Quantization conditions of eigenvalues for semiclassical Zakharov-Shabat systems on the circle

Bohr-Sommerfeld type quantization conditions of semiclassical eigenvalues for the non-selfadjoint Zakharov-Shabat operator on the circle are derived using an exact WKB method. The conditions are given in terms of the action associated with the unit circle or the action associated with turning points following the absence or presence of real turning points.


Introduction
We consider the eigenvalue problem (1.1) P (h)u = λu for the Zakharov-Shabat operator where u is a column vector, h is a small positive parameter, λ is a spectral parameter, and V is a real valued analytic function on S 1 = R/2πZ. The eigenvalue problem (1.1) appears in the inverse scattering method for the initial value problem for the focusing nonlinear Schrödinger equation as one half of the Lax pair [16]. It will also be written in the form The operator P (h) is not selfadjoint. To study the spectrum Spec P (h), let be the semiclassical symbol of the operator P (h). We define the closure of the set of eigenvalues of P (x, ξ) by where det(A) denotes the determinant of the matrix A. Thus, in our case where V 0 := max x∈S 1 |V (x)|. By Proposition 2.1 below, the spectrum of P (h) is discrete and concentrates on Σ(P ) as h → 0. Hence, to study its asymptotic form one can assume that the spectral parameter λ belongs to a small neighborhood of R i[−V 0 , V 0 ]. Before stating our main results, we recall that the roots of det(M (x, λ)) = 0, or equivalently, V (x) 2 + λ 2 = 0, are called turning points of the system (1.2). A zero of order n of V 2 + λ 2 is called a turning point of order n. When n = 1 or n = 2, the turning point is said to be simple or double, respectively. When λ = 0, the simple zeros of V are double turning points. Double (or higher order) turning points also occur when λ = iµ and ±µ is a local extreme value of V . There are no real turning points for non-zero real λ, and if V 1 := min x∈S 1 |V (x)| is strictly positive, then there are no real turning points for λ = iµ with −V 1 < µ < V 1 .
To describe the asymptotic distribution of eigenvalues in the semiclassical limit, we fix λ 0 ∈ R i[−V 0 , V 0 ] and study the quantization condition of eigenvalues in a small complex neighborhood of λ 0 . The form of the quantization condition depends on whether there are real turning points for λ 0 . We begin with the turning point free case, which is considerably easier. Let B ε (λ 0 ) denote the disc of radius ε > 0 centered at λ 0 . We then have the following Bohr-Sommerfeld type quantization condition associated with the action integral along S 1 given by Then there exists an ε > 0 and a function r 1 (λ, h), analytic with respect to λ in B ε (λ 0 ) and uniformly of O(h) as h → 0, such that λ ∈ B ε (λ 0 ) is an eigenvalue of P (h) if and only if (1.4) 2 cos(I/h) + r 1 (λ, h) = 2.
In particular, for any small h there is an integer k ∈ Z such that (1.5) I(λ) = 2πkh + O(h 2 ).
In particular, for any small h there are integers 1 ≤ j ≤ l and k ∈ Z such that This work was first inspired by the paper by Galtsev and Shafarevich [8] who treated the Schrödinger operator with complex potential D = −h 2 d 2 /dx 2 + i cos x on S 1 . They showed that the spectrum of this non-selfadjoint operator concentrates on a rotated 'Y' shape in the semiclassical limit h → 0 while the set Σ(D) is the half band {λ ∈ C : Re λ ≥ 0, −1 ≤ Im λ ≤ 1}. This fact has also been used by Dyatlov and Zworski [3] to provide a negative example of stochastic stability of resonances in the context of Anosov flows. The numerical range Σ(P ) of our operator P (h), on the contrary, is included in the real and imaginary axes and the spectrum Spec P (h) concentrates on the whole numerical range. Thus, P (h) does not share the behavior of D whose spectrum concentrates on a thin subset of Σ(D). On the other hand, the fact that Σ(P ) is just lines actually specifies the Stokes geometry near the real axis, which allows us to treat general potentials.
To obtain the quantization conditions, we use the exact WKB method along the lines of [6], first introduced by Ecalle [4] and used by Gérard and Grigis [9] to study the Schrödinger operator. An exact WKB solution is a convergent resummation of the WKB asymptotic expansion in a turning-point-free complex region, and the connections of such solutions in different regions via the Wronskian formula (see §2.2) enable us to get the global asymptotic behavior of solutions which leads to the quantization condition of eigenvalues. The asymptotic property of the exact WKB solution is valid only away from turning points, and this prevents us from computing the Wronskian between two exact WKB solutions whose common region of validity is pinched by two turning points close to each other (of distance O(h)). That is why we exclude energies near λ 0 for which V has double or higher order turning points.
