How Discrete Spectrum and Resonances Influence the Asymptotics of the Toda Shock Wave

We rigorously derive the long-time asymptotics of the Toda shock wave in a middle region where the solution is asymptotically finite gap. In particular, we describe the influence of the discrete spectrum in the spectral gap on the shift of the phase in the theta-function representation for this solution. We also study the effect of possible resonances at the endpoints of the gap on this phase. This paper is a continuation of research started in [arXiv:2001.05184].


Introduction
The Toda shock wave describes the motion of an infinite chain of particles with nonlinear nearest neighbor interactions when the chain is excited with shock type initial conditions. We are interested in the effect the eigenvalues in the spectral gap of the associated Lax operator have on the asymptotic behavior of the shock wave. The Toda shock wave is generated by the solution of the following initial value problem for the Toda lattice [23,24] d dtb (n, t) = 2 ã(n, t) 2 −ã(n − 1, t) 2 , d dtã (n, t) =ã(n, t) b (n + 1, t) −b(n, t) , with a steplike initial profile ã(n, 0),b(n, 0) such that a(n, 0) → a ± ,b(n, 0) → b ± , as n → ±∞, (1.2) where a ± > 0 and b ± ∈ R satisfy the condition This condition fixes the position of the background spectra relative to each other; their mutual location produces essentially different types of asymptotic solutions [20]. The notion of the Toda shock wave [4,5] was traditionally associated with symmetric initial datã a(n − 1, 0) =ã(−n, 0),b(n, 0) = −b(−n, 0), (1.4) and the background constants a − = a + = 1 2 , b + = −b − > 1. The asymptotic of the solution of (1.1) for the particular casẽ a(n, 0) = 1 2 ,b(n, 0) = b sgn n, n ∈ Z, where sgn 0 = 0, (1.5) was studied in the pioneering work [25] by Venakides, Deift, and Oba in 1991. By use of the Lax-Levermore approach they established that in a middle region of the half plane (n, t) ∈ Z × R + , the asymptotic of the shock wave (1.1), (1.5) is described by a 2-periodic solution of the Toda lattice. They also showed that the asymptotic undergoes a phase shift caused by the presence of a single eigenvalue at λ = 0. We refer to this middle region of periodic asymptotics as VDO region, 1 compare Figure 1.
In this paper, we offer a derivation and rigorous justification of the asymptotic for (1.1)-(1.3) in the VDO region using the vector Riemann-Hilbert problem (RHP) approach. We allow more general initial data (1.2) with arbitrary positive a ± and b ± satisfying (1.3). In particular, the novel features are: • an arbitrary discrete spectrum, • possible resonances at the edges of the continuous spectrum, • no symmetry assumption (1.4), • a partial revision of results in [25] including estimates on the error terms, • a finite gap (two band) asymptotic due to spectra of different length.
The vector RHP approach in the context of the Toda problem was proposed in [6] and further developed in [2,12,17,18,19]. We use standard conjugations/deformations such as the g-function technique [8] which proved its efficiency in steplike cases. A suitable g-function for the VDO region replaces the standard phase function and makes it possible to apply the lense mechanism. It also provides a characterization of the boundaries of the sectors (see Figure 2) where the asymptotics are given by a finite gap solution of (1.1) with unaltered phase. We describe the g-function for the VDO region as an Abel integral on the Riemann surface associated with the continuous two band spectrum of the underlying Jacobi operator of (1.1) in Section 3.
