1.39). Moreover, if the k-th gap length condition (1.9) holds, then we may define the system of functions {χ n (α)} 2k n=0 by the following asymptotic expansion: (1.30) In particular, χ 0 (α) = im α + (0).
Following Dubrovin et al. [6], we use the system of functions {χ n (α)} 2k n=0 to generate the k-th element of the KdV hierarchy KdV k (α). Specifically, we show that the generalized eigenfunction e R (p, α) itself satisfies the following differential relation We compute their coefficients by asymptotics (1.30) and consequently obtain Moreover, the partial derivatives in the η-direction of the coefficients exist; specifically, Then in the Fourier transforms (1.17) and (1.18) the multiplication operators by λ R and θ (k) R are of the form If the (k + 1)-th gap length condition (1.9) holds, the commutant relation between these operators corresponds to the Lax-pair representation for the k-th KdV hierarchy equation Respectively, the time evolution is given by α defined in the main theorem are indeed the Lax pair operators corresponding to the KdV hierarchy; in fact, this is relatively straightforward to recover from the definition of the coefficients χ n (cf. Dubrovin et al, [5,Section 30.2]). In the general setting, for a sufficiently smooth potential V , one can define coefficients χ n (V ) by formally expanding the m-function of L V . These coefficients are polynomial in V and its derivatives by the Ricatti equation. Then if one defines In our case, we have from the expansion (1.30) that P For example, one can verify directly from our results that An equivalent statement: there exists a function h of bounded characteristic in C + such that In this case g(z) = h(z), z ∈ C + . Using this notion we can say that e R (p, α) possesses a pseudocontinuation in the sense that e R (p, α), p ∈ ∂R + , can be extended in S + as a function of bounded characteristic. We can write this extension explicitly. We introduce the Widom function, which is the Blaschke product and denote its character by α W . In this case for almost all p ∈ ∂R w.r.t. dθ R . The last relation is easy to explain in the following way: W R (p)e R (p, α), p ∈ ∂R, has a form of the canonical product with a certain D * ∈ D(E). Therefore this is e R (p, α(D * )). It remains to note that by the definition the character of this function is The relation (1.38) is closely related with a description of the orthogonal complement of the Hardy spaces. Let us define In this case for an arbitrary Widom surface

and only if DCT holds.
A notion of pseudocontinuation is very closely related with the notion of the reflectionless property in the theory of ergodic operators. The role of this property in the spectral theory was completely understood in [26], see also [25]. Equation (1.38) implies that m α . This is exactly means that the Nevanlinna class functions m α ± (λ) possess reflectionless property on E, see (2.5).
With this discussion in hand, Theorem 1.1 follows almost immediately from our main theorem: Proof of Theorem 1.1. If V is ergodic such that L V has absolutely continuous spectrum E, then L V is reflectionless on E by Kotani theory. Since E is homogeneous, S + is a regular domain of Widom type with DCT; consequently, V = V α for some α ∈ Γ * . Since E satisfies the (k + 1)-th moment condition, V (·, t k ) := V α+η (k) t k satisfies (1.1), (1.2) and is almost-periodic in x and t k .

Preliminaries and elements of spectral theory
Let V : R → R be a continuous real-valued function which is bounded from below. Consider the associated one-dimensional Schrödinger operator and resolvent (or Green's) function By u 1,2 (x, x 0 , λ) we denote the unique fundamental system satisfying subject to the boundary conditions at The condition that V is bounded from below implies that L V is in the limit point case at +∞ and −∞, and hence we may define the Weyl-Titchmarsh functions m ± (λ; x 0 ) uniquely by the condition The diagonal of the resolvent function is given in terms of m ± by By translation, we may assume without loss of generality that inf σ( The functions R and m ± have asymptotic expansions of the form as λ → −∞.

Some inverse spectral theory
We recall from the introduction our preliminary assumptions on E: namely, E ⊂ R + is a closed set of positive Lebesgue measure of the form , and (DCT). We now provide an alternative characterization of the DCT property.
