1. Introduction and outline

In this work, we formulate the direct and inverse spectral theory for the self-adjoint Zakharov–Shabat (ZS) operator with periodic potentials, and we use the results to solve the initial value problem (IVP) for the defocusing non-linear Schrödinger (NLS) equation with periodic boundary conditions (BCs). More precisely, we study the eigenvalue problem

(1.1)\begin{equation} \mathcal{L}\,v = z\,v\,, \end{equation}

where the matrix-valued differential operator

(1.2)\begin{equation} \mathcal{L}:= \text{i}\sigma_{3}(\partial_{x} - Q)\, \end{equation}

with $v = v(x,z) = (v_1, v_2)^{T} $, the superscript $T$ denoting matrix transpose, where $\sigma_3$ is the third Pauli matrix (cf. Appendix A.1), and $Q(x)$ is the matrix-valued function

(1.3)\begin{equation} Q(x) = \begin{pmatrix} 0 & q(x) \\ q^*(x) & 0 \end{pmatrix}\,, \end{equation}

the asterisk denoting complex conjugation, and the ‘potential’ $q: {\mathbb R} \to {\mathbb C}$, is a function in $L^1_\mathrm{loc}({\mathbb R})$ with minimal period  $L$, i.e.,

(1.4)\begin{equation} q(x+L) = q(x)\,,\quad \forall x\in\mathbb{R}\,. \end{equation}

The operator $\mathcal{L}$ in (1.2) is a one-dimensional Dirac operator acting in $L^2(\mathbb{R},{\mathbb C}^2)$ with dense domain $H^{1}(\mathbb{R},{\mathbb C}^2)$. The Lax spectrum $\Sigma(\mathcal{L})$ of $\mathcal{L}$ is defined as the set

(1.5)\begin{equation} \Sigma(\mathcal{L}) := \big\{z\in{\mathbb C}: \mathcal L\phi=z\phi,\,\, 0 \lt \|\phi\|_{L^{\infty}({\mathbb R};{\mathbb C}^2)} \lt \infty \big\}\,, \end{equation}

i.e., the set of complex numbers $z$ such that (1.1) has at least one bounded non-zero solution. It can be proved that for $q$ locally integrable the Lax spectrum defined above equals the spectrum of the maximal operator associated with $\mathcal L$ in $L^2({\mathbb R};{\mathbb C}^2)$, the space of square-integrable two-component vector-valued functions, namely the set $\{z\in\mathbb{C} : z\notin\rho(\mathcal L)\}$, where $\rho(\mathcal L)$ is the resolvent set of $\mathcal L$ [Reference Rofe-Beketov and Khol’kin78]. Throughout this work, we will use the term ‘spectrum’ as a synonym for the Lax spectrum. Also, to avoid technical complications, we will require $q\in C^2([0,L])$ for simplicity, unless explicitly stated otherwise.

The eigenvalue problem (1.1) is of fundamental interest in the theory of completely integrable non-linear evolution equations, since it comprises the first half of the Lax pair for the defocusing NLS equation, namely the partial differential equation (PDE)

(1.6)\begin{equation} \text{i} q_t + q_{xx} - 2 |q|^2 q = 0\,, \end{equation}

with $q = q(x,t)$, and where subscripts $x$ and $t$ denoting partial differentiation. Indeed, Zakharov and Shabat [Reference Zakharov and Shabat85, Reference Zakharov and Shabat86] showed that (1.6) is the compatibility condition of the following overdetermined linear system of ordinary differential equations (ODEs)

(1.7a)\begin{equation} v_x = (-\text{i} z\sigma_3 + Q(x,t))\,v\,, \end{equation}

(1.7b)\begin{equation} v_t = ( -2\text{i} z^2\sigma_3 + H(x,t,z))\,v\,, \end{equation}

$v = v(x,t,z)$

$H(x,t,z) = 2zQ - \text{i}\sigma_3(Q^2-Q_x)$

$Q(x,t)$

$z\in{\mathbb C}$

$v$

In 1967, Gardner, Greene, Kruskal and Miura pioneered the use of direct and inverse spectral methods to solve the IVP for the Korteweg–de Vries (KdV) equation [Reference Gardner, Greene, Kruskal and Miura32], whose scattering problem is the time-independent Schrödinger equation. Then, using a similar approach, in [Reference Zakharov and Shabat85, Reference Zakharov and Shabat86], Zakharov and Shabat showed how the Lax pair (1.7a and 1.7b) can be used to solve the IVP for the NLS equation (1.6) via a technique, which is now referred to as the inverse scattering transform (IST), in which the direct and inverse spectral theory of the ZS problem (1.7a) play a crucial role. Shortly afterwards, similar ideas were shown by Ablowitz, Kaup, Newell and Segur to also apply to a rather large class of integrable non-linear PDEs [Reference Ablowitz, Kaup, Newell and Segur3]. Various significant further developments of the IST then followed (e.g., see [Reference Ablowitz and Segur1, Reference Ablowitz and Clarkson2, Reference Ablowitz, Prinari and Trubatch4, Reference Beals, Deift and Tomei7, Reference Deift and Trubowitz18, Reference Faddeev and Takhtajan29, Reference Konopelchenko55, Reference Novikov, Manakov, Pitaevskii and Zakharov74]), a key one among them being the reformulation of the inverse problem (i.e., the reconstruction of the potential from the scattering data) via a matrix Riemann–Hilbert problem (RHP) [Reference Ablowitz and Clarkson2, Reference Beals and Coifman6, Reference Deift17, Reference Its45, Reference Novikov, Manakov, Pitaevskii and Zakharov74, Reference Zhou87], which in turn made it possible to formulate powerful asymptotic techniques such as the Deift–Zhou method [Reference Deift and Zhou20], which has been used with great success in a variety of different settings.

Following the development of the direct and inverse spectral theory and IST for localized potentials (i.e., potentials decaying rapidly as $x\to\pm\infty$, the precise conditions depending on the particular PDE under consideration), a natural next step was the study of IVP with periodic BCs. The direct and inverse spectral theory for Hill’s equation (i.e., the time-independent Schrödinger equation with periodic potentials) and its application to the KdV equations with periodic BCs have, of course, a long and distinguished history, going back to the early 1970s, related to the so-called finite-genus formalism, e.g., see [Reference Belokolos, Bobenko, Enol’skii, Its and Matveev8, Reference Dubrovin24, Reference Its and Matveev47–Reference Its and Matveev49, Reference Kappeler and Pöschel53, Reference Lax59, Reference Magnus and Winkler64–Reference Matveev67, Reference McKean and van Moerbeke71, Reference Novikov, Manakov, Pitaevskii and Zakharov74, Reference Novikov75]. The theory was also extended to the case of infinite genus in [Reference Kappeler and Pöschel53, Reference McKean and Trubowitz69, Reference McKean and Trubowitz70], including the construction of global coordinate transformations which map the KdV flow to a periodic flow on an infinite-dimensional torus. The direct and inverse spectral theory of focusing and defocusing ZS operators and in particular their finite-gap formalism also have a long history [Reference Gesztesy and Holden35–Reference Gesztesy and Weikard37, Reference Harutyunyan and Ashrafyan43, Reference Its and Kotlyarov46, Reference Kotlyarov57, Reference Levitan and Sargsjan61–Reference Ma and Ablowitz63, Reference McKean68, Reference McLaughlin, Overman, Keller, McLaughlin and Papanicolaou72]. In particular, it is well known that, for the self-adjoint ZS spectral problem (1.1) with periodic potentials, $\Sigma(\mathcal L)$ is purely continuous [Reference Danilov15, Reference Eastham25, Reference Hislop and Sigal44, Reference Rofe-Beketov and Khol’kin78] (that is, essential without any eigenvalues and empty residual spectrum), and it is comprised of an at most countable collection of segments referred to as spectral bands, also often referred to as stability bands. All such bands lie along the real $z$-axis, separated by spectral gaps (also often referred to as instability bands), with the endpoints of the gaps lying at interlaced periodic and antiperiodic real eigenvalues.

