On the spectral problem associated with the time-periodic nonlinear Schr\"odinger equation

According to its Lax pair formulation, the nonlinear Schr\"odinger (NLS) equation can be expressed as the compatibility condition of two linear ordinary differential equations with an analytic dependence on a complex parameter. The first of these equations---often referred to as the $x$-part of the Lax pair---can be rewritten as an eigenvalue problem for a Zakharov-Shabat operator. The spectral analysis of this operator is crucial for the solution of the initial value problem for the NLS equation via inverse scattering techniques. For space-periodic solutions, this leads to the existence of a Birkhoff normal form, which beautifully exhibits the structure of NLS as an infinite-dimensional completely integrable system. In this paper, we take a first few steps towards developing an analogous picture for time-periodic solutions by performing a spectral analysis of the $t$-part of the Lax pair with a periodic potential.


Introduction
The nonlinear Schrödinger (NLS) equation is one of the most well-studied nonlinear partial differential equations. As a universal model equation for the evolution of weakly dispersive wave packets, it arises in a vast number of applications, ranging from nonlinear fiber optics and water waves to Bose-Einstein condensates. Many aspects of the mathematical theory for (1.1) are well-understood. For example, for spatially periodic solutions (i.e., u(x, t) = u(x + 1, t)), there exists a normal form theory for (1.1) which beautifully exhibits its structure as an infinite-dimensional completely integrable system (see [12] and references therein). This theory takes a particularly simple form in the case of the defocusing (i.e., σ = 1) version of (1.1). Indeed, for σ = 1, the normal form theory ascertains the existence of a single global system of Birkhoff coordinates (the cartesian version of action-angle coordinates) for (1.1). For the focusing (i.e., σ = −1) NLS, such coordinates also exist, but only locally [19]. The existence of Birkhoff coordinates has many implications. Among other things, it provides an explicit decomposition of phase space into invariant tori, thereby making it evident that an x-periodic solution of the defocusing NLS is either periodic, quasi-periodic, or almost periodic in time. The construction of Birkhoff coordinates for (1.1) is a major achievement which builds on ideas going back all the way to classic work of Gardner, Greene, Kruskal and Miura on the KdV equation [10,11] and of Zakharov and Shabat on the NLS equation [28]. Early works on the (formal) introduction of action-angle variables include [26,27]. More recently, Kappeler and collaborators have developed powerful methods which have led to a rigorous construction of Birkhoff coordinates for both KdV [14,15,17] and NLS [12,19] in the spatially periodic case.
The key element in the construction of Birkhoff coordinates is the spectral analysis of the Zakharov-Shabat operator L(u) defined by L(u) = iσ 3 d dx − U , where U = 0 u σū 0 and σ 3 = 1 0 0 −1 .
In particular, the periodic eigenvalues of this operator are independent of time if u evolves according to (1.1) and thus encode the infinite number of conservation laws for (1.1). The timeindependence is a consequence of the fact that equation (1.1) can be viewed as the compatibility condition φ xt = φ tx of the Lax pair equations [20,28] where λ ∈ C is the spectral parameter, φ(x, t, λ) is an eigenfunction, and we note that (1.2) is equivalent to the eigenvalue problem L(u)φ = λφ. Strangely enough, although the spectral theory of equation (1.2) (or, equivalently, of the Zakharov-Shabat operator) has been so thoroughly studied, it appears that no systematic study of the spectral theory of the t-part (1.3) with a periodic potential has yet been carried out (there only exist a few studies of the NLS equation on the half-line with asymptotically time-periodic boundary conditions which touch tangentially on the issue [4,5,21,23,24]). The purpose of the present paper is to perform such a study.
For the spectral analysis, it is appropriate (at least initially) to treat the four functions u, σū, u x , σū x in the definition of V as independent. We will therefore consider the spectral problem (1.3) with potential V given by where ψ = {ψ j (t)} 4 1 are periodic functions of t ∈ R with period one. Apart from the purely spectral theoretic interest of studying (1.3), there are at least three other reasons motivating the present study: − First, in the context of fiber optics, the roles of the variables x and t in equation (1.1) are interchanged, see e.g. [2]. In other words, in applications to fiber optics, x is the temporal and t is the spatial variable. Since the analysis of (1.3) plays the same role for the x-evolution of u(x, t) as the analysis of the Zakharov-Shabat operator plays for the t-evolution, this motivates the study of (1.3). − Second, one of the most important problems for nonlinear integrable PDEs is to determine the solution of initial-boundary value problems with asymptotically time-periodic boundary data [3,6,24]. For example, consider the problem of determining the solution u(x, t) of (1.1) in the quarter-plane {x > 0, t > 0}, assuming that the initial data u(x, 0), x ≥ 0, and the boundary data u(0, t), t ≥ 0 are known, and that u(0, t) approaches a periodic function as t → ∞. The analysis of this problem via Riemann-Hilbert techniques relies on the spectral analysis of (1.3) with a periodic potential determined by the asymptotic behavior of u(0, t) [4,23]. − Third, at first sight, the differential equations (1.2) and (1.3) may appear unrelated. However, the fact that they are connected via equation (1.1) implies that they can be viewed as different manifestations of the same underlying mathematical structure. Indeed, for the analysis of elliptic equations and boundary value problems, a coordinate-free intrinsic approach in which the two parts of the Lax pair are combined into a single differential form has proved the most fruitful [9,13]. In such a formulation, eigenfunctions which solve both the x-part (1.2) and the t-part (1.3) simultaneously play a central role. It is therefore natural to investigate how the spectral properties of (1.2) are related to those of (1.3). Since the NLS equation is just one example of a large number of integrable equations with a Lax pair formulation, the present work can in this regard be viewed as a case study with potentially broader applications.

1.1.