Recall that a meromorphic function f in C + is said to be of bounded characteristic if it can be represented as the ratio of two bounded analytic functions, f = f 1 /f 2 . The function is of Smirnov class if in addition f 2 is an outer function, cf. e.g. [9]. We say that a function F on S + belongs to the Smirnov class Here the contour integral is shorthand for integrating over both the "top" and "bottom" Later, we will add to these conditions (k-GLC) for higher k, but for now these conditions will suffice. Under these assumptions, a certain class of Schrödinger operator is particularly amenable to inverse scattering techniques: For a given set E, we define the set of potentials Note that (2.1) and (2.5) imply Re R(ξ + i0) := Re R(ξ + i0, 0, 0) = 0 for a.e. ξ ∈ E. (2.6) Since R(λ) is real and monotonic in the gaps (a j , b j ), for each j ∈ N there exists a unique "Dirichlet eigenvalue" λ j ∈ [a j , b j ] such that Therefore the resolvent function can be represented by Moreover, by the finite length gap condition and the normalization (2.2) we can compute C and obtain the product formula . (2.8) In particular, (2.2) and (2.7) yield the trace formula We now state a lemma which is fundamental in what follows; specifically, under the conditions above the infemum of the spectrum is a regular point for the Weyl m-functions: and m ± be the corresponding Weyl-Titchmarsh functions. Then the following limits exist and are finite: By the product representation (2.8), the function w(λ) = − 1 λR(λ) has the interlacing property and is of Nevanlinna class. Hence, the Nevanlinna measure dσ corresponding to the w has a mass point at the origin. But, since R is reflectionless (see (2.6)), On the other hand, if (DCT) holds, the singular part of dσ can not be supported on E [25, Theorem 1]. Consequently, it must be the case that lim λ→0 R(λ) = ∞. Since m + and m − are Nevanlinna class functions with Nevanlinna measures supported on R + , they are increasing functions on R − . Relation (2.1) then concludes the proof.
The following theorem was shown by Sodin and Yuditskii [29]: A similar result was obtained by Sodin and Yuditskii for Jacobi matrices [30] by associating to each reflectionless Jacobi matrix J a Hardy space of character automorphic functions. This description was carried over to continuous Schrödinger operators by Damanik and Yuditskii [3]. We will present and use their result in the following in a way that is convenient for our purpose.
First, we can explicitly describe the homeomorphism V(E) → D(E) mentioned above: , and denote by R(λ) and m + the corresponding resolvent and Weyl-Titchmarsh functions, respectively. Then there exists a divisor D ∈ D(E) such that Proof. By the reflectionless property, the absolutely continuous part of the measure corresponding to −R(λ) −1 should be equally divided between m + and m − . The λ j 's are given by (2.8). Set Again by the reflectionless condition, we encounter as in (2.7) that there exist λ . (2.12) Therefore, provided λ j ∈ (a j , b j ), it is not possible that m − and m + have a pole at λ j simultaneously. The j are chosen in order to add or cancel the pole at λ j . By the previous lemma it is possible to use 0 as a normalization point, which concludes the proof of (2.9) and (2.10). Finally, (2.11) follows by (2.2) and (2.3).
Both of these spaces are, in turn, homeomorphic to the Fuchsian dual Γ * by way of the generalized Abel map, which we now describe.
We defined the functions Φ(λ, λ 0 ) in (1.4). This function is related to the potential theoretic Green's function of S + by (1.4). Let E k = [b k , ∞)∩E and let γ k ∈ Γ correspond to the contour starting at −1 and containing the set E k . Moreover, let ω(λ, I) denote the harmonic measure on S + of the set I evaluated at the point λ. By (1.4) we see that the character ν λ 0 of Φ(λ, λ 0 ) can be given by means of the harmonic measure, i.e., The following generalized Abel map was introduced in [31].