Despite its undeniable success, the finite-genus approach also has certain drawbacks. For infinite gap potentials, computation of the map from the spectral variables to the infinite-genus torus requires solving an infinite-genus Jacobi inversion problem, and the infinite-dimensional Riemann period matrix must be explicitly computed in order to use the Its–Matveev formula. Another unsettled question is how effective is the finite-gap approach for the solution of the IVP for general periodic initial data. Finally, even in the case of finite genus, the fact that the finite-gap integration method is so different from the IST on the line is somewhat puzzling.

Recently, McLaughlin and Nabelek developed a Riemann–Hilbert approach to solve the inverse spectral problem for Hill’s operator in [Reference McLaughlin and Nabelek73]. Besides avoiding the above-mentioned technical difficulties arising from the finite-genus theory, the novel formulation of the inverse problem makes it possible for the first time to employ all the machinery available for RHPs in order to study a variety of physically interesting problems. The direct spectral theory for periodic self-adjoint operators, including first order systems, is classical and is naturally formulated in terms of Floquet theory, the monodromy matrix and the resulting band-gap decomposition of the spectrum, see [Reference Levitan and Sargsjan61, Reference Levitan and Sargsjan62]. Building on this framework, in this work, we generalize the approach of [Reference McLaughlin and Nabelek73] to formulate the inverse spectral theory for the self-adjoint Dirac operator (1.2) with periodic potentials via an RHP formalism and we use the resulting formalism to solve the IVP for the defocusing NLS equation (1.6) with periodic BC. In the process, we will show that a single set of Dirichlet eigenvalues is sufficient to completely determine the potential from the inverse problem. This result is in sharp contrast to the reconstruction formula for the potential from the finite-genus formalism, in which two different sets of Dirichlet eigenvalues appear (e.g., see [Reference Forest and Lee31, Reference Ma and Ablowitz63, Reference McLaughlin, Overman, Keller, McLaughlin and Papanicolaou72]).

Specifically, the outline of this work is as follows. In Section 2, we begin by recalling some well-known results from Bloch–Floquet theory, and we also introduce a similarity transformation of the fundamental solution of the scattering problem that leads to the definition of the Dirichlet spectrum. We then explicitly define the spectral data that will allow the reconstruction of the potential in the inverse problem. In Section 3, we define modified Bloch–Floquet solutions, we compute the asymptotics of all relevant quantities as $z\to\infty$, and we write down infinite product expansions for the Floquet discriminant and for one of the entries of the monodromy matrix associated with the similarity-transformed fundamental matrix solution that will be used in subsequent sections. All these results are used to formulate the inverse problem as a suitable matrix RHP in Section 4, where we also show how the potential can be recovered from the solution of the RHP. A key step in the process is the introduction of a matrix function $B(z)$, uniquely determined by the spectral data, which serves two purposes: first, to ensure that the solution of the RHP is unimodular, and, second, to eliminate the singularities of the RHP. Using $B(z)$ and the modified Bloch–Floquet solutions, we then define a matrix-valued function $\Phi(x,z)$ that satisfies RHP 4.1. We then also prove this RHP to have a unique solution. In Section 5, we use the results of Sections 2–4 to establish an RHP characterization for the solution of the IVP for the defocusing NLS equation with periodic, smooth, infinite-gap initial conditions. Namely, we compute the time evolution of the spectral data and study the evolution of the normalized Bloch–Floquet solutions to the Dirac equation. In Section 6, we use the results of Sections 4 and 5 to obtain conditions for the spatial and temporal periodicity of the solutions to the defocusing NLS equation. In Section 7, we connect the uniqueness results for the solutions to RHPs 4.1 and 5.1 to a uniqueness result for the corresponding spatially periodic infinite gap Baker–Akhiezer functions, and we discuss the relation of the infinite gap theory discussed in this paper to the well-known finite gap theory. Finally, in Section 8, we end this work with some final remarks and a discussion of some open questions motivated by the results of this work. Some technical results, various proofs, an explicit example and a few other considerations are relegated to the appendix.

2. Direct spectral theory for the periodic self-adjoint ZS problem

In this section, we begin formulating the direct spectral theory for the defocusing ZS spectral problem. Since the time dependence of the potential does not play any role in the direct and inverse spectral theory, in this section and the following ones we will temporarily omit the time dependence, which will then be restored when discussing the IVP for the NLS equation in Section 5.

We begin by briefly recalling some well-known results from Bloch–Floquet theory. These will be used to then define various necessary quantities as well as the spectral data that will allow us in Section 4 to uniquely reconstruct the potential. Recall that, for $A\in L^1_{{\rm loc}}({\mathbb R})$ an $n \times n$ matrix-valued function with $A(x+L) = A(x)$, Floquet’s theorem [Reference Brown, Eastham and Schmidt12, Reference Eastham25, Reference Floquet30] states that any fundamental matrix solution $Y(x)$ of (2.1) of the system of linear homogeneous ODEs

(2.1)\begin{equation} y_x = A(x)\,y \end{equation}

can be written in the Floquet normal form

(2.2)\begin{equation} Y(x) = W(x)\,\mathrm{e}^{Rx}\,, \end{equation}

where $W(x)$ is a nonsingular matrix with $W(x+L) = W(x)$, and $R$ is a constant matrix. Thus, all bounded solutions of the ZS system (1.7a) have the form

(2.3)\begin{equation} v(x,z) = \mathrm{e}^{\text{i}\nu x} w(x,z)\,, \end{equation}

where $w(x+L,z) = w(x,z)$, and $\nu\in\mathbb{R}$ is the Floquet exponent, or quasi-momentum. One also defines the so-called Bloch–Floquet solutions, or normal solutions, as the solutions of (1.7a) such that

(2.4)\begin{equation} \psi(x+L,z) = \rho\,\psi(x,z)\,, \end{equation}

where $\rho$ is the Floquet multiplier. Thus, a solution of (1.7a) is bounded if and only if $|\rho|=1$, in which case $\rho= \mathrm{e}^{\text{i}\nu L}$ with $\nu\in\mathbb{R}$. Moreover, the Floquet multipliers are the eigenvalues of the monodromy matrix $M(z)$, which is defined by