The main difference compared with the Schrödinger case is that for our Zakharov-Shabat operator there are two types of turning points, ones of which are the zeros of V (x)−µ and the others of which are the zeros of V (x)+µ. The phase function of the WKB solutions is a primitive function of the square root of their product V (x) 2 −µ 2 while the amplitude is a function of the fourth root of their quotient (see (2.4)). This means that the same Stokes geometry, determined by the phase function, may produce different so-called transition matrices (describing the connections between WKB solutions in different domains) according to the type of intermediate turning points. However, it turns out that the trace of the transition matrix is essentially unaffected (see Proposition 4.4) which explains why there is no mention in Theorem 1.2 of the type of turning points involved. This is not the case in the recent work by Hirota [11] about the eigenvalue problem for a semiclassical Zakharov-Shabat operator (corresponding to the defocusing nonlinear Schrödinger equation) on the real line. In [11], two cases are studied: a simple well potential (two turning points of the same type), and a monotone potential (two turning points of opposite type), resulting in different quantization conditions for the corresponding Zakharov-Shabat system. In our study, there are necessarily an even number of turning points by periodicity, which corresponds (in spirit) to the case of a simple well potential in [11].
Here we also mention the study of Grébert and Kappeler [10] of the periodic eigenvalues of a Zakharov-Shabat operator in the high-energy regime. The problem in the high-energy limit is equivalent to that in the semiclassical limit with a fixed positive energy and with a potential of order h. This small potential can be regarded as a small perturbation which does not affect the principal asymptotics of the eigenvalue distribution, and a slight modification of Theorem 1.1 would give (1.5) with I(λ) = 2πλ, the action for V = 0.
The study of non-selfadjoint Zakharov-Shabat systems on the real line has a long history in connection with inverse scattering theory. For the study in the semiclassical limit, we refer to the book by Kamvissis, McLaughlin and Miller [12]. Real energies belong to the continuous spectrum and the reflection coefficient is relevant. Energies near the imaginary axis consist of eigenvalues when V decays at infinity, and has a band structure when V is periodic (see for example Korotyaev and Kargaev [14]).
The paper is organized as follows. The exact WKB method is reviewed in Section 2, while Section 3 contains the proof of Theorem 1.1. The proof of Theorem 1.2 is the content of Section 4. A crucial detail is the computation of structure formulas for the transition matrices, which for Theorem 1.2 is a much more involved and delicate affair. These computations can be found in Section 5.

Preliminaries
We identify S 1 with the fundamental domain [0, 2π) ⊂ R, and V with a 2πperiodic function on R. In this context we regard the symbol P (x, ξ) in (1.3) as a function on T * R which is periodic in x, and P (h) as an operator acting on vectorvalued periodic functions on R belonging to L 2 ([0, 2π), C 2 ). As mentioned in the introduction, for small h > 0 it suffices to study the case when λ belongs to a small neighborhood of the set of eigenvalues of the symbol P (x, ξ). Proposition 2.1. For z in the resolvent set, (P (h) − z Id) −1 is a compact operator and P (h) has discrete spectrum. Moreover, if Ω is an open connected set such that Ω Σ(P ) = ∅, then (P (h) − z Id) −1 is a holomorphic function of z ∈ Ω provided that h is sufficiently small.
Proof. By [13,Theorem 6.29] the first statement follows if we show that (P (h) − z Id) −1 is a compact operator for some z in the resolvent set. For such z we have (P (h) − z Id) −1 ∈ Ψ −1 by the calculus, where Ψ k denotes the semiclassical pseudodifferential operators of order k. In view of the theorem of Rellich and Kondrachov (see e.g., [1, Section 6.3]) this implies that (P (h) − z Id) −1 is compact on L 2 ([0, 2π), C 2 ).
To prove the second statement we shall adapt the arguments of Dencker [2] to our situation. Note that the symbol P (x, ξ) is also the principal symbol of P (h), and that when we regard P (x, ξ) as a function on T * R the interior of Σ(P ) consists of those λ for which det(P (x, ξ) − λ Id) = 0 for some (x, ξ) ∈ T * R. Similarly, we introduce the eigenvalues at infinity, which is easily seen to be closed in C by taking a suitable diagonal sequence. We now claim that Σ(P ) = Σ ∞ (P ) in our case. Indeed, Σ(P ) ⊂ Σ ∞ (P ) by definition. Conversely, let λ ∈ Σ ∞ (P ), then |P (x j , ξ j )u j − λu j |/|u j | → 0 as j → ∞ for some (x j , ξ j ) ∈ T * R and 0 = u j ∈ C 2 . We cannot have |ξ j | → ∞ since this would imply that |P (x j , ξ j )u j − λu j |/|u j | > 1 if j is sufficiently large. Since {u j /|u j |} j and {V (x j )} j are also bounded, we find by restricting to a subsequence and passing to the limit as j → ∞ that where |u| = 1 and c belongs to the set of values of V . But then λ ∈ Σ(P ) which proves the claim. Next, using the weight m(x, ξ) = (1 + |ξ| 2 ) 1/2 we introduce the symbol classes where A denotes the norm of the matrix A. Then P ∈ S(m). Using the Frobenius norm which satisfies ≤ F , one easily checks that P (x, ξ) −1 ≤ C(1 + |ξ|) −1 when |ξ| 1. If z 1 / ∈ Σ(P ) we thus find by [2, Proposition 2.20] that P (h) − z 1 Id is invertible 1 for sufficiently small h, so we can define and by the calculus, the symbol of Q(h) is in S(1), i.e., the symbol and all its derivatives are bounded. We take z 1 / ∈ Ω which is possible by assumption. Now, the result follows.