(1.8) Figure 1 demonstrates the behavior of the Toda shock wave corresponding to the initial data a(n, 0) = 1 2 , n ∈ Z; b(n, 0) = −4, n < 0; b(0, 0) = −1.7, b(n, 0) = 0, n > 0, at a large but fixed time t = 200. Such initial data have one eigenvalue in the gap and the background spectra are of equal length. Hence the asymptotic of the shock wave in the VDO region is periodic with period 2 and exhibits one phase shift. In the left and right modulation regions (MR) the asymptotic is a modulated single-phase quasi-periodic Toda solution as discussed in [11]. The initial data (1.8) can have a finite discrete spectrum. We enumerate the eigenvalues λ j in the gap (b+2a, −1) increasingly starting from the leftmost; let ℵ be the number of eigenvalues in the gap. Given an arbitrary 2 small ε > 0, the VDO region consists of ℵ + 1 disjoint regions (n, t) : as depicted in Figure 2, where ξ j are the points ξ at which the level line Re g(λ, ξ) = 0 of the g-function (cf. Section 3) crosses R at λ j . We denote by ξ 0 and ξ ℵ+1 the points where Re g(b + 2a, ξ 0 ) = 0 and Re g(−1, ξ ℵ+1 ) = 0, and the rays n t = ξ ℵ+1 and n t = ξ 0 determine n t n t ∈ I j ε n t ∈ I 1 the outer boundaries of the VDO region. With each interval I j ε we associate a shift phase ∆ j (not depending on ε) expressed in terms of the initial scattering data for the solution of (1.6) and (1.8) (see (4.16) below). For each ∆ j one finds via Jacobi's inversion problem (3.12) the initial Dirichlet divisor and the unique finite gap solution â(n, t, ∆ j ),b(n, t, ∆ j ) from the isospectral set associated with the two-band spectrum S : ), b(n, t)} be the solution of the initial value problem (1.6)-(1.7), (1.8) and let n → ∞, t → ∞ with n t ∈ I j ε , where ε > 0 is an arbitrary, sufficiently small number. Let â(n, t, ∆ j ),b(n, t, ∆ j ) be the finite gap solution associated with the spectrum S and with the phase ∆ j given by (3.16), (4.16), (4.6), (4.12), (4.11), (2.23). Then there exists C(ε) > 0 such that The shift of the phases at the point λ j = z j +z −1 j 2 ∈ (b+2a, −1) of the discrete spectrum is given by where q and q 1 are defined by (2.6).
Remark 1.2. (i) For initial data (1.8) the scattering data consist of the modulo of the right . As expected, we see that R(λ) and the discrete spectrum to the right of λ j do not influence the asymptotic in the sector n t ∈ I j ε . (ii) The error terms in (1.9) are of order O e −C(ε)t and thus significantly better than the estimate O t −1 one would expect by analogy with the error estimates in the modulation regions [11]. The error terms in (1.9) were obtained by a careful analysis of the relations between the analytic continuation of the scattering functions. These relations allowed us to prove that there are no parametrix points [7] in the RHP for the VDO region.
(iii) We use vector RHP statements instead of matrix statements (as do [1,10,14,15,22] in the case of the KdV equation with steplike initial data), because the matrix statements for the shock wave are ill-posed for certain arbitrary large values of n and t in the class of invertible matrices with L 2 -integrable singularities on the jump contour, for both the initial and model RHPs. This fact for Toda shock can be established similarly as for the KdV case [13]. One would have to admit then additional poles for solutions outside the discrete spectrum in the matrix statements. This makes proving uniqueness of the solutions far more difficult. The statements of the RHPs in vector form together with additional symmetries to be posed on contours, jump matrices and on the solutions itself imply uniqueness almost straightforward. However, for the final small-norm arguments we need to construct an invertible matrix model RHP solution. It has poles and it might not be unique, but the corresponding error vector function has no poles. Such a solution is given in Lemma 5.4. (iv) Unlike to KdV, for the Toda equation with non-overlapping background spectra, the statements of the RHPs associated with left and right initial data look identical. The proper choice of the initial statement for the RHP can essentially simplify the further analysis in a given region of space-time variables Z×R + (cf. [12]). For the VDO region both choices are appropriate.

Notations and statement of the initial holomorphic RHP
To maintain generality of the presentation while keeping notations short, we formulate all preliminary facts on the inverse scattering transform for the steplike initial profile in terms of the spectral variables z ± associated with the initial data (1.2), (1.3). First of all, let us list some well known properties of the scattering data for the steplike Jacobi operatorH(t) involved in the Lax representation d dtH (t) = H (t),Ã(t) for the initial value problem (1.1), (1.2). This problem has a unique solution (cf. [23]). Assume that the coefficients of the initial Jacobi operatorH(0) tend to the limiting (or background) constants a ± , b ± with a first summable moment of perturbation, that is, n(ã(n, 0) − a ± ) ∈ 1 (Z ± ) and n(b(n, 0) − b ± ) ∈ 1 (Z ± ). Then the unique solution ã(t),b(t) of (1.1) satisfies With this condition fulfilled, introduce some notations and notions.