(2.5)\begin{equation} Y(x+L,z)=Y(x,z)M(z)\,, \end{equation}

where $Y(x,z)$ is any fundamental matrix solution of (1.7a). Hereafter, we choose $Y(x,z)$ as the principal matrix solution of (1.7a) that is, the matrix solution of (1.7a) normalized so that $Y(0,z)\equiv\mathbb{I}$, where $\mathbb{I}$ is the $2\times 2$ identity matrix. We then have

(2.6)\begin{equation} M(z) = Y(L,z)\,. \end{equation}

Standard techniques allow one to show that, under the above assumptions, $Y(x,z)$ can be expressed as the following Volterra integral equation

(2.7)\begin{equation} Y(x,z)=\mathrm{e}^{-\text{i} z\sigma_3x}+\int_0^x\mathrm{e}^{-\text{i} z\sigma_3(x-s)}Q(s)Y(s,z)\,\mathrm{d} s\,, \end{equation}

which also allows one to show that, for all $x\in\mathbb{R}$, $Y(x,z)$ is an entire function of $z$. Since the RHS of  (1.7a) is traceless, Abel’s formula implies $\det M(z) \equiv 1$. Hence the Floquet multipliers, i.e., the eigenvalues of $M(z)$, are given by roots of the quadratic equation

(2.8)\begin{equation} \rho^2 - 2\Delta(z)\rho+ 1 = 0, \end{equation}

where $\Delta(z)$ is the Floquet discriminant

(2.9)\begin{equation} \Delta:=\Delta(z)={\textstyle\frac12}\mathop{\rm tr}\nolimits M(z)\,, \end{equation}

which is also an entire function of $z$ [Reference Forest and Lee31, Reference Ma and Ablowitz63, Reference McLaughlin, Overman, Keller, McLaughlin and Papanicolaou72]. Moreover, the Schwarz symmetry of the ZS problem implies that $\Delta(z)$ satisfies a Schwarz reflection principle: $\Delta(z^*)=\Delta^*(z)$. As a result, $\Delta(z)$ is real-valued along the real $z$-axis. Accordingly, for $z\in\mathbb{R}$, the following possible cases arise: (i) if $\Delta^2(z) \gt 1$, the Floquet multipliers are real and the two Bloch eigenfunctions are unstable (i.e., they diverge either as $x\to\infty$ or as $x\to-\infty$). (ii) if $\Delta^2(z) \lt 1$, the Floquet multipliers are complex conjugate, have unit magnitude and the two Bloch eigenfunctions are stable (i.e., bounded for all $x\in\mathbb{R}$). (iii) if $\Delta^2(z)=1$, the Floquet multipliers $\pm 1$ and at least one of the Bloch eigenfunctions is periodic or antiperiodic. Thus, an equivalent representation of the Lax spectrum is:

(2.10)\begin{equation} \Sigma(\mathcal L) = \{z \in \mathbb{C} : \Delta(z) \in [-1,1]\}. \end{equation}

For the development of the direct and inverse spectral theory, it is convenient to have an explicit representation of the Floquet multipliers for all $z\in\mathbb{C}$ as follows. For $z\in\mathbb{C}$, denote the roots of (2.8) as

(2.11)\begin{equation} \rho_{1,2}(z) = \Delta(z) \mp \sqrt{\Delta^2(z)-1}, \end{equation}

for some appropriate choice of the complex square root, to be defined next. Obviously $\rho_{1,2}(z)$ satisfy the relation $\rho_1(z) = 1/\rho_2(z)$. For $z\in\Sigma(\mathcal{L})$, $\Delta(z)\in[-1,1]$ implies $|\rho_{1,2}(z)|=1$, whereas for $z\in\mathbb{C}\setminus\Sigma(L)$, $\Delta(z)\notin[-1,1]$ implies $|\rho_{1,2}(z)|\ne1$.

The main spectrum $\{\zeta_j\}_{j\in\mathbb{Z}}$ of the ZS problem (1.7a) is defined as the set $\{\zeta\in{\mathbb C}: \Delta^2(\zeta) = 1\}$. That is, the main spectrum comprises the eigenvalues $\zeta_j$ for which at least one eigenfunction is periodic ( $\Delta(\zeta) =1$, such that $\rho(\zeta)=1$) or antiperiodic ( $\Delta(\zeta) = - 1$, such that $\rho(\zeta)= -1$). Therefore, we can represent the Lax spectrum as $\Sigma(\mathcal{L})=\cup_{j\in\mathbb{Z}}[\zeta_{2j},\zeta_{2j+1}]$. The closed intervals $[\zeta_{2j},\zeta_{2j+1}]$ are the spectral bands (also often referred to as stable bands in the literature). The complementary open intervals $(\zeta_{2j-1},\zeta_{2j})$ are the spectral gaps (also often called unstable bands).

It is well known that, for the ZS problem with periodic potentials, knowledge of the main spectrum is not sufficient to uniquely recover the potential, and as a result one must introduce auxiliary spectral data in the form of Dirichlet eigenvalues. (The same is also true for the direct and inverse spectral theory for Hill’s operator.) Following [Reference Ma and Ablowitz63, Reference McLaughlin, Overman, Keller, McLaughlin and Papanicolaou72], we therefore define the Dirichlet spectrum associated with (1.7a) as follows:

(2.12)\begin{equation} \Sigma_{{\mathrm{Dir}}}(x_o) :=\{\mu \in {\mathbb C} : \exists~v \not\equiv 0 \in H^1([x_o,x_o+L],{\mathbb C}^2)~ ~\text{s.t.}~ ~ \mathcal{L}\,v = \mu\,v ~ \wedge~ v\in{\rm BC}_{{\rm Dir},x_o} \}\,, \end{equation}

where ‘BC $_{{\rm Dir},x_o}$’ denote the following Dirichlet BCs with base point $x_o$:

(2.13)\begin{equation} v_1(x_o)+v_2(x_o)=v_1(x_o+L)+v_2(x_o+L)=0\,, \end{equation}

where $v = (v_1,v_2)^T$. Any point $\mu\in\Sigma_{{\mathrm{Dir}}}(x_o)$ will be referred to as a Dirichlet eigenvalue of (1.1). Similarly to the Floquet spectrum, one can identify $\Sigma_{{\mathrm{Dir}}}(x_o)$ with the zero set of a suitable entire function. For the ZS problem (1.7a), however, additional complications arise compared to Hill’s operator. For this reason, we introduce the following similarity transformation, which will be instrumental not only for characterizing the Dirichlet spectrum but also for carrying out the inverse spectral theory:

(2.14)\begin{equation} \tilde{Y}(x,z)=U\,Y(x,z)\,U^{-1}, \end{equation}

where

(2.15)\begin{equation} U = \frac1{\sqrt2}\begin{pmatrix} 1 & 1 \\ -\text{i} & \text{i} \end{pmatrix}. \end{equation}

Then $\tilde{Y}(x,z)$ is the fundamental solution of the following modified scattering problem:

(2.16)\begin{equation} \tilde{y}_x = U(-\text{i} z\sigma_3 + Q)U^{-1}\,\tilde{y}\,. \end{equation}

For convenience, also let $\tilde{M}(z)=\tilde{Y}(L,z)$, which is also an entire function of  $z$ like $M(z)$. Moreover, since the trace and determinant are invariant under the transformation (2.14), the Floquet eigenvalues of the modified scattering problem (2.16), i.e., the eigenvalues of $\tilde{M}(z)$, coincide with $\rho(z)$. On the other hand, we now have

Proposition 2.2. The Dirichlet spectrum $\{\mu_j\}$ with base point $x_o=0$ coincides with the set

(2.17)\begin{equation} \Sigma_{{\mathrm{Dir}}}(0)= \left\{\mu \in {\mathbb C} : \tilde{y}_{12}(L,\mu) = 0\right\}. \end{equation}

The formulation of the inverse spectral theory in Section 4 will require the use of some known results on forward spectral theory, which are summarized for convenience in the following theorem [Reference Djakov and Mityagin23, Reference Grébert and Kappeler40, Reference Grébert, Kappeler and Mityagin42, Reference Ma and Ablowitz63, Reference McLaughlin, Overman, Keller, McLaughlin and Papanicolaou72]. (A proof of some of these results is also given in Appendix A.2.)

Theorem 2.3. The main spectrum and Dirichlet spectrum of the Dirac operator $\mathcal{L}$ (1.2) with $L$-periodic potential $q\in L^2({\mathbb R},{\mathbb C})$ satisfy the following properties:

1. (1) The main spectrum, $\{\zeta_j\}_{j\in\mathbb{Z}}$ is real and can be divided into non-degenerate band edges $E_j$ and degenerate band edges $\hat{\zeta}_j$, at which one has $\Delta'(E_j)\ne0$ and $\Delta'(\hat{\zeta}_j)=0$, respectively. Each $\zeta_j$ is either a simple or a double root of $\Delta^2(z)-1=0$, but not of higher order.

2. (2) The Dirichlet spectrum, $\{\mu_j\}_{j\in\mathbb{Z}}$, is real, and each $\mu_j$ lies inside an unstable band or on a band edge. Moreover, there is exactly one Dirichlet eigenvalue $\mu_j$ in each spectral gap. All the $\mu_j$ are simple roots of $\tilde{y}_{12}(L,z)$.

3. (3) $\Delta'(z)\neq0$ for all $z\in\mathbb{R}$ such that $-1 \lt \Delta(z) \lt 1$.

4. (4) The periodic, antiperiodic and Dirichlet eigenvalues $\zeta_j$ and $\mu_j$ have the following asymptotic behaviour as $|j|\to\infty$:

(2.18)\begin{equation} \zeta_{2j}, \zeta_{2j-1} = \frac{\pi j}{L}+O(1/j),\qquad \mu_j =\frac{\pi j}{L}+O(1/j). \end{equation}

For simplicity of presentation, we will assume that $\zeta_0$ is the first point of the main spectrum that is greater than or equal to  $0$, with $z=0$ lying in a degenerate or non-degenerate spectral gap, and that $\zeta_0$ and $\zeta_{-1}$ are the zeros of $\Delta-1$, so that we can order the $\zeta_j$ as

(2.19)\begin{equation} \cdots \lt \zeta_{-2j}\leq\zeta_{-2j+1} \lt \cdots \lt {\zeta}_{-1}\leq{\zeta}_0 \lt {\zeta}_1\leq\cdots \lt \zeta_{2j}\leq\zeta_{2j+1} \lt \cdots \end{equation}

(cf. Fig. 1). We also number the Dirichlet eigenvalues so that $\zeta_{2j-1}\le \mu_{j} \le\zeta_{2j}$. If $z=0$ lies in a spectral band instead of a spectral gap, we can simply set $\zeta_{-1}$ to be the first point of the main spectrum that is greater than or equal to zero. The Lax spectrum can still be expressed as above, and the whole formalism remains unchanged. (Moreover, if $\zeta_{-1} \gt 0$, one need not worry about the possibility that $\mu_0=0$ in Section 4.) Similarly, if $z=0$ lies in a spectral gap but $\zeta_1$ and $\zeta_0$ are zeros of $\Delta(z)+1$ instead of $\Delta(z)-1$, one just needs to switch the expressions for the infinite products in (3.11b) and (3.11c) in Section 4.

Figure 1.

Schematic representation of the behaviour of the Floquet discriminant $\Delta(z)$ (vertical axis) as a function of $z$ (horizontal axis), together with the periodic and antiperiodic spectrum, the spectral bands (in red) and gaps and the Dirichlet eigenvalues (blue dots). See text for further details.

1 Long description

The behavior of the Floquet discriminant on the vertical axis as a function of the spectral parameter on the horizontal axis. The x-axis is labeled with spectral parameter values, showing points such as zeta subscript -7 equals epsilon subscript -3, zeta subscript -5 equals epsilon subscript -2 and so on, up to zeta subscript 15 equals epsilon subscript 8. The y-axis represents the Floquet discriminant value, showing oscillating behavior with peaks and troughs. Periodic and antiperiodic spectra are highlighted, with spectral bands marked in red and gaps visible between them. Dirichlet eigenvalues are depicted as blue dots, positioned within the gaps or bands. The schematic demonstrates how periodic and antiperiodic points relate to band edges, with Dirichlet eigenvalues lying within these regions. The labeling scheme illustrates the relationship between spectral bands and gaps, providing insight into the distribution of eigenvalues across the spectrum.

We are now ready to define the spectral data that will eventually determine uniquely the potential  $q(x)$. For simplicity, we exclude the trivial potential $q(x)\equiv0$ from consideration.

Definition 2.6. (Non-degenerate spectral gaps) A spectral gap is called non-degenerate if $\zeta_{2j-1} \lt \zeta_{2j}$, degenerate if $\zeta_{2j-1}=\zeta_{2j}$. Let $I\subseteq {\mathbb Z}$ denote the set of indices $j$ for which the spectral gap For $j\in I$, we write

\begin{equation*} E_{2j-1}:=\zeta_{2j-1},\qquad E_{2j}:=\zeta_{2j}. \end{equation*}

Thus the Lax spectrum can be written as

\begin{equation*} \Sigma(\mathcal L) = \mathbb R\setminus\bigcup_{j\in I}(E_{2j-1},E_{2j}). \end{equation*}

Moreover, if $g_\pm$ are finite, one has $I=\{g_-,\dots,g_+\}$. Then, equivalently

\begin{equation*} \Sigma(\mathcal L) = (-\infty,E_{2g_- -1}] \cup \bigcup_{j=g_-}^{g_+-1}[E_{2j},E_{2j+1}] \cup [E_{2g_+},\infty). \end{equation*}

In this case, the number of non-degenerate spectral gaps is $g_+-g_-+1$.