Exact WKB solutions.
Here we recall the construction of a solution of (1.2) in a complex domain as a convergent series. Since V is real analytic and periodic, it follows that there is a δ > 0 such that V extends to a holomorphic function in the strip The exact WKB solutions of systems of type (1.2) are known to be of the form see [6]. The function z(x) is the complex change of coordinates Here, z(x) is defined on the Riemann surface of (V 2 + λ 2 ) 1/2 over D, and Q is the matrix valued function defined on the Riemann surface of H(z(.)) over D. These Riemann surfaces are defined by introducing branch cuts emanating from the zeros of x → det(M (x, λ)), i.e., of iV ± λ (the turning points of the system (1.2)), see Section 4. The amplitude vectors w ± in (2.2) are defined as the (formal) series with prescribed initial conditions w ± n (z) = 0 for some choice of base pointz = z(x) wherex is not a turning point. Here d/dz is defined through the chain rule, e.g., Note that these equations are the same as those obtained by an exact WKB construction for scalar Schrödinger equations [9,15]. When we want to signify the dependence on the base pointz = z(x) we write Let Ω be a simply connected open subset of D, free from turning points. Then z = z(x) is conformal from Ω onto z(Ω). For fixed h > 0, the formal series (2.5) converges uniformly in a neighborhood of the amplitude base pointx, and w ± even (x, h) and w ± odd (x, h) are analytic functions in Ω, see [6,Lemma 3.2]. It follows that the functions u ± given by (2.2) are exact solutions of (1.2). They shall henceforth be written as to indicate the particular choice of amplitude base pointx ∈ Ω, and phase base point x 0 ∈ D as it appears in (2.3). Note that these solutions are defined for example everywhere on R, although some of the expressions involved are only defined on Riemann surfaces of (V 2 + λ 2 ) 1/2 or H(z(.)).
For fixedx ∈ Ω, let Ω ± be the set of points x for which there is a path Γ(x, x) fromx to x along which t → ± Re z(t) is strictly increasing. In other words, x ∈ Ω ± if there is a path Γ(x, x) which intersects the Stokes lines (i.e., the level curves of t → Re z(t)) transversally in the appropriate direction. The calculation of the quantization condition will rely on the following asymptotic properties.
uniformly on compact subsets of Ω ± as h → 0, see [6,Proposition 3.3]. In particular, 2.2. The Wronskian formula. For vector valued solutions u and v of (1.2), we introduce the Wronskian For a phase base point x 0 ∈ D and different amplitude base pointsx,ỹ ∈ Ω, elementary computations show that , where u ± are given by (2.2) via (2.9), and we used the fact that det(Q) = −2i. Since the Wronskian is independent of x, we can choose x =ỹ, which in view of the initial conditions of the transport equations (2.6)-(2.7) means that the expression above reduces to . We may of course also choose x =x, thus obtaining This shows that such a pair of solutions is linearly independent if h is sufficiently small.
We shall also need the following formula, obtained by elementary computations, for pairs of solutions of the same type:

Conjugation.
Letx denote the scalar complex conjugate of x ∈ C. For a matrix A ∈ L(C n , C m ), letĀ denote the matrix with complex conjugated entries, so that A * = tĀ is the conjugate transpose (adjoint). For clarity we shall in this subsection write u(x, λ) for a solution to (1.2). By using (2.2) and (2.6)-(2.7) it is straightforward to check (see the proof of Lemma 5.4 below) that the WKB solutions enjoy the symmetry property This implies that it suffices to study the spectral problem P (h)u = λu for spectral parameter λ with nonnegative imaginary part. Proof. Let λ be an eigenvalue with 2π-periodic eigenvector u = u(x, λ). Introduce a pair of linearly independent WKB solutions u ± (x, λ). Since the solution space of Then 2.4. Periodic solutions. As a final preparation, we record a tractable condition for the existence of a nontrivial periodic solution of (1.2), i.e., an eigenvector of P (h) corresponding to λ.
Proposition 2.4. Let u and v be a pair of linearly independent solutions of (1.2), and setũ( . Let T be the transition matrix given by where (u v) is the 2 × 2 system with columns u and v. Then det(T ) = 1, and the existence of a nontrivial periodic solution of (1.2) is equivalent to the condition Proof. Taking the determinant of both sides of (2.14) we get and since the Wronskian is independent of x it follows that det(T ) = 1. Hence, (2.15) is equivalent to tr(T ) = det(T ) + 1, i.e., det(T − Id) = 0, which holds if and only if T c = c for some vector c = 0. If T c = c then a simple computation shows that x → (u(x) v(x))c is nontrivial and 2π-periodic. Conversely, if U is a nontrivial 2π-periodic solution then U can be expressed as a linear combination U = c 1 u+c 2 v, and the same computation as before shows that c = t (c 1 , c 2 ) satisfies T c = c.