This map is a homeomorphism [29].
It will be convenient to define the Abel map as being shifted by a fixed character corresponding to the divisor D c = {(c j , −1)} j≥1 . That is, we set We can thus put in correspondence potentials V ∈ V(E) and characters α ∈ Γ * ; we will later describe explicitly such a map (and various other important spectral quantities). M (λ) can also be represented in terms of a conformal mapping. Fix h j , η j > 0 with η j < η j+1 . To this data we associate the comb

Generalized eigenfunctions and a Wronskian identity
(2.13) By the Riemann mapping theorem, there exists a conformal map Θ : C + → Π + , which is unique under the normalization that Θ(0) = 0 and Θ(λ) = √ λ+o(1/ √ λ) as λ → −∞. Noting that Θ can be continuously extended to the boundary, we set E = Θ −1 (R + ). Moreover, Θ can be extended to S + as an additive automorphic function Θ(γ j (λ)) = Θ(λ) + 2πη j . (2.14) With this definition, one has Im Θ(λ) = M (λ). The tops of the needles η j + ih j correspond to the critical values M (c j ). Conversely, for every given set E there exists a corresponding conformal map Θ with the above properties for some comb Π + , cf. e.g. [35]. In fact, for E as in (2.4), Θ (λ) can be given explicitly. Specifically, We likewise define the Widom function such that W R (p) = W(λ R (p)); it is the inner part of Θ . With this notation, we can now define our generalized functions e α on S + and examine some of their properties: Theorem 2.6. Let j be the character generated by √ λ in S + and α = α(D). We define the character automorphic functions Moreover, for every D there exists α ∈ Γ * such that we have 18) and the following Wronskian identity holds for all α ∈ Γ * : Proof. Note that by the definition of the Abel map, e α indeed has character α; compare, e.g., [30,Section 10.2]. Let m ± (λ) = m ± (λ, D) and α(γ) = A(D, γ) − A(D c , γ). By (2.12), for ξ ∈ E we have By [32, Theorem 4.1] m + (λ(z)) is a function of bounded characteristic such that its inner part represents a ratio of Blaschke products. Therefore and (2.18) is proved. It remains to prove (2.19). We have Using (2.20) we obtain that which concludes the proof.
Consider the Riemann surface R = C + /Γ . R is called the (universal) Abelian covering for the Riemann surface C + /Γ. Points on R are denoted by p and the projection from R onto C + /Γ is denoted by π. For z ∈ C + we associate the point p = p(z) corresponding to the orbit { • γ(z)}• γ∈Γ . The group Γ/Γ acts on R in the natural way: Remark 3.1. We can describe R in terms of the covering maps λ R and θ R . First we fix a fundamental domain F for the covering λ. For a given set E there exists a system of non-intersecting half discs D + j = {z ∈ C + : |z − ζ j | < r j }, ζ j ∈ R + , r j > 0 such that C + can be mapped conformally onto F + = {z : Re z > 0, Im z > 0} \ ∪ j≥1 D + j , with the following properties: By the symmetry principle we extend φ as a conformal mapping from to C \ R + . By extending φ with respect to the gaps (a j , b j ) we obtain λ, respectively we can describe the action of Γ. We fix F as a fundamental domain of λ. LetΓ denote a system of representatives of Γ/Γ . Then is a fundamental domain for Γ . To describe R by means of the function λ R we take (Z ∞ ) 0 copies S γ of C \ R + cut along the gaps (a j , b j ). We fix the zero sheet S ι corresponding to F, where λ R is one-to-one. Let z ∈ F and γ j ∈ Γ corresponding to the closed loopγ j that passes through (a j , b j ). Passing from p(z) to p(γ j (z)) means that we pass from ι to γ j . Generally, two sheets corresponding to γ and γγ ±1 j are glued together at the gap (a j , b j ).