• The Joukovski maps z ± = z ± (λ) of the spectral parameter λ are given by The map z + → z − is one-to-one between the domains D + and D − , where The functions z ±n ± n∈Z are called the free exponents. They solve the background spectral equations H ± y(n) = λy(n).
• The operatorH(t) has an absolutely continuous spectrum on the set σ + ∪ σ − and a finite discrete spectrum σ d , which we divide into three parts, The points z ± j = z ± (λ j ) ∈ D ± ∩ (−1, 1), λ j ∈ σ d , are also called points of the discrete spectrum.
We can consider them as functions of z ± in the closures of D ± . Their Wronskiañ is an important spectral characteristic of the steplike scattering problem. It can be treated as an analytic function of z ± in clos D ± . The points z ± j are its simple zeros, and ψ ± (λ j , n, t) are the (dependent) eigenfunctions.
• The normalising constants are introduced by • The scattering relations hold on the sets |z ± | = 1.
• The time evolution of the scattering data is given by The function |T ± (λ, 0)| 2 cannot be continued analytically outside the domain |z ∓ | = 1. However, if the initial data tend to the limiting constants exponentially fast with a rate ρ > 0 (cf. (1.8)), then the right hand side of the ±-scattering relation continues analytically in the domain 1 − ρ < |z ± | ≤ 1, and the respective equality (2.3) is preserved. In particular, the reflection coefficient R ± (λ, 0) continues in the domain 1−ρ < |z ± | ≤ 1, and the function χ(λ) defined in (2.23) below can be continued analytically in both domains. The vector RHP connected with the scattering problem forH(t) can be stated in two ways, based either on the right or left scattering data. The correct choice of the scattering data which significantly simplifies the further analysis depends on the region of the (n, t) half-plane for which the asymptotic of (1.1)-(1.3) should be derived. In our situation, the VDO region on the Z × R + half plane could be analysed via left or right RHP and both cases are equivalent in structure and complexity of steps. To state a proper vector RHP we proceed as follows.
Let M be the two-sheeted Riemann surface associated with the function with glued cuts along σ + and σ − . Denote a point on M by p = (λ, ±). On the upper sheet of M introduce two 1 × 2 vector-functions M ± (p, n, t) := M ± (p) (here variables n and t are treated as parameters) by The first component of each function is a meromorphic function of p on the upper sheet on M with simple poles at points of σ d and known residues. At infinity M ± (p) have finite values and the product of components is equal to 1 [12]. Let us extend each function M ± to the lower sheet by where σ 1 = ( 0 1 1 0 ) is the first Pauli matrix and p * = (λ, −) is the involution point for p = (λ, +). With this extension, both functions have jumps along the boundaries of the sheets on M, and these jumps can be easily evaluated. The jump problems together with normalisation conditions In this paper, we use the traditional RHP statement based on the right scattering data in terms of the variable z + . As discussed in the introduction, we restrict ourselves to the case 1] the continuous spectrum of the Jacobi operator involved in (1.6). To ease notations, we omit from here on the subscript "+" in the notations and set respectively. Let us enumerate the points z j starting from σ gap d , that is, All remaining points of the discrete spectrum will lie outside of σ gap d . Further notations are where The domain D is in one-to-one correspondence with the upper sheet of M (we treat sheets as open sets) with The domain corresponds to the lower sheet by Therefore, the meromorphic RHP for M + (p) on M can be reformulated as an equivalent meromorphic RHP for m(z) = M + (p(z)) on the z-plane, with jumps along the unit circle T = {z : |z| = 1} and intervals [q 1 , q] and q −1 , q −1 1 . In this paper, we propose a slightly different (holomorphic) statement of the initial RHP, which is equivalent to the RHP for M + (p) on M, and therefore has a unique solution (cf. [12]). This statement is specific for the domain VDO, where we derive the asymptotics, and allows us to skip several of the standard transformations, such as the reformulation of the meromorphic problem as a holomorphic problem and one of two steps corresponding to opening of lenses.