Eigenvalues in the absence of real turning points
Here we prove Theorem 1.1 by computing the trace of the transition matrix T introduced above, then applying (2.15) and analyzing the result. We fix λ 0 satisfying the hypotheses of Theorem 1.1, then V (x) 2 + λ 2 0 > 0 for all x ∈ R and there are no turning points on the real axis. Choose a determination of so the real axis is a Stokes line. Since Stokes lines cannot intersect (see e.g. [5]) we find by restricting to a sufficiently small tubular neighborhood of R (which in particular should contain no turning points) that the Stokes lines are essentially parallel to the real axis there. Recall that away from turning points, the configuration of Stokes lines depends continuously on the parameter λ. Hence, we can find ε > 0 such that if |λ − λ 0 | < ε then there is a turning-point-free neighborhood of the real axis in which the imaginary axis is still transversal to the tangent vectors of any Stokes line. From now on, we fix λ ∈ B ε (λ 0 ). We also choose ε so small that (V (x) 2 + λ 2 ) 1/2 has positive real part at x = 0. This gives a determination of z(x) for this choice of λ which is consistent with the determination of z(x; x 0 , λ 0 ) above.
Recall that the number of Stokes lines starting from a turning point Here, x ± ) = ±(−1) j π/2. Hence, Stokes lines starting from turning points in the upper half plane have arguments π/2 mod 2π/3. Stokes lines starting from turning points in the lower half plane have arguments −π/2 mod 2π/3. Figure 1 shows the configuration of Stokes lines for λ = 1 and λ = 1 + 10 −1 i.
Note that if we let λ → 0 along the real line in Example 3.1, then the turning points collapse onto the real line to form double turning points, so that case is excluded from Theorem 1.1. For example, x + and x (2) + collapse to π/2. The Stokes lines starting from the resulting turning points have argument 0 mod π/2.
We now return to the general situation treated in Theorem 1.1. For x near 0 we get by Taylor's formula Let y 0 be an amplitude base point in the upper half plane near the real axis with Re y 0 = 0, and letȳ 0 be the complex conjugate. We will choose phase base points on the real line. With the previous discussion in mind, and because we want our WKB solutions to have asymptotic formulas valid in intersecting domains, we introduce the four WKB solutions , defined in accordance with (2.9). Inspecting the definition (see (2.6)-(2.7)) we find that u ± 1 (x) = u ± 0 (x − 2π), so Proposition 2.4 is applicable. Proposition 3.2. Let T be the transition matrix defined by (u + 0 u − 0 ) = (u + 1 u − 1 )T and let I denote the action integral I(λ) = 2π 0 (V (t) 2 + λ 2 ) 1/2 dt. Then where t 11 t 22 − t 12 t 21 = 1. Moreover, t jj = 1 + r j (λ, h) where r j depends holomorphically on λ, and Proof. Introduce the auxiliary solutions with I given above. We next determine T = (t ij ), noting that t 11 t 22 − t 12 t 21 = 1 since det(T ) = 1 by Proposition 2.4. Taking Wronskians we get .
We first compute W( u + 1 , u − 1 ). By the properties of Re z(x), we can find a curve from y 0 + 2π (in the upper half plane) toȳ 0 + 2π (in the lower half plane) along which Re z(x) is strictly increasing. Evaluating the Wronskian atȳ 0 + 2π (see (2.10)) we obtain Since we can also find curves from y 0 toȳ 0 + 2π and from y 0 +2π toȳ 0 along which Re z(x) is strictly increasing, the same arguments show that and both are equal to 4i + O(h). Since w + even depends analytically on λ we find that t jj = 1 + r j , where r j = r j (λ, h) is analytic in λ and r j = O(h) as h → 0.
End of Proof of Theorem 1.1. By Proposition 2.4 it follows that λ ∈ B ε (λ 0 ) is an eigenvalue of P (h) if and only if tr(T ) = 2, i.e., e iI/h t 11 − 2 + e −iI/h t 22 = 0, which is easily seen to yield (1.4). Multiplying with e iI/h /t 11 and completing the square, an elementary computation using t 11 t 22 − t 12 t 21 = 1 gives where c = (t 12 t 21 ) 1/2 . Hence, e iI/h = 1 + O(h). Taking logarithms we conclude that there is an integer k ∈ Z such that This gives the desired quantization condition (1.5).