Similarly, we can describe R by means of θ R . The sheet ι is given by see (2.13). Generally the γ-sheet represents the domain Π ι shifted by 2πη(γ). Two sheets enumerated corresponding to γ and γγ j are glued along the cuts with the common basis 2πη(γ) + πη j = 2πη(γγ j ) − πη j .
Note that all character automorphic functions, f , by lift to single-valued functions on R. For typographical simplicity both functions will henceforth be denoted by f , but we will keep using this notation for the special functions e R (p, α), λ R (p), µ R (p) and θ R (p), the functions on R corresponding to e α (λ), λ, √ λ and Θ(λ), respectively.
Recall the character automorphic Hardy space H 2 (α) introduced in Definition 1.8. On this space, the linear functional of point evaluation in C + is continuous. Thus, by the Riesz representation theorem, there exist reproducing kernels k α (p, Proposition 3.2. The reproducing kernels k α (p, p 0 ) can be given by Proof. We note that the given vector belongs to L 2 with respect to the measure dθ R and represents a function of Smirnov class. Therefore it belongs to H 2 (α). Using that µ R is real on ∂R and (2.17) we see that, for f α ∈ H 2 (α), one has By DCT we obtain

Thus by, (2.19) we have
We now clarify interrelations between the spaces H 2 (α) and H 2 Γ (α) H 2 S + (α). The scalar product in H 2 S + (α) was defined by (1.6) and also by Definition 1.3 is clearly related to the harmonic measure on E = ∂S + w.r.t. the point −1 ∈ S + . In other words this is a subspace of L 2 w.r.t. the measure d log Φ(λ, −1).
Recall that the Martin function M (λ) is actually defined up to a positive multiplier, its critical points were denoted by c j ∈ (a j , b j ). The critical points of the Green function G(λ, −1) were denoted by ξ j (−1) ∈ (a j , b j ). Lemma 3.3. The following ratio of two Abelian differentials is of the form It is a function of bounded characteristic in R, moreover its inner part φ in R (p) is represented by the ratio of the Blaschke products, see (1.20), Let ψ R (p) = φ R (p)/φ in R (p) and α ψ is the character generated by this function. Then if and only if g ∈ H 2 S + (α) and f H 2 (α+α ψ ) = g H 2 S + (α) . Respectively the reproducing kernels of the spaces are related by Proof. By the definition f (p) belongs to the Smirnov class, and its norm is finite and coincides with the norm of g, just because both measures are mutually absolutely continuous. Conversely, we considerg(z) as the lift of the ratio f (p)/ψ(p) on the universal covering C + . Theng is of Smirnov class in the upper half plane and it is square integrable w.r.t. to the harmonic measure on R. By the Smirnov maximum principleg(z) belongs to H 2 and therefore possesses a harmonic majorant. Sinceg(γ(z)) = e 2πiα(γ)g (z) we can interpretg(z) = g(λ(z)), where g ∈ H 2 S + (α).

From Fourier series to Fourier integral
Let L 2 +,x = {f ∈ L 2 (R) : suppf ⊂ (x, ∞)}. In particular L 2 + = L 2 +,0 . The goal of this subsection is to prove the following theorem: Forf ∈ L 2 +,x it is given explicitly by The corresponding Weyl-solution is of the form To the standard spectral theory for 1-D Schrödinger operators we have to add the following Theorem 3.5. See also [3], where a relationship between spectral theorems for Jacobi matrices and 1-D Schrödinger operators was shown in the given context. In fact, the resolvent (L V + 1) −1 becomes a Jacobi matrix with respect to the corresponding Fourier basis in the model space. We get (3.6) as a limit of the discrete Fourier representation found in [31], where B λ * = Φ(·, λ * ) is the complex Green function w.r.t. λ * ∈ S + and ν * is the character generated by this function.
Theorem 3.5. The reproducing kernels k α (p, p 0 ) of the Hardy spaces H 2 (α) are given by (3.8) Lemma 3.6. Fix p 0 ∈ R. The function is Lipschitz continuous. In particular the measure is absolutely continuous with respect to the Lebesgue measure.