If both $g_\pm$ are finite, the genus of the potential (and the associated solution of the defocusing NLS equation) is $g = g_+ - g_-$, and the number of spectral gaps is $g+1$. The case $g=0$ (i.e., $g_+=g_-$, implying that there is only one gap and no finite spectral band) yields a constant or plane-wave potential, while the case $g=1$ (i.e., two gaps and one finite band) yields the elliptic potentials of the NLS equation. Note that the above definition only identifies $g_\pm$ up to an identical arbitrary additive constant. We used the freedom to choose this constant when defining $\zeta_0$ above.

Definition 2.8. (Spectral data)

We define the spectral data associated to the potential $q$ as the set

(2.20)\begin{equation} S(q):=\{{E}_{2k-1},{E}_{2k},\gamma_k,\nu_k\}_{k=g_-,\dots,g_+}. \end{equation}

3. Modified Bloch–Floquet solutions

Some additional results are needed before we can begin to formulate the inverse spectral theory. Specifically, as usual one needs knowledge of the asymptotic behaviour of various relevant quantities as $z\to\infty$. It is straightforward to show the following (see Appendix A.3 for details):

Lemma 3.1. If $q\in C^2([0,L],\mathbb{C})$, then for all $z\in\mathbb{C}$ the fundamental matrix $Y(x,z)$ has the following asymptotic behaviour as $z\to\infty$:

(3.1)\begin{equation} Y(x,z)= \left[I-\frac{1}{2\text{i} z}\sigma_3 K[q](x)+\frac{1}{2\text{i} z} \begin{pmatrix} 0 & q(x)-q(0)\mathrm{e}^{-2\text{i} zx}\\ q^*(0)\mathrm{e}^{2\text{i} z x}-q^*(x) & 0 \end{pmatrix} +O(z^{-2}) \right]\mathrm{e}^{-\text{i} zx\sigma_3} \,, \end{equation}

where

(3.2)\begin{equation} K[q](x)=\int_0^x|q(s)|^2\mathrm{d} s\,. \end{equation}

Hereafter, for simplicity, we will always assume that $q\in C^2([0,L],\mathbb{C})$ unless explicitly stated otherwise. It is straightforward to show that the asymptotic behaviour of $Y(x,z)$ in Lemma 3.1 implies the following:

Proposition 3.3. The Floquet discriminant $\Delta(z)$ has the following asymptotic behaviour as $z\to\infty$:

(3.3)\begin{equation} \Delta(z)={\textstyle\frac12}\mathrm{e}^{\mp\text{i} zL}\left(1\mp\frac{1}{2\text{i} z}\int_0^L|q(x)|^2\mathrm{d} x+O\left(\frac{1}{z^2}\right)\right), \qquad z\in{\mathbb C}^\pm\,. \end{equation}

Next we introduce a particular family of Bloch–Floquet solutions of the modified ZS problem (2.16), which will be instrumental in the formulation of the inverse problem:

Definition 3.4. Let $\psi^\pm(x,z)$ be the Bloch–Floquet solutions of the modified ZS problem (2.16) uniquely determined by the conditions

(3.4)\begin{equation} \psi^{\pm}(x+L,z)=\rho^{\pm1}(z)\psi^{\pm}(x,z)\,,\qquad \psi_1^{\pm}(0,z)=1. \end{equation}

Also, let $\Psi(x,z)$ be the matrix Bloch–Floquet solution defined as

(3.5)\begin{equation} \Psi(x,z) = \big( \,\psi^\mp\,, \,\psi^\pm\, \big)\,,\qquad z\in\mathbb{C}^\pm\setminus\mathbb{R}\,. \end{equation}

In Appendices A.2 and A.4, we prove the following:

Proposition 3.5. The Bloch–Floquet solutions  $\psi^\pm(x,z)$ defined by (3.4) are given by

(3.6)\begin{equation} \psi^{\pm}(x,z)=\tilde{y}_1(x,z)+\frac{\rho^{\pm1}(z)-\tilde{y}_{11}(L,z)}{\tilde{y}_{12}(L,z)}\tilde{y}_2(x,z), \end{equation}

where $\tilde{Y}(x,z) = (\tilde{y}_1,\tilde{y}_2)$.

Proposition 3.6. For all $x \gt 0$, $\psi^-(x,z)$ has the following asymptotic behaviour as $z\to\infty$

(3.7a)\begin{equation} \psi^-(x,z)=\mathrm{e}^{-\text{i} zx}\left(\begin{array}{c} 1-\frac{1}{2\text{i} z}\int_0^x|q(s)|^2\mathrm{d} s-\frac1{2\text{i} z}(q^*(x)-q^*(0))+O(1/z^2) \\ -\text{i}+\frac{1}{2z}\int_0^x|q(s)|^2\mathrm{d} s-\frac1{2z}(q^*(x)+q^*(0))+O(1/z^2)\end{array}\right)\,,\quad z\in{\mathbb C}^+\,, \end{equation}

(3.7b)\begin{equation} \psi^-(x,z)=\mathrm{e}^{\text{i} zx}\left(\begin{array}{c}1+\frac{1}{2\text{i} z}\int_0^x|q(s)|^2\mathrm{d} s+\frac1{2\text{i} z}(q(x)-q(0))+O(1/z^2)\\\text{i}+\frac{1}{2z}\int_0^x|q(s)|^2\mathrm{d} s-\frac1{2z}({q(x)+q(0)})+O(1/z^2)\end{array}\right), \quad z\in{\mathbb C}^-\,. \end{equation}

Similarly, $\psi^+(x,z)$ has the following asymptotic behaviour as $z\to\infty$:

(3.8a)\begin{equation} \psi^+(x,z)=\mathrm{e}^{\text{i} zx}\left(\begin{array}{c}1+\frac{1}{2\text{i} z}\int_0^x|q(s)|^2\mathrm{d} s+\frac1{2\text{i} z}(q(x)-q(0))+O(1/z^2)\\\text{i}+\frac{1}{2z}\int_0^x|q(s)|^2\mathrm{d} s-\frac1{2z}(q(x)+q(0)+O(1/z^2)\end{array}\right),\quad z\in{\mathbb C}^+\,, \end{equation}

(3.8b)\begin{equation} \psi^+(x,z)=\mathrm{e}^{-\text{i} zx}\left(\begin{array}{c}1-\frac{1}{2\text{i} z}\int_0^x|q(s)|^2\mathrm{d} s-\frac1{2\text{i} z}(q^*(x)-q^*(0))+O(1/z^2)\\-\text{i}+\frac{1}{2z}\int_0^x|q(s)|^2\mathrm{d} s-\frac1{2z}(q^*(x)+q^*(0))+O(1/z^2)\end{array}\right)\,,\quad z\in{\mathbb C}^-\,. \end{equation}

Corollary 3.8. The matrix potential $Q(x)$ of the Dirac operator (1.2) can be recovered as

(3.9)\begin{equation} Q(x) = \lim_{z\to\infty} \sqrt2 iz[\sigma_3,U^{-1}\Psi(x,z)\mathrm{e}^{\text{i} zx\sigma_3}]\,. \end{equation}

Proposition 3.6 immediately gives the asymptotic behaviour of $\psi_2^\pm(0,z)$ as $z\to\infty$ up to $O(1/z)$. When considering the time dependence of the potential in Section 5, however, the next term in the asymptotic expansion of the Bloch–Floquet solutions at $z=0$ will also be needed (cf. Appendix A.4):