Eigenvalues in the presence of real turning points
We now turn to the proof of Theorem 1.2, and we let λ 0 = iµ 0 with µ 0 ∈ (V 1 , V 0 ) be fixed. By assumption, all turning points for λ 0 are simple. As in Section 3 we begin by describing the configuration of Stokes lines for λ 0 before turning to general parameter values λ close to λ 0 . 4.1. Turning points. As described in the introduction there are 2l turning points Since V (0) = V 0 is a local maximum, x 1 (µ 0 ) must be a simple zero of V − µ 0 . Thus, V (x 1 (µ 0 )) < 0 and by (3.1) we have so the Stokes lines starting at x 1 (µ 0 ) have arguments π/3 mod 2π/3. Basic calculus shows that Arg W (x 2 (µ 0 )) = π, so the Stokes lines starting at x 2 (µ 0 ) have arguments 0 mod 2π/3. A moments reflection shows that this pattern repeats itself, with the Stokes lines starting at odd numbered turning points having arguments π/3 mod 2π/3, and the Stokes lines starting at even numbered turning points having arguments 0 mod 2π/3; by periodicity, this includes the last turning point x 2l (µ 0 ) on (0, 2π). In particular, we see that there are bounded Stokes lines lying on R starting from even numbered turning points (on the left) and ending at odd numbered turning points (on the right). However, note that this is not a stable configuration and will not persist when λ 0 is perturbed off the imaginary axis.
We take where the choice of turning point x j (µ 0 ) will depend on the domain of interest, see (4.2)-(4.3) below. We define the Riemann surfaces of z(x) and H(z(x)) over D by introducing branch cuts from odd numbered turning points along the Stokes lines with argument −π/3, and from even numbered turning points along the Stokes lines with argument 2π/3. We choose branches so that Since V (0) ± µ 0 > 0, this also gives a determination of (V 2 − µ 2 0 ) 1/2 and therefore of z(x), namely, (V 2 − µ 2 0 ) 1/2 > 0 at the origin. By applying (3.2) with λ 0 = iµ 0 we see that Re z(x) is a strictly decreasing function of Im x for x near 0. This determines the behavior of Re z(x) in any simply connected open set that intersects the imaginary axis, does not contain any turning points, and does not pass through a branch cut. In particular, there is also for each j ≥ 1 a region between x 2j (µ 0 ) and x 2j+1 (µ 0 ) where Re z(x) is a strictly decreasing function of Im x.
When λ 0 is perturbed through rotation around the origin, the turning points are rotated around points on the real axis, e.g., x 1 and x 2 are rotated around the mid point (x 1 + x 2 )/2, see [7]. Each bounded Stokes line lying on R will then split into two unbounded Stokes lines, but with the topology of the Stokes configuration otherwise unchanged. If the perturbation is allowed to continue, then at a rotation angle of around π/4 new bounded Stokes lines will appear between simple turning points, and these lines will coincide with the integration paths of the action integrals S j (µ) defined by (1.6). The first change is not significant for the proof of Theorem 1.2, while the second change completely alters the behavior of Re z(x) and is not permitted. From now on we therefore fix ε > 0 such that the integration paths of S j (µ) are not bounded Stokes lines when λ ∈ B ε (λ 0 ), and we make sure that We also take ε so small that if λ ∈ B ε (λ 0 ) is purely imaginary, then the turning points are simple and the Stokes configuration is the same as for λ 0 ; in particular this means that B ε (λ 0 ) does not contain 0 and iV 0 . Moreover, the arguments of Stokes lines at the turning points will be almost unchanged, so we place branch cuts as described for λ = λ 0 , modified in the obvious manner. We also use the inherited determination of H(z(.)) and (V 2 − µ 2 ) 1/2 , namely the one which has positive real part at the origin. Figure 2 illustrates the configuration of Stokes lines near three consecutive turning points, including the location of branch cuts.

The transition matrix.
Under the assumptions of Theorem 1.2, the transition matrix T , as defined in Proposition 2.4, will consist of a product of intermediate transition matrices. These matrices will be of (at most) four types, which we now describe. Let λ = iµ ∈ B ε (λ 0 ). We fix amplitude base points y 1 , . . . , y l ∈ D in the upper half plane independent of λ as in Figure 3 in such a way that x 2j (µ) < Re y j < x 2j+1 (µ) and so that y j is always in the same region bounded by Stokes lines when x 2j x 2j+1 x 2j+2 x 2j+3 y j y j y j+1 y j+1 Figure 3. The location of amplitude base points relative the neighboring turning points x k (µ) over a partial period for generic V and λ = iµ ∈ B ε (λ 0 ). Branch cuts are indicated by dashed lines.
λ varies in B ε (λ 0 ). We then set y 0 = y l − 2π. Introduce the WKB solutions . The transition from the pair u + j−1 , u − j−1 to the pair u + j , u − j is one of the following four kinds: That there are no other kinds follows from the fact that since V (0) = V 0 , the nature of the zeros x 2j−1 and x 2j also determines the nature of x 2j+1 , e.g., if both x 2j−1 and x 2j are zeros of V (x)−µ then so is x 2j+1 . Introduce also the auxiliary solutions x 2j (µ),ȳ j ), j = 1, . . . , l. Let T j (resp. T j ) be the transition matrix between u + j−1 , u − j−1 and u + j , u − j (resp. between u + j−1 , u − j−1 and u + j , u − j ): For each for j = 1, . . . , l it is clear that T j is of the same transition type as T j . The transition matrix T as defined in Proposition 2.4 is given by Recall the definition of the action integrals I j (µ) in (1.6). By straightforward computation we immediately obtain the following relationship between T j and T j .