Comparison with the analysis of the x-part. Compared with the analysis of the xpart (1.2), the spectral analysis of the t-part (1.3) presents a number of novelties. Some of the differences are: − Whereas equation (1.2) can be rewritten as the eigenvalue equation L(u)φ = λφ for an operator L(u), no (natural) such formulation is available for (1.3) due to the more complicated λ-dependence. Nevertheless, it is possible to define spectral quantities associated with (1.3) in a natural way. − Asymptotically for large |λ|, the periodic and antiperiodic eigenvalues of (1.2) come in pairs which lie in discs centered at the points πn, n ∈ Z, along the real axis [12]. In the case of (1.3), a similar result holds, but in addition to discs centered at points on the real axis, there are also discs centered at points on the imaginary axis (see Lemma 3.12). Moreover, the spacing between these discs shrinks to zero as |λ| becomes large. − For so-called real type potentials (the defocusing case), the Zakharov-Shabat operator is selfadjoint, implying that the spectrum associated with (1.2) is real. No such statement is true for the t-part (1.3). This is clear already from the previous statement that there exist pairs of eigenvalues tending to infinity contained in discs centered on the imaginary axis. However, it is also true that the eigenvalues of (1.3) near the real axis need not be purely real and the eigenvalues near the imaginary axis need not be purely imaginary. This can be seen from the simple case of a single-exponential potential. Indeed, consider the potential where α, c ∈ C, ω ∈ 2πZ, and σ = ±1. For potentials of this form, equation (1.3) can be solved explicitly (see Section 5) and Fig. 1 shows the periodic and antiperiodic eigenvalues of (1.3) for two choices of the parameters. − Whereas the matrix U in (1.2) is off-diagonal and contains only the function u and its complex conjugateū, the matrix V in (1.3) is neither diagonal nor off-diagonal and involves also u x andū x . This has implications for the spectral analysis-an obvious one being that (1.5) involves four instead of two scalar potentials ψ j (t). − The occurrence of the factor λ 2 in (1.3) implies that the derivation of the asymptotics of the fundamental solution as |λ| → ∞ requires new techniques (see the proof of Theorem 2.7). For the x-part, the analogous result can be established via an application of Gronwall's lemma [12]. This approach does not seem to generalize to the t-part, but instead we are able to perform an asymptotic analysis inspired by [7,Chapter 6] (see also [22]). − In Theorem 4.4 and its corollaries, we will, for sufficiently small potentials, establish the existence of analytic arcs which connect the eigenvalues in a pairwise manner and along which the discriminant is real. An analogous result for (1.2) can be found in [19,Proposition 2.6]. In both cases, the proof relies on the implicit function theorem. However, the proof of (1.3) is quite a bit more involved and requires, for example, the introduction of more complicated function spaces, see (4.11).

1.2.
Outline of the paper. In order to facilitate comparison with the existing literature on the x-part (1.2), our original intention was to follow [12] in terms of notation and exposition. However, as it became more and more evident that the analysis of (1.3) is quite different from that of (1.2), we were forced to deviate from this plan. Nevertheless, some resemblance to the first two chapters of [12] remains. In Section 2, we define and study the fundamental matrix solution of (1.3). The main result is Theorem 2.7, which establishes the asymptotic behavior of the fundamental solution for large λ. In Section 3, we consider the spectrum and derive asymptotic localization results for the eigenvalues. The main result is the Counting Lemma (Lemma 3.12). In Section 4, potentials of real and imaginary type (corresponding to the defocusing and focusing NLS, respectively) are investigated. The main results are Theorem 4.4 and Corollaries 4.9 and 4.10. In Section 5, we consider the special, but important, case of single-exponential potentials. In Section 6, we derive useful formulas for the gradients of the fundamental solution and the discriminant.  Figure 1. Plots of the periodic and antiperiodic eigenvalues for two single exponential potentials with different sets of parameters σ, ω, α and c; cf. (1.6). Fig. 1a shows the periodic and antiperiodic eigenvalues for the real type potential given by σ = 1, ω = −2π, α = 6 15 + 11 4 i, c = 1 10 ; Fig. 1b shows the spectrum of the imaginary type potential with σ = −1, ω = −2π, α = 1 2 , c = iα √ 2α 2 − ω, which arises from an exact plane wave solution of the focusing NLS.

Fundamental solution
In Section 2.1, we introduce the framework for the study of (1.3) and establish basic properties of the fundamental solution. In Section 2.2 we derive estimates for the fundamental matrix solution and its λ-derivative for large |λ|. These estimates will be used in in Section 3 to asymptotically localize the Dirichlet, Neumann and periodic eigenvalues and the critical points of the discriminant of (1.3).
On the space M 2×2 (C) of complex valued 2 × 2-matrices we consider the norm | · |, which is induced by the standard norm in C 2 , also denoted by | · |, i.e.
For given λ ∈ C and ψ ∈ X, let us write the initial value problem corresponding to (1.3) as Equation (2.1) reduces to (1.3) if we identify (ψ 1 , ψ 2 , ψ 3 , ψ 4 ) = (u, σū, u x , σū x ). In analogy to the conventions for the eigenvalue problem (1.2) for the x-part of the NLS Lax pair, we say that the spectral problem (2.1) is of Zakharov-Shabat (ZS) type. The corresponding equation written in AKNS [1] coordinates (q 0 , p 0 , q 1 , p 1 ) reads It is obtained by multiplying the operator equation D = R + V from the right with T and from the left with T −1 , where and by writing that is, In what follows we show the existence of a unique matrix-valued fundamental solution M of (2.1), that is, a solution of DM = RM + V M, M (0) = I, (2.5) where I ∈ M 2×2 (C) denotes the identity matrix. The proof relies on a standard iteration technique. We first observe that the fundamental matrix solution for the zero potential ψ = 0 is given by Indeed, E λ solves the initial value problem For λ ∈ C, ψ ∈ X and 0 ≤ t < ∞ we inductively define Using that |E λ (t)| = e 2| λ 2 |t for t ≥ 0, we estimate where one can choose C(ψ, t) := max(|ψ 1 ψ 2 |, |ψ 1 | + |ψ 3 |, |ψ 2 | + |ψ 4 |) t as a uniform bound for bounded sets of [0, ∞) × X. Therefore the matrix exists and converges uniformly on bounded subsets of [0, ∞)×C×X. By construction, M solves the integral equation hence M is the unique matrix solution of the initial value problem (2.5). Since each M n , n ≥ 0 is continuous on [0, ∞)×C×X and moreover analytic in λ and ψ for fixed t ∈ [0, ∞), M inherits the same regularity due to uniform convergence. Thus we have proved the following result. The fundamental solution M is in fact compact: For any sequence (ψ k ) k in X which converges weakly to an element ψ ∈ X as k → ∞, i.e. ψ k ψ, one has Proof. It suffices to prove the statement for each M n , since the series (2.7) converges uniformly on bounded subsets of [0, ∞) × C × X. The assertion is true for M 0 = E λ , which is independent of ψ. To achieve the inductive step, we assume that the statement holds for M n , n ≥ 1, and consider an arbitrary sequence ψ k ψ in X. Then Furthermore, M satisfies the Wronskian identity: In particular, the inverse M −1 is given by Proof. The fundamental solution M is regular for all t ≥ 0. Therefore a direct computation yields The solution of the inhomogeneous problem corresponding to the initial value problem (2.1)-(2.2) has the usual "variation of constants representation": Proposition 2.4. The unique solution of the inhomogeneous equation (2.9) Proof. Differentiating (2.9) with respect to t and using that M is the fundamental solution of (2.5), we find that and f (0) = v 0 .