Proof. Fix α ∈ Γ * . Note that for a character automorphic inner function w with character β we have, Taking w(p) = e iθ R (p)x and noting that k α (p 0 , p 0 ) is real-valued, we have Taking the log we see that Our goal is to construct an inner function e iΘ(λ) with inverse character −η. By the same trick we will then obtain Since we have proved this for arbitrary α ∈ Γ * and this concludes the proof. Note that the function has positive imaginary part in S + . Due to the behaviour at infinity in the given domain, if we lift it to the universal cover, the corresponding Nevanlinna measure has a mass point at infinity on the universal covering. We note also that in our case the measure corresponding to the lifting of the Martin function M (λ) = Im Θ(λ) is pure point, see e.g. [33]. Thus, there is a constant κ > 0 such that is represented by a positive measure, where θ = Θ • λ. Hence, we can setΘ := κH − Θ, and, in fact, κ = 1. It has positive imaginary part, and, since H is single valued in S + , Θ is additive character automorphic with the character −η.
Note that since our domain is Dirichlet regular and M (−1) > 0 we can choose a sequence {λ N } ⊂ R − such that λ N → −∞ and Let p * ∈ R be such that λ R (p * ) = −1, and similarly let p N ∈ R be such that λ R (p N ) = λ N defined above. For notational brevity, we henceforth denote λ = λ R (p) (λ 0 = λ R (p 0 )) unless otherwise noted. Since, for fixed N , form an orthonormal basis of H 2 (α), we obtain that We will show that the Fourier series (3.9) converges to the Fourier integral (3.8) as N → ∞.
Lemma 3.7. Suppose n N is such that n N /N → x as N → ∞ and λ N as above. Then Proof. This follows from the fact that and that M (λ) = Im (Θ(λ)).

Lemma 3.8. Let
and let n N be such that n N /N converges as N → ∞. Then is absolutely continuous.
Proof. Evaluating (3.9) at p = p 0 = p * , we have Let x > 0 be such that n N /N → x as N → ∞. Then we obtain for the right hand-side that On B([0, x]) we define the compact family of measures By (3.11) we obtain that all subsequences converge to the same limit and hence Finally, υ is absolutely continuous by Lemma 3.6.
Proof of Theorem 3.5. In general we write That is, Let us now again take a sequence n N /N increasing to x. We consider a function g N with and linear in between. This family is equicontinuous, uniformly bounded, and converges pointwise for k = n N to the continuous function Hence, by Arzela-Ascoli it converges uniformly on [0, x]. Thus, the expression in brackets is less than for N sufficiently large and By Lemma 3.8 the second term converges to Thus, to conclude, we have shown that (3.12) with some integrable functions f α . It remains to show that f α = 1 a.e. Recall that That is, Hence, Therefore, by using the expansion of Θ(λ) and m β + (λ) as λ → −∞, we obtain that In [33] it is shown that if B is a Blaschke product, whose zeros are in R + \ E. We have, Hence we conclude that for all x > 0, The theorem is proved.