We now describe the different types of transition matrices.
where the matrix R j (λ, h) depends analytically on λ in B ε (λ 0 ) for some positive ε and is uniformly of O(h) there as h → 0, and We postpone the proof of Theorem 4.3 to Section 5, where it will be an immediate consequence of Theorem 5.1 and Theorems 5. to indicate that T j is a transition matrix of type m • , then (4.5) cannot contain any of the factors  Proof. For k ≤ j we let g(j, k) = 2 j−k j−1 n=k cos(I n /h) with the convention that g(j, j) = 1, and set In view of Theorem 4.3, the result follows if we show that It is straightforward to check that for each k ≤ n ≤ j − 1 we have g(j, n + 1) · 2 cos(I n /h) · g(n, k) = g(j, k), and that as a result thereof (4.7) A ± (j, n + 1)A ± (n, k) = A ± (j, k), k ≤ n ≤ j − 1.
Now, (4.6) is clearly true when l = 1, since E m1 is necessarily of type 1 • or 4 • then, i.e., m 1 = 1 or m 1 = 4. When all the E mj are of type 1 • , it is easy to see that (4.6) holds with A + (l, 1) on the right by using induction with respect to l and applying (4.7) with j = l, n = l − 1 and k = 1. If all the E mj are of type 4 • one obtains (4.6) with A − (l, 1) on the right in the same way.
It remains to prove (4.6) when {E m1 , . . . , E m l } contains at least one pair of matrices of type 2 • and 3 • . Write and say that F j is of type m • if m j = m. Since the trace is invariant under cyclic permutations, we may without loss of generality assume that F 1 is of type 2 • . To the left of F 1 there must be a block of type 4 • matrices of length k − 1 ≥ 0, followed by a type 3 • matrix. If k ≥ 2 then the first paragraph shows that this block F k · · · F 2 of type 4 • matrices is equal to A − (k, 2). It is then easy to see that Now, to the left of this block there can be a block of type 1 • matrices of length ≥ 0, followed by another block of the same kind as F k+1 · · · F 1 of length ≥ 0, and this is repeated a finite number of times until the left-hand side of (4.6) is exhausted. But since both types of blocks have already been treated, (4.6) follows by virtue of (4.7).
Hence, for some 1 ≤ j ≤ l we must have cos(I j /h) = O(h 1/l ), so I j /h = ( 1 2 + k)π + O(h 1/l ) which yields the quantization condition (1.8) and the proof is complete.

Structure results for transition matrices
Here we prove Theorem 4.3 by studying each matrix of transition 1 • -4 • separately, starting with transition 2 • . In view of Lemma 4.2 it suffices to consider the auxiliary transition matrices T j .
Theorem 5.1. Let T j be the transition matrix defined by (4.4). If T j is of type 2 • then where a j , b j , c j , d j depend analytically on λ in B ε (λ 0 ) and are there equal to 1+O(h) uniformly as h → 0, and where S j (µ) is the action integral given by (1.6).
Before providing the full details of the proof, which requires some preparation, we briefly sketch the main idea: Write T j = (t mn ). Taking Wronskians in (4.4) we get (5.1) , .
Here, W(u + j−1 , u − j ) and W( u + j , u − j ) can easily be computed using (2.10) and then estimated using Remark 2.2. However, in contrast to the case studied in Theorem 1.1, we will not be able to use the Wronskian formula (2.12) for solutions of the same type to handle W( u + j , u + j−1 ) and W(u − j−1 , u − j ), for these will no longer exhibit the same rapid decay. Instead, we shall express one of the WKB solutions in each Wronskian in the coordinates of a different sheet of the Riemann surface, thereby changing the type from u ± to u ∓ . We can then use (2.10) to compute the Wronskian as normal, and estimate the result using Remark 2.2. For the computation of W( u + j , u − j−1 ), the presence of branch cuts means that although the WKB solutions already are of different type, similar techniques have to be used to ensure that Remark 2.2 is applicable.
The following observations are stated in sufficient generality to be useful in the sequel, but to anchor the discussion we use the assumptions of Theorem 5.1 as starting point, with primary goal of rewriting u + j (x) = u + (x; x 2j , y j ) as a solution of type u − in order to allow for the computation of W( u + j , u + j−1 ). Let R(x 0 , θ) denote the operator acting through rotation around x 0 by θ radians, so that, e.g., i.e., when t is rotated 2πk radians anticlockwise around x 0 then V (t) − µ is rotated 2πk radians anticlockwise around the origin. (Negative k results in clockwise rotation by 2π|k| radians.) We of course have similar behavior for V + µ when V (x 0 ) + µ = 0.
Definition 5.2. Let x 2j be a turning point such that V (x 2j ) + µ = 0 (transitions 2 • and 4 • ). The point over y j that is obtained when rotating y j clockwise once around x 2j will be denoted byŷ j , i.e., More generally, the sheet reached (from the usual sheet) by entering the cut starting at x 2j from the left will be referred to as thex-sheet. The point over y j that is obtained when rotating y j anticlockwise once around x 2j will be denoted byy j , i.e., The sheet reached (from the usual sheet) by entering the cut starting at x 2j from the right will be referred to as thex-sheet. If instead V (x 2j ) − µ = 0 (transitions 1 • and 3 • ) then all directions are to be reversed.