As a corollary we obtain a formula for the λ-derivativeṀ of M .
Proof. Differentiation of DM = (R + V )M with respect to λ gives and Proposition 2.4 yields (2.10). The second claim is a consequence of Proposition 2.2.
The fundamental solution M of the ZS-system is related to the fundamental solution K of the AKNS-system by The fundamental solution for the zero potential in AKNS coordinates is therefore given by e 2iλ 2 σ 2 t = cos 2λ 2 t sin 2λ 2 t − sin 2λ 2 t cos 2λ 2 t , σ 2 = 0 −i i 0 .
Remark 2.6. It is obvious that all results in this section possess an analogous version in which the space X of 1-periodic potentials is replaced by the space X τ of potentials defined on the interval [0, τ ], τ > 0.

2.2.
Leading order asymptotics. The results in this section hold for 0 ≤ t ≤ 1 and hence apply to the time-periodic problem we are primarily interested in. It was pointed out in [23] that the fundamental matrix solution M of (2.5) for a potential with sufficient smoothness and decay admits an asymptotic expansion as |λ| → ∞ of the form where the matrices Z i , W i , i = 1, 2, . . . , can be explicitly expressed in terms of the potential and therefore only depend on the time t ≥ 0, and satisfy Z i (0) + W i (0) = 0 for all integers i ≥ 1. This suggests that M satisfies for t within a given bounded interval. These considerations suggest the following result.
Theorem 2.7 will be established via a series of lemmas. We first introduce some notation and briefly discuss the idea of the proof.
For λ ∈ C and ψ ∈ X, let M be the fundamental solution of (2.5), which will be considered on the unit interval [0, 1]. We set θ := 2λ 2 and define M + and M − by we denote by A d its diagonal part and by A od its off-diagonal part, i.e.
We finally define M p by . Letting Q j , j = 1, 2, 3, 4, denote the four quadrants of the complex λ-plane, we set For an arbitrary complex number λ = x + iy with x, y ∈ R and t > 0, it holds that We will prove Theorem 2.7 by establishing asymptotic estimates for the distance between the fundamental solution M and the explicit expression M p that approximates M . For this purpose we will consider the columns of M + and M − separately and compare them with the columns of Z p and W p , respectively, after restricting attention to either D + or D − . By combining all possible combinations, we are able to infer asymptotic estimates for the full matrix M valid on the whole complex plane.
Remark 2.8. For a given smooth potential ψ, the matrices Z i and W i can be determined recursively up to any order i ≥ 0 by integration by parts. Indeed, note that V = V 0 + λV 1 where Assuming that the formal expression , one infers the following recursive equations for the coefficients Z k and W k : which turns out to be enough to prove the asymptotic estimates of M asserted in Theorem 2.7.
Proof. By applying the product rule, assuming that (2.5) holds and noting that σ 3 commutes with diagonal matrices, we obtain Conversely, if (2.15) holds, we similarly obtain and a multiplication with e −iθtσ 3 from the right yields that M satisfies the differential equation in (2.5). The statement concerning the initial conditions holds because M (0, λ) = M + (0, λ).
The following lemma is concerned with the invertibility of Z p and W p . We let C K := {λ ∈ C : |λ| > K} for K > 0 and let B r (0, X 1 ) denote the ball of radius r > 0 in X 1 centered at 0. Furthermore, we definė We use the general fact that if an element A of a Banach algebra (A, · ) satisfies A < 1, then I − A is invertible and its inverse is given by the Neumann series n≥0 A n .
To prove assertion (1), we choose K r > 0 so large that for all t ∈ [0, 1], λ ∈ C Kr and ψ ∈ B r (0, X 1 ). This can always be achieved, because the functions {ψ j } 4 1 , and hence also the functions |Z 1 (t, ψ)| and |Z od 2 (t, ψ)|, are uniformly bounded To prove assertion (2), we fix ψ ∈ X 1 and consider the determinant of W p : We see that both ψ 1 0 and ψ 2 0 have to be nonzero in order for W p to be invertible. If ψ 1 0 , ψ 2 0 = 0, then W p is invertible iff the expression within the square brackets is nonzero. This expression is of order 1 uniformly for t ∈ [0, 1] and ψ ∈ X 1 ; hence there exists a constantK r > 0 such that this expression (and hence also det W p (t, λ, ψ)) is nonzero for t ∈ [0, 1], λ ∈ CK r and ψ ∈ B r (0,Ẋ 1 ). In this case, we can write Thus we can -by the argument used in the proof of (1) -find a constant K r > 0 such that the inverse of W p is given by Lemma 2.10 and its proof suggest the introduction of the following notation.