Let us choose a large natural number N 1 and set We can always assume that δ < ρ, where ρ is the decay rate from (1.8). Then the right Jost solutions ψ(z, n, t) and ψ(z −1 , n, t) are holomorphic functions in an δ-vicinity of the unit circle T, and the standard scattering relation is continued analytically in the open ring With our choice of δ, there are no points of the discrete spectrum inΩ δ , moreover, where we denoted In particular, the continuation of the initial reflection coefficient R(z) is an analytic function inΩ δ . Set with I := [q 1 , q], introduce the vector-function m(z) = (m 1 (z, n, t), m 2 (z, n, t)) by 3 (2.14) Here 12]). We have Formula (2.17) implies that the vector function (2.14), considered as a piecewise-analytic function in C, has jumps along the circle C δ , along the interval I and the small circles T δ,j , as well as along their images C * δ , I * and T * δ,j under the map z → z −1 . However, m(z) does not have a jump along the unit circle |z| = 1, i.e., it is holomorphic in the ring 1 − δ < |z| < (1 − δ) −1 . The fact that m(z) does not have any singularities at z j ∈ σ d is established in [6] and [19].
The symmetry condition (2.17) plays a crucial role in establishing uniqueness of the solution for RHPs, and we cannot violate it. For this reason, the initial RHP and all its further transformations (deformations and conjugations) should satisfy the following symmetry constraints (i) and (ii). Let Σ be the jump contour of a generic RHP.
(i) The jump contour Σ should be symmetric with respect to the map z → z −1 , i.e., with every point z it also contains z −1 .
(ii) Symmetric parts of Σ are oriented in such a way that the jump matrixṽ(z) of the problem m + (z) =m − (z)ṽ(z) and the solution itself satisfy the symmetries . On all such conjugations we pose the symmetry constraint (iii): (iii) The contour Σ of a non-analyticity for d(z) should be symmetric with respect to z → z −1 .
Moreover, the function d(z) should satisfy either the property or the property Recall that m(z) in (2.14) has bounded positive limits of both components at 0 and ∞, moreover, by Lemma 2.2, This is a normalization condition. The properties of the conjugation matrices [d(z)] −σ 3 listed above allow to preserve the normalisation condition for all transformations.
The phase function of our problem is given by Set ξ = n t and consider the cross points of the 0-level lines for the function Re Φ(z, ξ), that is, the lines described by for different values of ξ ∈ R. One of the level lines for all ξ ∈ R is evidently the unit circle |z| = 1.
To state the RHP for which m(z) in (2.14) is the unique solution, 4 we introduce orientations on the jump contour Σ δ ∪ Σ * δ according to the symmetry requirements above. The contour C δ is oriented counterclockwise, C * δ is oriented clockwise. On the two symmetric parts the orientation 5 is taken from right to left on I and from left to right on I * . Moreover, all T δ,j and T * δ,j are supposed to be oriented counterclockwise. Then m + (z) (resp. m − (z)) will denote the limit from the positive (resp. negative) side of the contour. We assume that these limits exist and m(z) extends to a continuous function on the sides of Σ δ ∪ Σ * δ except possibly at the end points of I and I * , where the square root (not L 2 -integrable!) singularities are admissible.
2)) is the Wronskian of the Jost solutions, z ∈ clos D. If the Wronskian vanishes atq, we callq a resonant point. The general situation is non-resonant, that is, W (q) = 0. Note that as a function of z, the Wronskian takes complex conjugated values on the sides of the contours (2.19). 11,12]). Let δ > 0 be given as in (2.10). For all (n, t) ∈ Z × R + and z ∈ C, the function (2.14), (2.17) is the unique solution of the following RH problem: to find a vector-

22)
where Φ(z) = Φ(z, n t ) is the phase function, matrices A j (z) are defined by (2.15) and R(z) is the holomorphic continuation of the initial right reflection coefficient (2.7). The function χ(z) is defined by

23)
with ζ(z), z ∈ D, connected with z by the Joukovski map λ = b + a(ζ + ζ −1 ). Here W (z) =W (λ(z)) is the Wronskian of the Jost solutions defined in (2.2), • in vicinities of the points in J , m(z) has the following behavior: Remark 2.4. According to (1.8), W (z) admits an analytic continuation in a small vicinity of the interval I (see [11, equation (2.11)]). Respectively, in this vicinity there exists an analytic continuation X(z) of χ(z) such that X ± (z) = ±i|χ(z)| for z ∈ I. The function X(z) does not have other jumps in this vicinity.