When winding this way around a turning point we always assume that the path is appropriately deformed so as not to be obstructed by other branch cuts. Informally, we think ofx as lying in the sheet "above" the usual sheet, andx as lying in the sheet "below" the usual sheet. The next lemma describes the relative direction of the branch cut starting at x 2j−1 . The reason for wanting to reverse the directions in Definition 5.2 when V (x 2j ) − µ = 0 is to make sure that (5.3)-(5.4) are always in force. In fact, these identities can be taken as definitions of the sheets.
Proof. Assume first that V (x 2j ) + µ = 0. Fix a point x on the line segment from x 2j−1 to x 2j in the area between the cuts, thenx = R(x 2j , −2π)x. When x is rotated −2π radians around x 2j , V (x) + µ is rotated −2π radians around the origin, so 2π)x, which by (5.2) again yields Hence, (5.3) is still in force. One proves (5.4) in the same way.
Next, suppose that V (x 2j−1 ) − µ = 0. To see that thex-sheet is reached by entering the cut starting at x 2j−1 from the left, take x on the line segment from x 2j−1 to x 2j and note that thus H(z(x)) = H(z(R(x 2j−1 , 2π)x)), which means thatx = R(x 2j−1 , 2π)x. (For a fixed point x, H(z(.)) takes distinct values at each of the four points over x.) In the same way one proves the statement concerning thex-sheet, as well as the reverse statements when V (x 2j−1 ) + µ = 0. We omit the details.
In order to compute W( u + j , u + j−1 ) we want to express u + j in the coordinates of a different sheet over y j , and continue the resulting function through the branch cut into the domain of u + j−1 in the usual sheet (containing the amplitude base point y j−1 ), so that their domains intersect and (2.10) and Remark 2.2 are applicable. Under the assumptions of Theorem 5.1 (transition 2 • ), entering the branch cut starting at x 2j from the left leads to thex-sheet by Definition 5.2, so this means rewriting u + j in the coordinates of thex-sheet. (The same is true for transition 4 • ; for transitions 1 • and 3 • it means rewriting u + j in the coordinates of thex-sheet.) Recall that u + j (x) = u + (x; x 2j , y j ), which by definition means that for x near y j in the usual sheet we have The change of variables s = R(x 2j , −2π)t gives Proof. It suffices to prove (5.9) for µ = µ 0 , for then the general result follows by continuity. To evaluate the right-hand side of (5.9) when µ 0 is real, note that the line segment from x 2j−1 (µ 0 ) to x 2j (µ 0 ) is an interval on the real line. Next, recall that we have chosen a determination of (V 2 − µ 2 0 ) 1/2 so that it is positive at the origin, or in fact at any point x ∈ R in the same sheet as the origin where V (x) 2 − µ 2 0 > 0. In particular, Arg(V (s) 2 − µ 2 0 ) 1/2 = 0 for real s < x 2j−1 close to x 2j−1 . For real t > x 2j−1 close to x 2j−1 we write ) 1/2 with s < x 2j−1 as above (rotation in the opposite direction is incorrect since it means passing through a branch cut). Hence, the right-hand side of (5.9) equals where the integrand in the rightmost integral is non-negative by (5.10). The lemma now follows by inspecting the definition of S j (µ 0 ).
Proof of Theorem 5.1. Recalling (5.1), we first compute W( u + j , u − j ). According to the behavior of Re z(x), there is a curve Γ(y j ,ȳ j ) from y j toȳ j along which Re z(x) is strictly increasing. By evaluating the Wronskian atȳ j (see (2.10)) we get and note that the phase base points of u + j−1 and u − j differ. We therefore rewrite u + j−1 as = e Sj /h u + (x; x 2j , y j−1 ), (5.12) where the last identity follows by virtue of Lemma 5.5. Since we can find a curve Γ(y j−1 ,ȳ j ) along which Re z(x) is strictly increasing we can evaluate the Wronskian atȳ j (see (2.10)) and get W(u + j−1 , u − j ) = 4ie Sj /h w + even (ȳ j , h; y j−1 ), where w + even (ȳ j , h; y j−1 ) = 1 + O(h) by Remark 2.2. Thus, by (5.1) and (5.11) we have t 11 = e Sj /h a j , a j = w + even (ȳ j , h; y j−1 ) w + even (ȳ j , h; y j ) It is clear that a j depends analytically on λ since this is true for the amplitude functions w + even . We now compute W( u + j , u + j−1 ). By Lemma 5.4 we have u + j (x) = −iu − (x; x 2j ,ŷ j ) forx nearŷ j . Take the function on the right and continue it through the branch cut starting at x 2j into the domain in the usual sheet containing y j−1 . We remark that at y j−1 it takes the value −iu − (y j−1 ; x 2j ,ŷ j ). Similarly, writing u + j−1 (x) = e Sj /h u + (x; x 2j , y j−1 ) as before, we see that u + j−1 (ŷ j ) = e Sj /h u + (ŷ j ; x 2j , y j−1 ), so by evaluating the Wronskian atŷ j (see (2.10)) we get W( u + j , u + j−1 ) = −W(u + j−1 , u + j ) = −4e Sj /h w + even (ŷ j , h; y j−1 ).