For t ∈ [0, 1], λ ∈ C and ψ ∈ X 1 , we define whenever the inverses Z −1 p and W −1 p exist. By Lemma 2.10, the inverse of Z p exists uniformly on [0, 1] and on bounded sets in X 1 provided that |λ| is large enough (in order for W −1 p to exist one also needs ψ 1 0 , ψ 2 0 = 0). For a t-dependent matrix A with entries in L p , we define Lemma 2.13. Let V be an arbitrary bounded subset of X 1 and let 1 ≤ q ≤ 2. Then Proof. The case q = 2 follows directly from the definitions of Z 1 , Z od 2 , W 2 and W d 3 , the continuity of the operator · L 2 (0,1) • ∂ t : H 1 (0, 1) → R and the fact that H 1 (0, 1) is an algebra. The cases 1 ≤ q < 2 follow from the case q = 2 in view of the continuous embeddings L 2 (0, 1) → L q (0, 1), 1 ≤ q < 2.
Lemma 2.14. For λ ∈ C and ψ ∈ X 1 , let M be the fundamental solution of (2.5) on the unit interval.
(1) If |λ| is so large that Z −1 p exists for all t ∈ [0, 1], then Proof. Equation (2.20) is obtained by a direct calculation: The proof of (2.21) is similar.
For z ∈ C, we define the linear map e zσ 3 on the space of complex 2 × 2-matrices by e zσ 3 (A) := e zσ 3 Ae −zσ 3 . Lemma 2.14 yields Volterra equations for M + and M − . Lemma 2.15. For λ ∈ C and ψ ∈ X 1 , let M be the fundamental solution of (2.5) on the unit interval.
(1) If |λ| is so large that Z −1 p exists for all t ∈ [0, 1], then M + satisfies (2.23) Proof. Using the first part of Lemma 2.14, we obtain In order to obtain (2.22), we first integrate (2.24) from 0 to t and use that M + (0, λ) = I to determine the integration constant. Applying e −iθtσ 3 to both sides of the resulting integral equation and multiplying by Z p from the left, we find (2.22). The Volterra equation for M − follows in an analogous way from the equation which is a consequence of the second part of Lemma 2.14.

Let [A] 1 and [A] 2 denote the first and second columns of a 2
Lemma 2.17. Let r > 0.
(1) There exists a constant C > 0 such that Proof. For λ ∈ C Kr , the functions are well-defined on their domains [0, 1]×C Kr ×B r (0, X 1 ) and [0, 1] 2 ×C Kr ×B r (0, X 1 ) respectively, where the inverse Z −1 p is given by (2.16) and is uniformly bounded by Lemma 2.10 and Corollary 2.12. Due to Lemma 2.15, M satisfies the following Volterra equation for t ∈ [0, 1], λ ∈ C Kr and ψ ∈ B r (0, X 1 ): where the ψ-dependence has been suppressed for simplicity. Thus M admits the power series representation which converges (pointwise) absolutely and uniformly on [ . Therefore, in view of Corollary 2.12 and Lemma 2.16, there exists a constant C > 0 such that . This proves (2.36); the proofs of (2.37)-(2.39) are similar.
To prove (2.14), we recall Cauchy's inequality: the derivative f of a holomorphic function f : C ⊇ G → C satisfying |f (z)| ≤ C on a disc D(r, a) ⊆ G of radius r centered at a in the open domain G can be estimated at the point a by |f (a)| ≤ Cr −1 . By using that uniformly for t ∈ [0, 1] and locally uniformly for ψ ∈ X as |λ| → ∞, and by applying Cauchy's inequality, we obtain (2.14).
Theorem 2.19. For any potential ψ ∈ X 1 and any sequence of complex numbers z n = ± nπ/2+ The estimates (2.44) and (2.45) hold uniformly on bounded subsets of X 1 and on sequences (z n ) n∈Z satisfying |z n ∓ nπ/2| ≤ C 1/|n| for all n ∈ Z for some constant C > 0; estimate (2.46) holds uniformly on bounded subsets of X 1 and on sequences (z n ) n∈Z where |z 2 n ∓ nπ/2| ≤ C 1/|n| for all n ∈ Z for some constant C > 0.
Proof. The estimates (2.44) and (2.45) follow directly from Theorem 2.7, because z 2 n = O(1) as |n| → ∞ by assumption, and therefore e 2| z 2 n |t = O(1) uniformly in t ∈ [0, 1] as |n| → ∞. To prove (2.46) we note that |e z − 1| ≤ |z| e |z| for arbitrary z ∈ C, thus the additional restriction on z n implies that uniformly for t ∈ [0, 1] as |n| → ∞. The triangle inequality implies that Remark 2.20. For convenience, the above results are stated for the space X τ with τ = 1 (which contains the periodic space X as a subspace). It is clear that analogous results hold for an arbitrary fixed τ > 0.

Spectra
We will consider three different notions of spectra associated with the spectral problem (2.1): the Dirichlet, Neumann and periodic spectrum. These spectra are the zero-sets of certain spectral functions, which are defined in terms of the entries of the fundamental solution M evaluated at time t = 1. We introduce the following notation: , as A D corresponds to D D under the transformation T . For a given potential ψ ∈ X, we say that λ ∈ C lies in the Dirichlet spectrum if there exists a φ ∈ D D which solves (2.1). 1 We arbitrarily choose to define the Dirichlet spectrum in terms of f2 and the Neumann spectrum in terms of f1. Theorem 3.1. Fix ψ ∈ X. The Dirichlet spectrum of (2.1) is the zero-set of the entire function . (3.1) In particular, χ D (λ, 0) = sin 2λ 2 .
Proof. Due to the definition of D D , a complex number λ lies in the Dirichlet spectrum of (2.1) if and only if the fundamental solution M maps the initial value (1, 1) to a collinear vector at t = 1. That is, if and only ifm 1 +m 2 =m 3 +m 4 .
It is now obvious that there are no other roots, because the number of roots of χ D (·, ψ) in each of the discs B N +k , k ≥ 1, is exactly 2(N + k + 1) due to the same argument as we used before. But these roots correspond to the 2(2N + 1) roots of χ D (·, ψ) inside B N ⊆ B N +k plus the 2k roots inside the discs D l ⊆ B N +k with N < |l| ≤ N + k that we have already identified earlier in the proof.