Note that in the VDO region, the off-diagonal matrix elements of the jump matrix v(z) grow exponentially with respect to t for z ∈ I∪I * . The same is true for v(z) on those contours T δ,j ∪T * δ,j which correspond to the origins 0 > z j > z 0 (ξ). The remaining parts of the jump matrices are asymptotically close to the identity matrix as t → ∞. In the next sections we perform a series of conjugation/deformation steps which transform the initial RHP of Theorem 2.3 to the equivalent problem with a jump matrix which is asymptotically close as t → ∞ to a piecewise constant matrix with respect to z. This limiting matrix also depends on ξ as a piecewise constant matrix, and the respective (so called model) RHP has a unique solution which can be found explicitly in terms of the Riemann theta-function. Let us emphasize that for the VDO region we propose transformations which lead to the absence of any additional parametrix problems. The first transformation of the initial problem solution is associated with the so called g-function method first introduced for the KdV equation in [8]. In our case, this g-function is a normalized Abel integral associated with the two-sheeted Riemann surface glued via the cuts along the continuous 1]. In fact, the g-function is a linear combination (dependent on ξ) of the normalized Abel differentials of the second and third type which are involved in the exponential part of the Baker-Akhiezer function corresponding to the finite gap solution of the Toda lattice associated with the two-band spectrum S. It is crucial for our endeavor to understand in detail the properties of the g-function as a function of the spectral parameter λ. The next section is devoted to this subject.

g-function as an Abel integral and its connection with the Baker-Akhiezer function
Let M be the two-sheeted Riemann surface associated with S, i.e., with the function Let Ω 0 be the Abel differential of the second kind on M with second order poles at ∞ + and ∞ − and let ω ∞ + ,∞ − be the Abel differential of the third kind with logarithmic poles at ∞ + and ∞ − , both normalized as As it is known, where ν i ∈ R for i = 1, 2, 3. Moreover, ν 3 ∈ (b + 2a, −1), and at least one of the points ν 1 or ν 2 also lies in the gap (b + 2a, −1). Consider the Abel integral given by where ξ ∈ R is a parameter. On Π U we denote it by g(λ, ξ), that is, then µ i (ξ) ∈ R, these points do not coincide, and at least one of them belongs to the gap. By definition of Ω 0 we have that is, which implies that is, µ i (ξ) are the zeros of the quadratic equation With the notations we infer Lemma 3.1. The functions µ i (ξ), i = 1, 2, are monotonically decreasing with respect to ξ ∈ R. For ξ ∈ (ξ ℵ+1 , ξ 0 ), 6 where Proof . Differentiating (3.6) with respect to ξ implies We observe that holds if Γ 2 2 < Γ 1 . The last one follows from the Cauchy inequality Thus, µ i (ξ) are monotonically decreasing with respect to ξ. Assume that µ 1 (ξ) < µ 2 (ξ). A trivial analysis shows that the value ξ 0 corresponds to the location µ 1 (ξ 0 ) = b + 2a, that is, This implies the second equation in (3.7). The location µ 2 (ξ ℵ+1 ) = −1 provides the first formula in (3.7). From (3.4) it follows that that is, Let ε > 0 be an arbitrary small number and let ξ 0 and ξ ℵ+1 be defined by (3.7) and (3.5). For any ξ ∈ [ξ ℵ+1 + ε, ξ 0 − ε], both points µ 1 (ξ) and µ 2 (ξ) are inner points of the gap. The set of level lines Re g = 0 consists of the two intervals [b − 2a, b + 2a] and [−1, 1] and an infinite contour which intersects the real axis at µ 0 (ξ) such that (3.8) Lemma 3.2. The real-valued function µ 0 (ξ) implicitly given by Re g(µ 0 (ξ), ξ) = 0 is monotonic with d dξ µ 0 (ξ) < 0 for ξ ∈ (ξ ℵ+1 , ξ 0 ). Moreover, Proof . By (3.6) we have µ 1 (ξ)µ 2 (ξ) = (b − ξ)Γ 2 − Γ 1 . This implies with (3.4) that µ 0 (ξ) is given implicitly by Differentiating with respect to ξ implies Since the second multiplier is negative, it is sufficient to prove that the integral is positive. But that is, we have to prove that . This inequality is true by the mean value theorem. Equalities (3.9) were proven in [12].