Recall the space H 2 R defined in Definition 1.7. Since S + is of Widom type there exists a measurable fundamental set E 0 ⊂ R for Γ . The set E 0 for the action of Γ is of course related to the fundamental set E for the action of Γ by Viewing R as the quotient C + /Γ , we can equivalently define H 2 R as follows: Definition 3.9. The space H 2 R is formed of those analytic functions F in C + such that: (i) F is of Smirnov class, Remark 3.10. Condition (ii) means that we consider in fact single-valued functions on the Riemann surface R. For this reason we may also write F (p) and We denote the reproducing kernels of H 2 R by K(p, p 0 ) = K p 0 (p). We have the following fundamental relationship between the space H 2 R and the character automorphic Hardy spaces H 2 (α) discussed above: belongs to L 2 dα as a function on Γ * and f (p, α) ∈ H 2 (α) as a function of p for a.e. α. Vice versa, if f (p, α) is a function with these properties then (3.14) belongs to H 2 R . Moreover, The reproducing kernels are related by and To a function F ∈ H 2 R we associate the vector function By L 2 ( 2 (Γ/Γ )) we denote the space of 2 (Γ/Γ ) -valued functions f (ϑ) ∈ 2 Γ/Γ , with the norm Proof. Since dθ R is invariant w.r.t. the action of the group Γ, we obtain Thus Lax-Halmos Theorem, see e.g. [22, p. 17] suggests an existence of the following representation H 2 R = ΘH 2 (E), where E is a subspace of 2 Γ/Γ and Θ a measurable operator valued function R whose values Θ(ξ) are isometric operators on E. Below we present an explicit form of such a representation (iii). But, before to proceed we note that 2 Γ/Γ and L 2 dα are unitarily equivalent, see (3.3). We will show that one of them can be chosen as the scale space E. For definitiveness, we denote E = L 2 dα . Lemma 3.13. Let V : H 2 R → H 2 (E) be defined by where θ R (p 0 ) = ϑ 0 . Then V defines an isometry on H 2 R .
Proof. Due to (3.16), (3.8), Fubini's theorem and the shift-invariance of dα we see that (i) Recall that is a unitary operator.

Generalized Abelian integrals and the KdV hierarchy
It is well known that in the finite gap setting the direction of the time shift is generated by Abelian integrals of the second kind, see e.g. [11]. The following functions will serve as generalized Abelian integrals on R; cf. [33,Theorem 5] for the case M = M 0 .
Proof. Clearly M k (λ) is harmonic in C \ R + . On the gaps (a j , b j ) we have that v k (λ + i0) = v k (λ − i0) but the derivative ∂v k ∂y has a jump. Due to the Cauchy Riemann equation we find that the generalized Laplacian of v k is given by Hence, M k is harmonic in S + .
Remark 4.2. Note that the condition (4.1) follows immediately from the assumptions (PW) and (k-GLC).
Note that in particular M 0 (λ) is a positive harmonic function which, since we assumed that E is Dirichlet regular, vanishes on the boundary. Hence, M 0 = M defines the Martin function of S + . Let Θ k define the analytic function in the domain such that Im Θ k = M k . Definition 4.3. Let Θ k be defined as above. We define the generalized normalized Abelian integral of order k as the function on R of the form By η (k) we denote the additive character generated by this function, i.e., R (p) can be treated as a function on this surface with the group action (4.2). Moreover, the two dimensional flow in this case, is ergodic with respect to dα.

Relation to the finite-gap case
Proof. k α S + (λ, λ 0 ) has a harmonic majorant in S +,N and therefore belongs to H 2 S +,N (α N ), moreover k α We have On the other hand the family {k α N S +,N (λ N (z), λ 0 )} N is compact in the standard H 2 (w.r.t. the harmonic measure). We choose a subsequence N j so that We note that this function can be understood as an element of H 2 S + (α), i.e., f (z) = g(λ(z)), where g ∈ H 2 S + (α) and moreover As a combination of (4.3) and (4.4) we have That is, in fact, the sequence Corollary 4.6. Uniformly on compact subsets in S + e α N (λ) → e α (λ) We wish to explain further the relationship between the generalized Abelian integrals θ We fix a basis {A j , B j } N j=1 for the homology of S N : let A j denote the equivalence class of loops forming a clockwise circle around [a j , b j ] on the upper sheet S +,N , and let B j denote the equivalence class of loops beginning at −1, passing through the gap (a j , b j ) from the upper sheet to the lower sheet, and then returning to −1.