We will in the sequel denote the Dirichlet eigenvalues, i.e. the roots of χ D (ψ), by µ i n ≡ µ i n (ψ), i = 1, 2, n ∈ Z. The Dirichlet spectrum of the zero potential ψ = 0 consists of two bi-infinite sequences on the real and imaginary axes in the complex plane; here sgn denotes the sign function for integers n ∈ Z: Lemma 3.2 tells us that µ i n ∈ D i n for sufficiently large |n|. Proposition 3.3. Uniformly on bounded subsets of X, µ i n (ψ) = µ i n (0) + p n , i = 1, 2 for 2 < p ≤ ∞, where µ i n (0) is given by (3.4). Proof. This result follows directly from Lemma 3.2, because the radius of the disc D i n centered at sgn(n) (−1) i−1 nπ/2, i = 1, 2, which contains the Dirichlet eigenvalue µ i n , is of order O(|n| −1/2 ) as |n| → ∞. By Theorem 2.7, the characteristic function χ N satisfies uniformly on bounded subsets of X. For ψ ∈ X we set σ N (ψ) := {λ ∈ C : χ N (λ, ψ) = 0}.
We denote the Neumann eigenvalues, i.e. the roots of χ N (ψ), by ν i n = ν i n (ψ), i = 1, 2, n ∈ Z. The Neumann spectrum of the zero potential ψ = 0 coincides with the corresponding Dirichlet spectrum: As in the Dirichlet case, we obtain the following results for the Neumann spectrum.
Lemma 3.6. Let B be a bounded subset of X. There exists an integer N ≥ 1, such that for every ψ ∈ B, the entire function χ N (λ, ψ) has exactly one root in each of the two discs D i n , i = 1, 2, for n ∈ Z with |n| > N , and exactly 2(2N + 1) roots in the disc B N when counted with multiplicity. There are no other roots.
Proposition 3.7. Uniformly on bounded subsets of X, given by (3.7). Corollary 3.8. There exists a neighborhood W of the zero potential in X such that for every ψ ∈ W and n ∈ Z, σ N (ψ) ∩ D i n = {ν i n (ψ)}, i = 1, 2. The sum of the off-diagonal entries ofM is referred to as the anti-discriminant: δ ≡ δ(λ, ψ) :=m 2 +m 3 .
Theorem 3.9. The discriminant ∆, the anti-discriminant δ and their respective λ-derivativeṡ ∆ andδ are compact analytic functions on C × X. At the zero potential, ∆(λ, 0) = 2 cos 2λ 2 , λ ∈ C.  For ψ ∈ X, we set σ P (ψ) := {λ ∈ C : χ P (λ, ψ) = 0}. For the zero potential ψ = 0, we obtain where each root has multiplicity two, except the root λ = 0 which has multiplicity four. Thus the periodic spectrum of the zero potential consists of two bi-infinite sequences of double eigenvalues on the real and imaginary axes in the complex plane. The λ-derivative of the discriminant at the zero potential is given by∆ (λ, 0) = −8λ sin 2λ 2 , and its roots, denoted byλ i n (0), i = 1, 2, n ∈ Z, coincide with the periodic eigenvalues in the zero potential case:λ i n (0) = sgn(n) Note that λ = 0 has multiplicity three; all the other roots of∆(·, 0) are single roots. These asymptotic estimates hold uniformly on bounded subsets of X. For the zero potential they hold without the error terms.
The following result provides an asymptotic localization of the periodic eigenvalues.
Lemma 3.12 (Counting Lemma). Let B be a bounded subset of X. There exists an integer N ≥ 1, such that for every ψ ∈ B, the entire function χ P (λ, ψ) has exactly two roots in each of the two discs D i n , i = 1, 2, and exactly 4(2N + 1) roots in the disc B N , when counted with multiplicity. There are no other roots.
Proof. Let B ⊆ X be bounded. By Lemma 3.11, for |λ| → ∞ with λ / ∈ Π uniformly for ψ ∈ B. Hence there exists an integer N ≥ 1 such that, for all ψ ∈ B, |χ P (λ, ψ) − χ P (λ, 0)| < |χ P (λ, 0)| on the boundaries of all discs D i n with |n| > N , i = 1, 2, and also on the boundary of B N . As in the proof of Lemma 3.2, the result follows by an application of Rouché's theorem.
The discs B N and D i n , i = 1, 2, |n| > N , are illustrated in Fig. 2 (see also Fig. 5). Figure 2. Localization of the periodic eigenvalues according to the Counting Lemma. The first 4(2N + 1) periodic eigenvalues lie in the large disc B N in the center. The remaining periodic eigenvalues lie in discs centered at the corresponding double eigenvalues λ ±,i n (0), i = 1, 2, |n| > N , of the zero potential. The radii of these discs shrink to zero at order O(|n| −1/2 ) as |n| → ∞.
Corollary 3.16. There exists a neighborhood W of the zero potential in X such that for every ψ ∈ W and n ∈ Z the following assertions hold: We close this section with an identity that relates the values of the discriminant and the anti-discriminant at a Dirichlet or a Neumann eigenvalue.

Potentials of real and imaginary type
For potentials ψ ∈ X, we define We say that a potential ψ of the ZS-operator is of real type if ψ * = ψ. In this case, ψ 2 =ψ 1 and ψ 4 =ψ 3 , that is, ψ = (q 0 + ip 0 , q 0 − ip 0 , q 1 + ip 1 , q 1 − ip 1 ) for some real-valued functions {q j , p j } 1 j=0 . Hence a potential is of real type iff all coefficients of the corresponding AKNS system are real-valued. The subspace of X of all real type potentials will be denoted by Note that this is a real subspace of X, not a complex one; it consists of those potentials that are relevant for the defocusing NLS.