Let us recall the Baker-Akhiezer function for a finite gap solution â(n, t),b(n, t) of the Toda lattice equation associated with the spectrum S = [b − 2a, b + 2a] ∪ [−1, 1] and the initial Dirichlet divisor p 0 = (λ(0, 0), σ(0, 0)), σ(0, 0) ∈ {+, −}. In our case the divisor consists of one point on the Riemann surface M with projection on the closed gap of the spectrum and it will later depend on the slow variable ξ.
As is known, for each n and t fixed, the Baker-Akhiezer function is a meromorphic function of p on M with a simple pole at p 0 . Respectively, the vector function (Ψ(p * , n, t), Ψ(p, n, t)) does not have jumps on M, and the vector function m(p) := Ψ(p * , n, t) exp −tg p * , n t , Ψ(p, n, t) exp −tg p, n t , has an evident jump along a, Here we took into account that for p ∈ J U ∪ J L , Note that since g(p * , ξ) = −g(p, ξ), The function has a simple pole at the branch point b + 2a and a simple zero at p 0 . Moreover, Note that (3.11) can be rewritten as the Jacobi inversion problem and allows us to compute uniquely the divisor point p 0 for any given real valued ∆. Summing up the considerations above, we proved the following • the symmetry conditioñ m(p * ) =m(p)σ 1 for p ∈ M \ J U ∪ J L , (3.14) • the normalization conditionm 1 (∞ + )m 2 (∞ + ) = 1.
• Both components ofm(p) have simple poles at the branch point p = b + 2a and no other singularities.
Together with Theorem 9.48 of [23] it implies Corollary 3.4. For the vector functionm(p) =m(p, n, t) the following holds whereã = Cap S is the logarithmic capacity of the spectrum S.
Introduce the product and let h(λ) :=h(p) for p = (λ, +). The functionh(p) has a double pole on M at the branch point b + 2a, that is, h(λ) has a simple pole at b + 2a. Moreover, θ(Z(p, n, t)) has the only zero at p(n, t) = (λ(n, t), ±) ∈ J U ∪ J L , which is the unique solution of the Jacobi inversion problem p(n,t) θ(Z(p * , n, t)) has a simple zero at the involution point p * (n, t). Thus h(λ(n, t)) = 0, and it is a simple zero of h. We observe that from the jump and symmetry conditions it follows thath(p) does not have jumps on M, moreover,h(p) =h(p * ), p ∈ M. This means that h(λ) does not have jumps along the spectrum S and on the gap [b + 2a, −1]. The normalisation condition implies lim λ→∞ h(λ) = 1. Hence h(λ) is a meromorphic function on C, i.e., h(λ) = λ−λ(n,t) λ−b−2a .
Corollary 3.5. Let λ(n, t) ∈ [b + 2a, −1] be the projection on C of the Dirichlet eigenvalue p(n, t) given by (3.15). Then We recall that the trace formula in our case looks likê b(n, t) Therefore, Since the problem (3.12) has a unique solution p 0 for any real ∆, we can treat ∆ as the initial data to choose the representative â(n, t),b(n, t) for the isospectral set of finite gap potentials with spectrum S. To emphasise this dependence we denote the representative as â(n, t, ∆),b(n, t, ∆) . In turn, the solution of the RHP with jump (3.13) we denote as m(p, n, t, ∆). We proved the following Remark 3.7. For convenience of the reader, we recall from [23] that the finite gap solution {â(n, t, ∆),b(n, t, ∆)} corresponding to the initial phase ∆ and spectrum S is given bŷ a(n, t, ∆) =ã where ζ is the normalized holomorphic Abel differential,ã = Cap S, and U , Λ are defined by (3.10).