Consider now the usual Abelian integrals of the second kind ω k,N 2 with pole at ∞ of order k in N , written in local coordinates near P = ∞ as where f k,N is holomorphic and k ≥ 2. This form is unique up to the addition of Abelian differentials of the first kind; we normalize by assuming f k,N ( )d has vanishing A-periods. We denote by Θ (k) N the generalized Abelian integrals corresponding to Ω N ; that is, Im (Θ which is given in local coordinates at ∞ by (2k + 1)f 2k+2,N ( )d . This is an Abelian differential of the first kind, and is thus determined by its period class; since f 2k+2,N ( )d has vanishing A-periods, we conclude that the B-periods of dΘ We can represent S N as a regular 4N -gon R by cutting along representative loops (A j , B j ) and identifying the corresponding sides. Using the residue theorem and the normalizations of dω l,N 1 and dω 2k+2,N 2 , we have Because the B-periods of dΘ where we take the limit for N ≥ N 0 such that λ ∈ S +,N 0 .
Proof. Let N 0 be such that λ ∈ S +,N 0 . By monotonicity of the domains, we have that for N ≥ N 0 , N , and let η k,N ∈ Γ * be the character formed by including η k,N into the larger group Γ * by Corollary 4.9. Suppose (4.1) holds. Then the characters η k,N converge to η (k) .
With these facts in hand, we now have the tools to prove the following Theorem 4.10. There exist polynomials such that for the generalized Abelian integral θ (k) Proof. For a fixed N and t k > 0, we write the corresponding relation to (4.5) in the integral form: This representation is classical in the finite gap case (cf. e.g. [11]). Then we pass to the limit as N → ∞, using the discussion of convergence above. A compactness argument shows that the representation (4.5) exists.
Remark 4.11. Let ε(λ, α) ∈ H ∞ S + (α) be the extremal function in the following sense By the DCT ε(λ, α) → 1 as α → 0 Γ * [12, (21) p. 205] uniformly on compact subsets in S + . Despite the similar notation, we emphasize that it is not necessarily the case that η k,N agree with the restriction η (k) N ). For this reason in the approximation procedure above one has to consider the correction functions ε(λ, (η k,N − η

An explicit map Γ * → V(E)
We wish to recover the potential function V and its derivatives not via the traditional trace formulas, but rather by way of asymptotic expansion of the m-function. Recall that (DCT) guarantees that m α + (λ 0 ) is continuous on Γ * for internal points λ 0 ∈ S + . We will show that this also holds for λ 0 = 0. As an immediate consequence, we will obtain an explicit expression for V (x) in terms of our special functions on Γ * . Theorem 4.12. Let η ∈ Γ * be the direction corresponding to the function Θ; cf. (2.14). Define Then χ 0 is continuous on Γ * . Moreover, χ 0 is differentiable in direction η. Defining the potential V corresponding to m α + is given by Proof. First we prove continuity of χ 0 . Since for fixed α, m α ± (0) exists and m α ± is increasing on R − , for fixed > 0 there exist ξ < 0 such that Let {β j } such that lim β j = α and {k j } a subsequence such that By continuity of m α ± (ξ), there exists j 0 such that for all j ≥ j 0 we have That is, Using that m α which proves continuity of χ(α). Let ξ < 0. Due to Theorem 3.4 Recall the Riccati equation for m α + , That is, Since m α + (λ) is continuous on Γ * and m α + (λ) has no pole on R − we can apply the dominated convergence theorem and pass to the limit to obtain which proves differentiability of χ 0 and (4.8).

(PW
where λ belongs to an arbitrary gap (a j , b j ), j ≥ 1.
As a consequence, we have the following corollary: Corollary 4.14. The condition (4.9) implies the Widom condition (1.11).
Moreover, we can explicitly compute the second term in asymptotics.
Theorem 4.15. Assume that (4.9) holds. Let D = {(λ j , j )} ∈ D(E), to which we associate the Blaschke product assuming the normalizations Φ λ j (λ) ≥ 0 in R − . Then the following limit exists Proof. For a fixed λ < −1, we have Note that M (c j ) ≥ M (λ j ) for an arbitrary λ j ∈ (a j , b j ). By (4.10) and (4.9) the form an integrable majorant . Therefore we can pass to the limit in (4.12) and we obtain (4.11).