Proposition 3.9. If $q\in C^2([0,L],\mathbb{C})$, in addition to (3.7a), (3.7b), (3.8a) and (3.8b), we also have, as $z\to\infty$:

(3.10a)\begin{equation} \psi_2^-(0,z)=\begin{cases}\displaystyle -\text{i}-\frac{q^*(0)}{z}-\frac{q^2(0)+q_x(0)}{2\text{i} z^2}+O(1/z^3)\,, &z\in{\mathbb C}^+\,, \\[1ex]\displaystyle \text{i}-\frac{q(0)}{z}+\frac{q^2(0)+q_x(0)}{2\text{i} z^2}+O(1/z^3)\,, & z\in{\mathbb C}^-\,,\end{cases} \end{equation}

(3.10b)\begin{equation} \psi_2^+(0,z)=\begin{cases}\displaystyle \text{i}-\frac{q(0)}{z}+\frac{q^2(0)+q_x(0)}{2\text{i} z^2}+O(1/z^3)\,, &z\in{\mathbb C}^+\,, \\[1ex]\displaystyle -\text{i}-\frac{q^*(0)}{z}-\frac{q^2(0)+q_x(0)}{2\text{i} z^2}+O(1/z^3)\,, & z\in{\mathbb C}^-\,.\end{cases} \end{equation}

Remark 3.10. Using the asymptotics in (3.1) and the corresponding expressions for the entries of $\tilde{Y}(x,z)$ (cf. Appendix A.3), one can prove that $\Delta(z)$ and $\tilde{y}_{12}(L,z)$ are entire functions with order 1 (cf. Appendix A.5). Thus, Hadamard’s factorization theorem (e.g., see [Reference Ahlfors5, Reference Krantz58]) allows us to write $\Delta(z)\pm1$ and $\tilde{y}_{12}(L,z)$ as the following infinite products: if $\mu_0\ne0$ and $\zeta_{0}\ne0$,

(3.11a)\begin{equation} \tilde{y}_{12}(L,z)=\mathrm{e}^{A_1z+B_1}\prod_{j\in\mathbb{Z}}\left(1-\frac{z}{\mu_j}\right)\mathrm{e}^{\frac{z}{\mu_j}}\,, \end{equation}

(3.11b)\begin{equation} \Delta(z)-1=\mathrm{e}^{A_2z+B_2}\prod_{j\in{\mathbb Z}\atop j\text{ even}}\left(1-\frac{z}{\zeta_{2j-1}}\right)\left(1-\frac{z}{\zeta_{2j}}\right)\mathrm{e}^{z\big(\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big)}\,, \end{equation}

(3.11c)\begin{equation} \Delta(z)+1=\mathrm{e}^{A_3z+B_3}\prod_{j\in{\mathbb Z}\atop j\text{ odd}}\left(1-\frac{z}{\zeta_{2j-1}}\right)\left(1-\frac{z}{\zeta_{2j}}\right)\mathrm{e}^{z\big(\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big)}\,. \end{equation}

If $\mu_0=0$, (3.11a) is replaced by

(3.12a)\begin{equation} \tilde{y}_{12}(L,z)=z\mathrm{e}^{A_1z+B_1}\prod_{j\neq 0}\bigg(1-\frac{z}{\mu_j}\bigg)\,\mathrm{e}^{\frac{z}{\mu_j}}. \end{equation}

Similarly, if $\zeta_{0}=0$, (3.11b) is replaced respectively by

(3.12b)\begin{equation} \Delta(z)-1 = z\bigg(1-\frac{z}{\zeta_{-1}}\bigg)\,\mathrm{e}^{A_2z+B_2+\frac{z}{\zeta_{-1}}}\prod_{j\neq 0\atop j\text{ even}}\left(1-\frac{z}{\zeta_{2j-1}}\right)\left(1-\frac{z}{\zeta_{2j}}\right)\mathrm{e}^{z\big(\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big)}\,, \end{equation}

while the factorization of $\Delta(z)+1$ remains unchanged. The presence of the exponential terms in (3.11) is a significant difference between the spectral theory for the ZS operator compared to that for Hill’s equation, and it is a consequence of the fact that the solutions of the ZS problem (1.7a) have order of growth 1 as $z\to\infty$, as opposed to those of Hill’s equation, whose order of growth is 1/2. A corresponding difference will appear in the definition of the jump matrix for the RHP in Section 4. Note also that, by virtue of [Reference Grébert and Kappeler41] (see Lemma 5.4 and Appendix C), we can also express $\Delta^2(z)-1$ and $\tilde{y}_{12}(L,z)$ via the following infinite products:

(3.13a)\begin{equation} \tilde{y}_{12}(L,z)=-\prod_{j\in\mathbb{Z}}\frac{\mu_j-z}{\pi_j}, \end{equation}

(3.13b)\begin{equation} \Delta^2(z)-1=-\prod_{j\in\mathbb{Z}}\frac{(\zeta_{2j}-z)(\zeta_{2j-1}-z)}{\pi_j^2}, \end{equation}

where $\pi_j$ is defined as

(3.14)\begin{equation} \pi_j=\begin{cases} j\pi,& j\neq 0\\ 1, & j=0. \end{cases} \end{equation}

While the fundamental matrices $Y(x,z)$ and $\tilde{Y}(x,z)$ are entire, the Bloch–Floquet eigenfunctions $\psi^\pm$ are singular at $z=\mu_j$, as is easily seen from (3.6). For completeness, we next characterize this singular behaviour.

Proposition 3.12. If $\nu_j=1$, then $\psi^-(x,\mu_j)$ is finite, while $\psi^+(x,z)$ has a simple pole at $\mu_j$ with residue

(3.15a)\begin{equation} \mathop{\rm Res}\limits_{z=\mu_j}\psi^+=\frac{\rho(\mu_j)-\rho^{-1}(\mu_j)}{\tilde{y}'_{12}(L,\mu_j)}\tilde{y}_2(x,\mu_j). \end{equation}

Conversely, if $\nu_j=-1$, then $\psi^+(x,\mu_j)$ is finite, while $\psi^-(x,z)$ has a simple pole at $\mu_j$ with residue

(3.15b)\begin{equation} \mathop{\rm Res}\limits_{z=\mu_j}\psi^-=\frac{\rho^{-1}(\mu_j)-\rho(\mu_j)}{\tilde{y}'_{12}(L,\mu_j)}\tilde{y}_2(x,\mu_j). \end{equation}

Proof. For the purposes of the present calculation, it is convenient to define $\tilde{\Psi}(x,z) = (\psi^-,\psi^+)$ for all $z\in\mathbb{C}$. (Obviously one has $\tilde{\Psi} = \Psi$ for all $z\in\mathbb{C}^+$ and $\tilde{\Psi} = \Psi\,\sigma_1$ for all $z\in\mathbb{C}^-$.) We can write

(3.16)\begin{equation} \tilde{\Psi}(x,z) = \tilde{Y}(x,z)\,C(z),,\qquad C(z) = \begin{pmatrix} 1 & 1 \\ c_-(z) & c_+(z) \end{pmatrix}\,, \end{equation}

with $c_\pm(z) = (\rho^{\pm1}(z) - \tilde{y}_{11}(L,z))/\tilde{y}_{12}(L,z)$.