We can write the Zakharov-Shabat t-part (1.3) as or, in other words, If the eigenfunctions v, w lie in the periodic domain D P , we can integrate by parts without boundary terms and find that Therefore, if the potential ψ is of real type and v is a periodic eigenfunction with eigenvalue λ, and thus we find that According to the Counting Lemma, the periodic eigenvalues of type λ 2 n (ψ) for arbitrary ψ ∈ X necessarily possess non-vanishing imaginary parts for sufficiently large |n|. In analogy with the x-part (1.2), one might expect that λ 1,± n (ψ) = 0 for real type potentials. However, we will see in Section 5 that this is not the case: there are single exponential potentials of real type for which some λ 1,± n are nonreal. Proof. Since ψ = ψ * , the AKNS coordinates (p i , q i ), i = 0, 1, are real. If in addition λ ∈ R, the system (2.3) has real coefficients, so its fundamental solution K is real-valued. The relation M = T KT −1 , cf. (2.11), then implies that m 4 =m 1 and m 3 =m 2 . To prove the second claim, we note thatT = σ 1 T and hencē If v is real in AKNS coordinates, it has real initial data v 0 and v = M T v 0 . Thereforev = We say that a potential ψ ∈ X is of imaginary type if ψ * = −ψ. The subspace X I := {ψ ∈ X : ψ * = −ψ} of potentials of imaginary type is relevant for the focusing NLS. In particular, ∆(λ, ψ) = ∆(λ, ψ) and∆(λ, ψ) =∆(λ, ψ) (4.5) for all ψ ∈ X R ∪ X I and λ ∈ C, so that ∆ and∆ are real-valued on R × (X R ∪ X I ).
Proof. Let us first assume that ψ ∈ X R and λ ∈ C. Then a computation using (4.2) shows that where v * := σ 1v = (v 2 ,v 1 ). The symmetry (4.3) follows from uniqueness of the solution of (4.1) and the initial condition M (0, λ, ψ) = I. Evaluation of (4.3) at t = 1 gives (4.5). This finishes the proof for the case of real type potentials. If ψ ∈ X I , we instead have , which leads to (4.4) and (4.5).

Corollary 4.3.
There exists a neighborhood W of the zero potential in X such that for each ψ ∈ W ∩ (X R ∪ X I ) and each n ∈ Z, Proof. We already know from Corollary 3.16) that there exists a neighborhood W of the zero potential such that, for all general potentials ψ ∈ W and all n ∈ Z, {λ ∈ C :∆(λ, ψ) = 0} ∩ D 1 n = {λ 1 n (ψ)}. Due to the symmetry (4.5) we infer that, for all potentials ψ ∈ W ∩ (X R ∪ X I ) and n ∈ Z, is the only root of∆(·, ψ) in D 1 n , we conclude thatλ 1 n (ψ) is real. Theorem 4.4. There exists a neighborhood W of the zero potential in X and a sequence of nondegenerate rectangles with ε, δ ∈ p,1/2 R , 2 < p < ∞, such that for all ψ ∈ W ∩ (X R ∪ X I ) and all n ∈ Z \ {0}, {λ ∈ C : ∆(λ, ψ) ∈ R} ∩ R ε,δ n = γ n (ψ) ∪ (R ε,δ n ∩ R). The sets γ n (ψ) are analytic arcs transversal to the real axis, which cross the real line in the critical pointsλ 1 n (ψ) of ∆(·, ψ). Moreover, these arcs are symmetric under reflection in the real axis and the orthogonal projection of γ n (ψ) to the imaginary axis is a real analytic diffeomorphism onto its image.
We refer to Fig. 3a for an illustration of the analytic arc γ n (ψ) within the rectangle R ε,δ n centered at the critical pointλ 1 n (0) of the discriminant ∆(·, 0). The p -based spaces p,s R are defined below in (4.11).
Remark 4.5. For potentials ψ in X R ∪ X I near 0, Theorem 4.4 improves the asymptotic localization of the critical pointsλ 1 n (ψ) established in Lemma 3.14, cf. Remark 4.8. The proof of Theorem 4.4 is based on an application of the implicit function theorem for real analytic mappings in an infinite dimensional setting. This level of generality is necessary in order to treat the arcs γ n in a uniform way. Let us first briefly discuss the strategy of the proof. Writing λ = x + iy with x, y ∈ R, we split ∆(λ; ψ) ≡ ∆(x, y; ψ) into its real and imaginary parts and write ∆ = ∆ 1 + i∆ 2 with ∆ 1 (x, y; ψ) := (∆(λ; ψ)), ∆ 2 (x, y; ψ) := (∆(λ; ψ)).
We want to use the implicit function theorem to study the zero-set of ∆ 2 for λ close to λ 1,± n (0) and for ψ close to the zero potential. Letλ 1 n :=λ 1 n (0) denote the critical points of ∆(·, 0) given in (3.11) and recall from (3.11) that λ 1,± n (0) =λ 1 n . We clearly have ∂∆ 2 ∂x (λ 1 n , 0; 0) = 0 for all n ∈ Z ("R bifurcates into the arcs γ n "). Therefore we introduce a functionF with the same zeros on R × (R \ {0}) × (X R ∪ X I ) as ∆ 2 by We observe thatF has a real analytic extension Indeed, since ∆ is analytic on C × X, ∆ 2 is real analytic on R × R × X, and thereforeF is real analytic on R × (R \ {0}) × (X R ∪ X I ). By Proposition 4.2, ∆ 2 is odd in the y-argument for each ψ ∈ X R ∪X I ; hence the constant term in the Taylor expansion of ∆ 2 at y = 0 vanishes. Therefore ∆ 2 (x, y; ψ)/y admits a Taylor series representation at y = 0, which converges absolutely locally around y = 0 to an analytic extension F . For ψ ∈ X R ∪ X I and real sequences u = (u n ) n∈Z and v = (v n ) n∈Z , we define the map F = (F n ) n∈Z by F n (u, v; ψ) := F (λ 1 n + u n , v n ; ψ). (4.9) For the zero potential and the zero sequence, both denoted by 0, we calculate In order to determine ∂F ∂u at the origin (0, 0; 0), we first observe that ∂F ∂u has diagonal form because F j is independent of u n for j ∈ Z with j = n. On the diagonal, we obtain thus ∂F n ∂u n (0, 0; 0) = −32(λ 1 n ) 2 cos[2(λ 1 n ) 2 ], and therefore ∂F ∂u (0, 0; 0) = 16π|n| diag((−1) n+1 ) n∈Z . (4.10) Consequently, ∂F ∂u (0, 0; 0) is at least formally bijective in a set-theoretic and algebraic sense, for example as a mapping from the space of real sequences {u = (u n ) n∈Z : Z → R | u 0 = 0} to itself. In order to give these formal considerations a rigorous justification, we need to consider appropriate subspaces of sequences equipped with appropriate topologies.