Corollary 4.16. Assume that (4.9) holds and Then lim y→∞ y log e R (p(iy), α(D)) = 1 2 Proof. By (4.11) we have that an arbitrary Blaschke product in the product representation (1.20) has this property.
Recall that the function R(λ), see (2.7), was defined as This decomposition is valid for an arbitrary set of gaps and positions of λ j ∈ [a j , b j ]. We wish to consider specifically the exponential part of R(λ). Under the assumption (4.13) we can specify C by the normalization condition where µ goes to infinity along imaginary axis. Moreover, in this case for the integral due to the fact that 2τ Im µ τ 2 +Im µ 2 ≤ 1. As a combination of (4.11) and (4.15) we get (4.14). Moreover the corresponding leading term depends only on the Blaschke factor component Φ D (λ). Proof. For λ < −1 we have

Remark 4.18.
Recall that the complex Martin function Θ(λ) is closely related to the integral density of states ρ(ξ), which can be defined by That is, ρ(ξ) for almost every ξ ∈ E can be defined via the boundary values of Θ(λ), λ = ξ + i and In this term for w M we get
We will use the even-odd decomposition of this space C 2k+2 = C k+1 ⊕ C k+1 . In particular, with respect to this decomposition where s is the standard shift in C k+1 and Lemma 4.19. Assume that for the generalized Abelian integral θ Proof. Let α(ξ) = α − η (k) ξ. We claim that for an arbitrary t k the following integral tends to zero where µ goes to infinity along the imaginary axis. Indeed, by (4.5) this integral is equal to By Corollaries 4.16 and 4.17 we have (4.22). Thus, the polynomial part of the relation (4.5) is of the form Separating odd and even parts, we have which corresponds to (4.21).
The computations for the second term are based on Lemma 4.21, see (4.26). First, we note that On the other hand, for a functionĝ(x, α) in the domain, we have ∞ 0 λ R (p) n A n (α − ηx)e R (p, α − ηx)e iθ R (p)xĝ (x, α)dx Finally, we note that −i∂ x A α n (x) = i∂ η A n (α − η x ), and, basically, repeat the above computations Combining these three remarks, we obtain Remark 4.23. We note that P α k is self-adjoint in L 2 on the whole axis, and therefore can be rewritten into the form

The k-th gap length condition
Recall that the function m α + is given as, see ( where and in the asymptotic expansion we have χ 0 (α) = im α + (0) and m α + (µ 2 ) = iµ + i That is, for the even terms we get (4.28) Note, that we have shown continuity of χ 0 (α) in a quite fashionable way, see Section 4.2.
The situation with the odd terms is much simpler. They can be given in terms of the function Under k + 1 2 -GLC condition the right hand sides are continuous functions in D ∈ D(E). Therefore τ m (α), and respectively χ 2m+1 (α) are continuous in α ∈ Γ * for all m ≤ k.
In this section we prove that under the condition (k-GLC) the coefficient χ 2k (α) is well-defined as the corresponding term in asymptotics for m α + (λ), that is, (4.28) has sense.  That is, the moment χ 2k (α) is well defined as a bounded function on Γ * .
Proof. We define a Nevanlinna class function Z(µ) by its argument πκ(ξ) on the real axis, where To be more precise, with a suitable choice of the positive multiplier, we have An additive representation of this function is of the form Note that ρ + n ρ − n = (b n − λ n )(λ n − a n ) λ n (1 + b n /λ n )(1 + a n /λ n ) j =n λ n − b j λ n − λ j λ n − a j λ n − λ j We fix C such that b n − a n ≤ 1/2 for a n ≥ C. Then b n /λ n ≤ b n /λ n ≤ b n /a n ≤ 1/(2a n ) + 1 ≤ 1/(2C) + 1.