The result then follows because, by Lemma 2.5, one has $\tilde{y}_{11}(L,\mu_j) = \rho^{-\nu_j}(\mu_j)$. Incidentally, note that

(3.17)\begin{equation} C^{-1}(z)= \frac1{\rho(z) - \rho^{-1}(z)} \begin{pmatrix} \rho(z)-\tilde{y}_{11}(L,z) & - \tilde{y}_{12}(L,z) \\ - \rho^{-1}(z)-\tilde{y}_{11}(L,z) & \tilde{y}_{12}(L,z) \end{pmatrix} \end{equation}

for all $z$ such that $\rho(z)\ne1$.

Proposition 3.13. The fundamental matrix solutions $Y$ and $\tilde{Y}$ satisfy the following symmetries:

(3.18)\begin{equation} Y^*(z^*)=\sigma_1 Y(z)\sigma_1\,,\quad \tilde{Y}^*(z^*)= \tilde{Y}(z)\,,\qquad z\in{\mathbb C}\,. \end{equation}

Proposition 3.14. The Bloch–Floquet solutions $\psi^\pm$ and the matrix Bloch–Floquet solution satisfy the symmetries

(3.19)\begin{equation} \psi^{\pm*}(z^*)=\psi^\pm(z)\,,\quad \Psi^*(z^*)=\Psi(z)\,\sigma_1\,,\qquad \forall z\in{\mathbb C}\setminus{\mathbb R}\,. \end{equation}

Proof. Using (3.6), (3.18), and the fact that $\rho^*(z^*)=\rho(z)$, we have

(3.20)\begin{equation} (\psi^+)^*(z^*)= \tilde{y}_1^*(z^*)+\frac{\rho^{*}(z^*)-\tilde{y}^*_{11}(L,z^*)}{\tilde{y}^*_{12}(L,z^*)}\tilde{y}^*_2(x,z^*)=\tilde{y}_1(x,z)+\frac{\rho(z)-\tilde{y}_{11}(L,z)}{\tilde{y}_{12}(L,z)}\tilde{y}_2(x,z)=\psi^+(z). \end{equation}

The result for $\psi^-$ is proved in a similar way. The result for $\Psi$ then follows immediately from the definition (3.5).

Proof. By an argument analogous to the symmetry reduction used in the focusing case (cf. [Reference Biondini, Oregero and Tovbis10]), in the present self-adjoint reduction, for any purely imaginary and even potential, the symmetries of the ZS problem allow the monodromy matrix $M(z)$ to be written as

(3.21)\begin{equation} M(z) = \Delta(z) I + c(z) \sigma_3 + \text{i} s(z)\sigma_2\,, \end{equation}

where $I$ and $\sigma_2$ are the identity matrix and the second Pauli matrix. In terms of $M(z)$, the Dirichlet eigenvalues are the zeros of $\frac{i}{2} [ M_{11} - M_{22} + M_{21} - M_{12} ] = {i}(c(z) - s(z))$. Thus, the Dirichlet eigenvalues are the points for which $c(z)-s(z)=0$. Next, note that, since $\det M(z) =1$, (3.21) yields

(3.22)\begin{equation} \Delta^2(z)- 1 = c^2(z) - s^2(z)\,. \end{equation}

The main spectrum is therefore located at the points for which $c^2(z)=s^2(z)$. We then see that each Dirichlet eigenvalue must coincide with a main eigenvalue.

4. Inverse spectral theory for the periodic self-adjoint ZS problem

Using the results of Section 3, we now construct an RHP for the ZS eigenvalue problem (1.7a) with periodic potentials that also applies in the case of infinite genus. We then establish a unique solution of this RHP and we obtain a reconstruction formula for the potential from the solution of this RHP.

Definition 4.1. Let the $2\times2$ matrix-valued function $B(z)$ be defined as

(4.1a)\begin{equation} B(z) = b(z) \begin{cases} \text{i}\,\mathop{\rm diag}\nolimits(f^-,f^+),&z\in\mathbb{C}^+\setminus\Sigma(\mathcal{L})\,,\\[1ex] \mathop{\rm diag}\nolimits(f^+,f^-),&z\in\mathbb{C}^-\setminus\Sigma(\mathcal{L})\,, \end{cases} \end{equation}

with

(4.1b)\begin{equation} b(z) = \frac{(f^0(z))^{1/2}}{(\Delta^2(z)-1)^{1/4}}\,, \end{equation}

and where $f^0(z)$ and $f^\pm(z)$ are defined as

(4.1c)\begin{equation} f^0(z)= - \prod_{\nu_j=0}\frac{\mu_j-z}{\pi_j}\,,\quad f^+(z)=\prod_{\nu_j=1}\frac{\mu_j-z}{\pi_j}\,,\quad f^-(z)=\prod_{\nu_j=-1}\frac{\mu_j-z}{\pi_j}\,. \end{equation}

Note that the functions $f^0(z)$, $f^\pm(z)$ defined by the infinite products in (4.1c) are entire.

Moreover, one has $f^0(z) f^+(z) f^-(z) = \tilde{y}_{12}(L,z)$. As usual, if any of the products in (4.1c) contain no terms, the corresponding functions are defined to be equal to 1. The function $\sqrt{f^0(z)}$ is defined to be holomorphic everywhere except where $f^0(z)\le0$, and the branch for the square root is chosen accordingly (cf. Fig. 2). The function $\sqrt[4]{\Delta^2-1}$ is defined so that it is holomorphic for $z\in{\mathbb C}\setminus\big(\cup_{n=-\infty}^{\infty}[\zeta_{4n-4},\zeta_{4n-1}]\big)$ (also see Fig. 2).

Figure 2.

The branch cuts along the real $z$-axis for the functions $\sqrt{\Delta^2-1}$ (in red) and for $\sqrt[4]{\Delta^2-1}$ (in blue), as well as the segments (thick black) where $\sqrt{\Delta^2-1}$ and $\sqrt[4]{\Delta^2-1}$ are both positive. Top: $g=0$. Middle: $g=1$. Bottom: $g=2$.