The proof of Theorem (4.4) uses techniques from the theory of analytic maps between complex Banach spaces. We therefore review some aspects of this theory. Let (E, · E ), (F, · F ) be complex Banach spaces. Furthermore, we denote by L(E, F ) the Banach space of bounded (complex) linear operators E → F endowed with the operator norm · L(E,F ) , where In this case we call A(u) the derivative of f at u and write df (u) for A(u). In the special case E = F = C, we simply write df (u) = f (u) ∈ C ∼ = C . We call f weakly analytic on O if for every u ∈ O, h ∈ E and L ∈ F the function is analytic in some neighborhood of zero. (1) f is analytic in O.
(2) f is weakly analytic and locally bounded on O.
(3) f is infinitely many differentiable on O and for all u ∈ O the Taylor series of f at u converges to f uniformly in a neighborhood of u.
Let us consider the following p -based spaces of (bi-infinite) sequences, which we will use to prove Theorem 4.4. For 1 ≤ p ≤ ∞ and s ∈ R, we introduce the linear spaces One easily checks that these spaces are Banach spaces. Furthermore, defining Λ n := (1 + n 2 ) 1 2 , n ∈ Z, the map Λ r : p,s R → p,s−r R , u n → Λ r n u n , is an isometric isomorphism for each r ∈ R. In particular Λ s maps p,s R isometrically onto p R . For s ∈ R and 1 < p < ∞, the topological dual of p,s R is isometrically isomorphic to q,−s R , i.e., ( p,s R ) ∼ = q,−s R , where q is the Hölder conjugate of p defined by 1/p + 1/q = 1. The isomorphism is given by the dual pairing ·, · p,s;q,−s : p,s and can be deduced directly from the well-known pq -duality. Henceforth, we will identify the dual of p,s R with q,−s R by means of ·, · p,s;q,−s . In particular, p,s R is a reflexive Banach space for 1 < p < ∞. We will also consider the corresponding complex versions p,s C based on p C . All the above properties hold true for p,s C in an analogous way. We will also use the closed subspaceš p,s R := {u ∈ p,s R : u 0 = 0},ˇ p,s C := {u ∈ p,s C : u 0 = 0}, which inherit reflexivity for 1 < p < ∞: The linear operator T defined by T n u n → |n|u n , T :ˇ p,s C →ˇ p,s−1 C is a topological isomorphism. Likewise, T r : u n → T r n u n = |n| r u n is an isomporhism p,s C → p,s−r C for real r.
(1) As |λ| → ∞, the partial derivative ∂ y ∆ 2 satisfies the asymptotic estimate x 2 + y 2 (4.12) uniformly for ψ in bounded subsets of X R ∪ X I , wherè (2) For each p > 2, the mapping (x n , y n ) → ∂ y ∆ 2 (λ i n + x n , y n ; ψ) Proof. In order to prove part (1), we recall from Theorem 2.7 that In the proof of Theorem 2.7, we gained additional information on the remainder term: it is of the form where the diagonal part of the 1/λ-terms is given by Thus, for any potential ψ ∈ X, ∆(λ, ψ) = 2 cos 2λ 2 − iΓ λ sin 2λ 2 + O |λ| −2 e 2| λ 2 | , (4.14) SinceΓ(ψ) ∈ iR for ψ ∈ X R ∪ X I , the asymptotic estimate (4.12) follows by taking the real part of (4.15). We prove part (2) of the lemma in the complex setting; this includes the real setting as a special case. The proof relies on the asymptotic estimate (4.12). First note that (x n , y n ) → (λ i n + x n )y n clearly maps bounded sets in and see that (x n , y n ) → cos[2((λ i n + x n ) 2 − y 2 n )] maps bounded sets in p,1/2 C × p,1/2 C to bounded sets in ∞ C . Concerning the first term in (4.12), we keep in mind that (λ i n y n ) n∈Z ∈ p C ⊂ ∞ C for y ∈ p,1/2 C , thus | sinhλ i n y n | ≤ L|λ i n y n |, where the Lipschitz constant L depends only on |(λ i n y n ) n∈Z | ∞ and can therefore be chosen uniformly on bounded sets in p C ; analogously | sinλ i n y n | ≤ L|λ i n y n |. Therefore, (x n , y n ) → (λ i n + x n ) sin[2((λ i n + x n ) 2 − y 2 n )] cosh[4(λ i n + x n )y n ] maps bounded sets in is bounded in p C on bounded sets in Indeed, where ∆ 2 (x, 0; ψ) = 0 because, by Proposition 4.5, ∆ is real-valued on R × (X R ∪ X I ). We obtain from (4. 16  × (X R ∪ X I ), and an extension F C : U C →ˇ p,−1/2 C of F, which is locally bounded. Furthermore F C is weakly analytic. To see this, we note that for each u ∈ U C , h ∈ p,1/2 R ×ˇ p,1/2 R × (X R ∪ X I ) and every L ∈ ( where q is the Hölder conjugate of p, z → LF C (u + zh) (4.18) is a well-defined complex valued function on some neighborhood of the origin in the complex plane due to Lemma 4.7 and (4.17). Since the function F is real analytic on R × R × (X R ∪ X I ), it admits an analytic extension to some open set in C × C × (X R ∪ X I ) that contains R × R × (X R ∪ X I ). Therefore all terms in the infinite series that represents the right-hand side of (4.18) are analytic on some neighborhood of the origin in the complex plane. Since F C (u + zh) is uniformly bounded inˇ p,−1/2 C for z locally around the origin in C, it follows that the infinite series LF C (u + zh) converges absolutely and uniformly on some neighborhood of 0 ∈ C. This shows that F C is indeed weakly analytic. We conclude from Lemma 4.6 that F C is analytic on U C ; in particular F is real analytic.