Reduction of the initial RHP to the model RHP
From now we work again in the variable z. Let us identify the upper sheet Π U of the Riemann surface M with the domain (2.8) and the lower sheet with (2.9). The image of J U under the map p → z we denote by J, preserving the orientation, i.e., J = [q 1 , −1] is oriented from right to left. As for the image J * of J L , we change its orientation in accordance with our symmetry requirements, i.e., J * = q −1 1 , −1 is oriented from left to right. The other contours used here are already defined by (2.19), (2.11), (2.12), (2.13). In this section, we perform two transformations (steps) which transform the solution of the initial RHP (Theorem 2.3) to the solution of the RHP with the jump matrix which is close as t → ∞ to a piecewise constant jump matrix everywhere on the jump contour, without exceptional points (parametrices).
The signature table for Re g(z, ξ) is given in Figure 4. Recall that we enumerated the eigenvalues of the problem (1.6), (1.8) starting from the gap (b + 2a, −1) in ascending order, i.e., λ 1 is the minimal eigenvalue and λ ℵ is the maximal one. The respective z j are enumerated in descending order. Choose a small ε 1 > 0 such that Smaller values of ε or δ do not affect these conditions. From Lemma 3.2 it follows that there are unique values ξ i ∈ (ξ ℵ + ε 1 , ξ 0 − ε 1 ) such that Evidently, ξ ℵ+1 < · · · < ξ j+1 < ξ j < · · · < ξ 0 . We denote From the considerations above it is straightforward to get the following For any arbitrary small positive ε < ε 1 one can choose δ > 0 such that for all ξ ∈ I ε the following inequalities are valid Note that the infimum in (4.3) is taken along the circles around all points of the discrete spectrum σ d . The VDO region (n, t) ∈ Z × R + : n t ∈ I ε consists of ℵ + 1 nonintersecting sectors separated by arbitrary small sectors as depicted in Figure 2. If σ gap d = ∅, then ℵ = 0 and the VDO region is the simply connected region For each ξ ∈ I ε we divide the eigenvalues based on their relative location with respect to the point µ 0 (ξ) (respectively, y 0 (ξ)) and introduce the Blaschke product In fact, Set also (cf. (2.12)) The matrix E is not an identity matrix only in small vicinities of those z j which lie in the domain where Re g(z, ξ) > 0. Here With our choice of the VDO region we evidently have Here · is a norm of 2 × 2 matrices. Recall that these singularities essentially depend on the presence or absence of resonances at points (2.20). In the next step we apply the lense mechanism around I and I * , which will at the same time weaken these singularities. We will use = −1, 0, 1 to indicate singularities as follows. In C \ [q −1 , q] introduce a function Q(z) such that s + z s − z can be considered as the Cauchy kernel for symmetric contours, because Ω(z, s) = 1 z−s (1 + o(1)) as z → s, and Ω z, s −1 d s −1 = Ω z −1 , s ds.
Using this kernel allows us to preserve the symmetry condition. Set . (4.12) This function satisfies the symmetries P(z −1 ) = P(z) for z ∈ C \ (I ∪ I * ) and P − (z) = −P + (z) for z ∈ I ∪ I * . Define It is straightforward to verify that S(z) solves the scalar RH problem The above considerations imply that the function is the unique solution of the following RHP with jump along I ∪ I * ∪ J ∪ J * , Since χ(s)Q −2 (s) = 0 as s ∈ I ∪ I * and it is a continuous function on I ∪ I * , then F(z) also has nonzero finite limiting values as z →q ∈ J (cf. [21]). It is straightforward to obtain Lemma 4.5. The function F (z) := F(z)Q(z), defined for z ∈ C \ q −1 , q by (4.11)-(4.13), solves the following RHP (4.14) In a vicinity ofq ∈ {q, q 1 } we have F (z) = C(z −q) 1/4 (1 + o(1)) ifq is a nonresonant point, and F (z) = C(z −q) −1/4 (1 + o(1)) ifq is a resonant point. The jumps of F along the contours I * and J * as well as its behavior at q −1 , q −1 1 are uniquely defined by the symmetry (iii).