Since Re z(x) is a strictly increasing function of Imx nearx =ŷ j , we can find a curve from y j−1 toŷ j , passing through the branch cut at x 2j , along which t → Re z(t) is strictly increasing (compare with Figure 2). Hence, by (5.1), (5.11) and Remark 2.2 we get t 21 = ie Sj /h c j , c j = w + even (ŷ j , h; y j−1 ) w + even (ȳ j , h; y j ) Let us consider W(u − j−1 , u − j ) next. We fix the domain of u − j−1 and express u − j in the coordinates of the sheet reached when passing through the branch cut at x 2j−1 from the left. For transition 2 • this is thex-sheet according to Definition 5.2. First note that compare with (5.12). Applying Lemma 5.4 we get u − j (x) = −ie Sj /h u + (x; x 2j−1 ,ŷ j ). We continue the expression on the right through the branch cut at x 2j−1 , and note that atȳ j−1 , it takes the value −ie Sj /h u + (ȳ j−1 ; x 2j−1 ,ŷ j ). As above we can find a curve fromŷ j toȳ j−1 , passing through the branch cut at x 2j−1 , along which t → Re z(t) is strictly increasing. Hence, by evaluating the Wronskian atȳ j−1 (see (2.10)) we obtain W(u − j−1 , u − j ) = −4e Sj /h w + even (ȳ j−1 , h;ŷ j ). In view of (5.1), (5.11) and Remark 2.2 we conclude that t 12 = ie Sj /h b j , b j = w + even (ȳ j−1 , h;ŷ j ) w + even (ȳ j , h; y j ) Finally, let us consider W( u + j , u − j−1 ). To get an asymptotic estimate we will need to connect the amplitude base points of u + j and u − j−1 by a curve passing through both the branch cut at x 2j−1 and the branch cut at x 2j . In view of Definition 5.2 this will be possible if we (in the present case of transition 2 • ) express u + j in the coordinates obtained by rotating anticlockwise twice around x 2j . Let the coordinates thus obtained be denotedx. Applying Lemma 5.4 two times we get u + j (x) = −u + (x; x 2j ,ŷ j ) = −e (z(x;x2j )−z(x;x2j−1))/h u + (x; x 2j−1 ,ŷ j ) for x near y j . We can find a path fromŷ j toȳ j−1 along which t → Re z(t) is strictly increasing (compare with Figure 2), so evaluating the Wronskian atȳ j−1 we get (see (2.10)) W( u + j , u − j−1 ) = −4ie (z(ȳj−1;x2j )−z(ȳj−1;x2j−1))/h w + even (ȳ j−1 , h;ŷ j ), where w + even (ȳ j−1 , h;ŷ j ) = 1 + O(h) as h → 0 by Remark 2.2. Here, where the path of integration is homotopic to a curve starting at x 2j and following a curve in thex-sheet through the branch cut at x 2j−1 , then arriving at x 2j−1 viā y j−1 . In other words, it is homotopic to the path from x 2j to x 2j−1 in thex-sheet.
If we rotate back to the usual sheet using (5.2) and then reverse the integration direction, we find that which by Lemma 5.5 is equal to the action integral S j given by (1.6). Recalling (5.1), (5.11) we get t 22 = −e Sj /h d j , d j = w + even (ȳ j−1 , h;ŷ j−1 ) w + even (ȳ j , h; y j ) Γ(y j ,ȳ j−1 ) goes through the branch cut at x 2j from the right (thereby entering thê x-sheet), then goes through the branch cut at x 2j−1 from the right, reentering the usual sheet and then ending atȳ j−1 . In view of the discussion following (5.13), this implies that z(ȳ j−1 ; x 2j ) − z(ȳ j−1 ; x 2j−1 ) = S j , which gives = e Sj /h w + even (ȳ j−1 , h; y j ) w + even (ȳ j , h; y j ) .
The proof is complete.
Theorem 5.8. Let T j be the transition matrix defined by (4.4). If T j is of type 4 • then where a j , b j , c j , d j depend analytically on λ in B ε (λ 0 ) and are there equal to 1+O(h) uniformly as h → 0, and where S j (µ) is the action integral given by (1.6).
Proof. Write T j = (t mn ), then t mn is given by (5.1). The computations of t 11 and t 22 are the same as for transition 1 • (except that for t 22 , the path from y j toȳ j−1 goes via thex-sheet instead), while the computations of t 12 and t 21 are reversed: Since the nature of x 2j−1 is the same as in transition 3 • , the computation of t 12 (rewriting u − j by rotating around x 2j−1 ) is the same as for transition 3 • . Since the nature of x 2j is the same as in transition 2 • , the computation of t 21 (rewriting u + j by rotating around x 2j ) is the same as for transition 2 • .