The partial derivative ∂ u F(0, 0; 0), which is given by (4.10), is a topological isomorphism  , ψ), v, ψ) = 0, and such that the map We may assume that the sequences ε = (ε n ) n∈Z and δ = (δ n ) n∈Z satisfy ε n > 0 and δ n > 0 for n ∈ Z \ {0} and ε 0 = δ 0 = 0. Clearly, if −1 ≤ τ n ≤ 1 for each n, then (τ n ε n ) n∈Z ∈ B p,1/2 ε and (τ n δ n ) n∈Z ∈ B p,1/2 δ . Thus we can run through the intervals in each coordinate in a uniform way. Let R ε,δ n , n ∈ Z \ {0}, be the associated sequence of nondegenerate rectangles defined in (4.6). Our considerations show that, for every ψ ∈ W and n ∈ Z \ {0}, the zero-set of F can be  Figure 3. Fig. 3a shows an illustration of the path γ n within the rectangle R ε,δ n which is contained in the disc D 1 n . The critical pointsλ 1 n =λ 1 n (ψ) = γ n ∩ R anḋ λ 1 n (0) are marked with dots. Fig. 3b shows a plot of the zero-set of ∆ 2 (·, 0) in the complex λ-plane; the boundaries of the discs D i n , i = 1, 2, are indicated by dashed circles, the periodic eigenvalues (which coincide with the critical points of ∆(·, 0) and the Dirichlet and Neumann eigenvalues) are indicated by dots.
Corollary 4.10. There exists a neighborhood W * of 0 in X such that for all potentials ψ ∈ W * ∩ X N ∩ (X R ∪ X I ) and all n ∈ Z, there exists an analytic arc γ * n ≡ γ * n (ψ) ⊆ C which connects the two periodic eigenvalues λ 1,± n (ψ) and which satisfies all the properties listed in Corollary 4.9.
Remark 4.11. The Counting Lemma tells us that the eigenvalues λ 1,± n lie in the disc D 1 n for all sufficiently large |n| uniformly on bounded sets in X, where the radius of D 1 n is of order O(1/ |n|). In other words, λ 1,± n (ψ) = λ 1,± n (0) + ∞,1/2 C uniformly on bounded sets in X. If we had a stronger localization result, say of the form λ 1,± n (ψ) = λ 1,± n (0) + p,1/2 C for some 2 < p < ∞ uniformly for ψ near the zero potential in X R ∪ X I , then we could prove a uniform version (in n ∈ Z \ {0}) of Corollary 4.9 valid for all real and imaginary type potentials near zero (not just for finite gap potentials). We note in this regard that for many spectral problems the localization of the eigenvalues can be improved when the potential possesses higher regularity, see e.g. [8,16,18,25].
In Fig. 4c and Fig. 4d, the spectral gaps are larger than in Fig. 4a and Fig. 4b, because the parameter α is larger (α = 1/2). Fig. 5 shows the zero-set of ∆ 2 (·, ψ) for the real type single exponential potential (5.1) with parameters σ = 1, ω = −2π, α = 6 15 + 11 4 i and c = 1 10 . This example clearly demonstrates that Corollaries 4.9 and 4.10 fail to remain true for potentials with sufficiently large X-norms. We further notice that some arcs γ n do not only "leave" the discs D i n (and hence also the rectangles R ε,δ n ), but the zero-set differs qualitatively from the previous examples: certain arcs "merge" with other arcs and subsequently split into new components. Such bifurcations yield discontinuities in the lexicographic ordering of the periodic eigenvalues. Note also that for this potential (and consequently all potentials in X with smaller X-norm), the assertion of the Counting Lemma holds for N = 3: there are 4(2 · 3 + 1) = 28 periodic eigenvalues contained in the disc B 3 (when counted with multiplicity) and each disc D i n , i = 1, 2, |n| > 3, contains precisely one double eigenvalue.  Figure 5. A plot of the zero level set of ∆ 2 (·, ψ) in the complex λ-plane for the real type single exponential potential (5.1) with σ = 1, ω = −2π, α = 6 15 + 11 4 i, c = 1 10 ; periodic eigenvalues are indicated with dots, the large dashed circle is the boundary of the disc B 3 and the remaining dashed circles are the boundaries of the discs D i n .
6. Formulas for gradients 6.1. Gradient of the fundamental solution. Let dF denote the Fréchet derivative of a functional F : Y → C on a (complex) Banach space Y . If it exists, dF : Y → Y is the unique map from Y into its topological dual space Y such that for u ∈ Y . The map dF h : Y → C (also denoted by ∂ h F ) is the directional derivative of F in direction h ∈ Y . For any differentiable functional F : X → C and h ∈ X, we have for some uniquely determined function ∂F = (F 1 , F 2 , F 3 , F 4 ) : X → X. We denote the components of ∂F by ∂ i F , i = 1, 2, 3, 4, and define the gradient ∂F of F by ∂F = (∂ 1 F, ∂ 2 F, ∂ 3 F, ∂ 4 F ) = (F 1 , F 2 , F 3 , F 4 ).
The following notation is useful to express the gradient of M more compactly. Let M 1 and M 2 denote the first and second columns of M , and denote by V 1 the first two components, and by V 2 the last two components of the four-vector ψ: Analogously, let The star product of two 2-vectors a = (a 1 , a 2 ) and b = (b 1 , b 2 ) is defined by Moreover, recall that γ = m 1 m 4 + m 2 m 3 . With this notation, we obtain Corollary 6.2. For any t ≥ 0, the gradient of the fundamental solution M is given by In the special case when ψ = 0 and λ is a periodic eigenvalue corresponding to the zero potential (i.e. λ = λ i,± n (0), i = 1, 2, n ∈ Z), we find At the zero potential, ∂∆(λ, 0) = 0 for all λ ∈ C.
Proof. The formula for the gradient follows directly from Corollary 6.2. In the case of the zero potential, m 2 = m 3 = 0; hence M 1 M 2 = 0 and therefore ∂∆(λ, 0) = 0 for all λ ∈ C.
The following formulas for the derivative of the anti-discriminant are derived in a similar way.