Step 2. Introduce two symmetric contours L δ and L * δ = z : z −1 ∈ L δ surrounding I and I * counterclockwise at a small distance such that where Ω δ and Ω * δ are the enclosed regions so that L δ = ∂Ω δ and L * δ = ∂Ω * δ , see Figure 5. Condition (4.17) ensures that L δ is away from the level line Re g(., ξ) = 0 and from any point of the discrete spectrum. Let X(z), z ∈ Ω δ , be the continuation of χ(z) as described in Remark 2.4. Set Define m (2) (z) by Theorem 4.6. For every ξ ∈ I ε , the vector function m (2) (z) = m (2) (z, ξ) is the unique solution of the following RHP: to find a holomorphic function in the domain which has continuous limits on the sides of the contour Σ δ ∪ Σ * δ ∪ L δ ∪ L * δ ∪ J ∪ J * except possibly at points J (2.20) and satisfies: • the jump condition m where Σ δ is defined by (2.13) and v mod (z) = • the symmetry condition m (2) • the normalization condition m • at points of the set (2.20), m (2) (z) has at most a fourth root singularity, Proof . The proof of this theorem is completely analogous to the proof of [11,Theorem 3.6] except for a small contour On the symmetric contour I δ, * (oriented from left to right) we get v (2) = v mod by the symmetry.
Remark 4.7. According to (4.16) and (4.19) we see that Let us label the jump contour which appears in Theorem 4.6 by We extend the matrix v mod (z) to the whole contour K δ by defining it as the identity matrix on the remaining part K δ \ (I ∪ I * ∪ J ∪ J * ).

Solution of the model problem and conclusive analysis
In Section 3, Lemma 3.3, we constructed the vector-functionm(p), which solves the jump problem (3.13) for ∆ given by (3.12). One can treat this result in the following way: let ∆ be an arbitrary real value and let p 0 be the unique solution of the Jacobi inversion problem (3.12). Consider this point as the initial Dirichlet divisor and letâ(n, t, ∆),b(n, t, ∆) be the finite gap solution associated with this divisor and with the spectrum S. In particular, we can construct m(p) associated with ∆ given by (4.14). Being considered on the z-plane, the vector-functioñ m(z) =m(p(z)) has additional jumps on I ∪ I * due to (3.14) and solves the jump problem m + (z) =m − (z) σ 1 , z ∈ I ∪ I * , v mod (z), z ∈ J ∪ J * , with v mod (z) given by (4.19) (or by (4.20)) on J ∪ J * . Introduce the function Recall that according to Lemma 4.8, uniformly with respect to z ∈ Ξ δ and n, t, j : n t ∈ I j ε .
In turn, it implies the following Lemma 5.6. Uniformly with respect to n t ∈ I ε , Now we are ready to apply the technique of singular integral equations. Let C denote the Cauchy operator associated with Ξ δ , where h = h 1 h 2 ∈ L 2 (Ξ δ ) and satisfies the symmetry h(s) = h s −1 σ 1 . Let (C + h)(z) and (C − h)(z) be the non-tangential limiting values of (Ch)(z) from the left and right sides of Ξ δ , respectively. As usual, we introduce the operator C W : L 2 (Ξ δ ) ∩ L ∞ (Ξ δ ) → L 2 (Ξ δ ) by C W h = C − (hW ), where W is the error matrix (5.11). Then as well as where E 1 (z) is a holomorphic vector function in a vicinity of z = 0, uniformly bounded with respect to n, t, j as n t ∈ I j ε . The normalization conditions for m (2) and m mod imply that τ 2 1 + O e −C(ε)t = 1, that is, τ = 1 + O e −C(ε)t .
Using ( This finishes the proof of our main result, Theorem 1.1.