Figure 2 Long description

Organized into three horizontal rows, each labeled on the left as g equals 0, g equals 1 and g equals 2 respectively. Each row contains parallel horizontal lines with marked points and segments indicating energy states and thick black segments connecting specific labeled points. The top row, g equals 0, shows two labeled states: E subscript minus 1 and E subscript 0, positioned close together near the center, with a thick black segment connecting them and an arrow pointing rightward from E subscript minus 1 toward E subscript 0. The middle row, g equals 1, shows four labeled states: E subscript minus 1, E subscript 0, E subscript 1 and E subscript 2. A thick black segment connects E subscript minus 1 to E subscript 0. To the right of center, E subscript 1 and E subscript 2 are connected by a dashed segment. The bottom row, g equals 2, shows six labeled states: E subscript minus 1, E subscript minus 0, E subscript 1, E subscript 2, E subscript 3 and E subscript 4. A thick black segment connects E subscript minus 1 to E subscript minus 0 and a second thick black segment connects E subscript 3 to E subscript 4. A dashed segment connects E subscript 1 to E subscript 2. Two parameter-regime conditions are printed above the diagram. On the left, the condition reads: left parenthesis delta superscript 2 minus 1 right parenthesis superscript 1 over 2 greater than 0 and left parenthesis delta superscript 2 minus 1 right parenthesis superscript 1 over 4 greater than 0. On the right, the condition reads: left parenthesis delta superscript 2 minus 1 right parenthesis superscript 1 over 2 greater than 0 and left parenthesis delta superscript 2 minus 1 right parenthesis superscript 1 over 4 less than 0. Diagonal arrows connect states across rows, indicating correspondences between branches in different g rows. One arrow descends from the g equals 0 row toward E subscript minus 1 in the g equals 1 row and another descends further toward E subscript minus 1 in the g equals 2 row. Additional arrows cross from E subscript 0 in the g equals 1 row downward to E subscript 2 and E subscript 3 in the g equals 2 row.

Proposition 4.2. The matrix $B(z)$ is completely determined by the spectral data $S(q)$ defined in (2.20). Explicitly, we have

(4.2a)\begin{equation} b^4(z) = - \prod\limits_{\nu_j=0} \frac{(\gamma_j-z)^2}{(E_{2j}-z)(E_{2j-1}-z)} \prod\limits_{\nu_j\neq0}\frac{\pi_j^2}{(E_{2j}-z)(E_{2j-1}-z)}\,, \end{equation}

(4.2b)\begin{equation} f^+(z)=\prod_{j=g^-,\atop\nu_j=1 }^{g^+}\frac{\gamma_j-z}{\pi_j}\,, \qquad f^-(z)=\prod_{j=g^-\atop\nu_j=-1 }^{g^+}\frac{\gamma_j-z}{\pi_j}\,. \end{equation}

Proof. Taking the fourth power of the coefficient of $B(z)$, we have

(4.3)\begin{equation} \begin{aligned} b^4(z) = - \frac{ \prod\limits_{\nu_j=0}(\mu_j-z)^2}{\prod\limits_{j\in\mathbb{Z}}(\zeta_{2j}-z)(\zeta_{2j-1}-z)} = \frac{ \prod\limits_{\nu_j=0}(\mu_j-z)^2}{\prod\limits_{\nu_j=0}(\zeta_{2j}-z)(\zeta_{2j-1}-z)}\frac{\pi_j^2}{\prod\limits_{\nu_j\neq0}(E_{2j}-z)(E_{2j-1}-z)} \\ = \prod\limits_{\nu_j=0}\frac{(\gamma_j-z)^2}{(E_{2j}-z)(E_{2j-1}-z)} \prod\limits_{\nu_j\neq0}\frac{\pi_j^2}{(E_{2j}-z)(E_{2j-1}-z)}, \end{aligned} \end{equation}

where in the last equality we used the fact that the fixed Dirichlet eigenvalues coincide with the double main eigenvalues $\zeta_j$, allowing them to cancel each other in the first fraction.

Definition 4.3. Let the sectionally meromorphic $2\times2$ matrix $\Phi(x,z)$ be defined as

(4.4)\begin{equation} \Phi(x,z)=\Psi(x,z)B(z)\mathrm{e}^{\text{i} zx\sigma_3}\,, \quad z\in\mathbb{C}\,. \end{equation}

Also, let

(4.5)\begin{equation} V(x,z) = \begin{cases} (-1)^{n + m(z)} \begin{pmatrix}{f^-(z)}/{f^+(z)} & 0 \\ 0 & {f^+(z)}/{f^-(z)}\end{pmatrix}\,, &z\in[E_{2n-2},E_{2n-1}]\,,\\ (-1)^{n + m(z)}\text{i}\mathrm{e}^{-2\text{i} zx\sigma_3}\sigma_1\,, &z\in[E_{2n-1},E_{2n}]\,. \end{cases} \end{equation}

The counting function $m(z)$ in (4.5) is defined as follows: If there are no $\gamma_k$ with $\nu_k=0$, let $m(z) = 0$. Alternatively, if there is at least one $\gamma_k$ with $\nu_k=0$, let $\gamma_*$ be the value of $\gamma_k$ closest to zero with $\nu_k=0$, and let

(4.6)\begin{equation} m(z) = \begin{cases} \big|\{\gamma_k:~\nu_k=0 ~\wedge~ \gamma_k \in (z,\gamma_*) \} \big|\,, &z \lt \gamma_*\,, \\[0.2ex] \big|\{\gamma_k:~\nu_k=0 ~\wedge~ \gamma_k \in [\gamma_*,z] \} \big|\,, &z\geq\gamma_*. \end{cases} \end{equation}

In other words, for $z\geq \gamma_*$, $m(z)$ counts the number of Dirichlet eigenvalues $\gamma_k$ with $\nu_k=0$ lying in the interval $[\gamma_*,z]$, whereas for $z \lt \gamma_*$, $m(z)$ counts the number of Dirichlet eigenvalues $\gamma_k$ with $\nu_k=0$ lying in the interval $(z,\gamma_*)$.

Definition 4.4. (Disk sequence)

Let $R=\mathop{\rm max}\nolimits_{g^-\leq k\leq g^+}\left(E_{2k}-E_{2k-1}\right)$, and let $D_j$ be the open disc of radius  $R$ centred at $c_j=\frac12(\zeta_{2j}+\zeta_{2j-1})$. We define the set $\mathcal{D}$ as

(4.7)\begin{equation} \mathcal{D}=\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^-}^{g^+}\bar{D}_{k}\right). \end{equation}

Condition 5 is needed to obtain the global boundedness required in the Liouville argument below. It allows us to apply the Phragmén–Lindelöf argument and obtain the required growth control. Combined with the asymptotics at infinity, this finally gives the uniqueness of the solution. See Lemma 4.6 and Theorem 4.7.

Proof. We prove each condition separately.

Condition 1. It follows from (3.6) that $\psi^\pm$ are meromorphic functions of $z$ for $z\in\mathbb{C}\setminus\Sigma(\mathcal{L})$, so $\Psi(x,z)$ is meromorphic in $\mathbb{C}\setminus\Sigma(\mathcal{L})$. From (4.1a), we can derive that $\Phi(x,z)$ could only be singular on $\{\mu_j\}\subset\mathbb{R}$ and $\{\zeta_j\}\subset\mathbb{R}$. Then $\Phi(x,z)$ is a holomorphic function of $z$ for $z\in\mathbb{C}\setminus\mathbb{R}$.

Condition 2. From the definition of the Bloch–Floquet solutions (3.6), we can derive directly that $\Psi_\pm(x,z)$ can only be singular at $\mu_j$. From (4.1a), we can also get that $\Phi_\pm(x,z)$ can only be singular at $\mu_j$ and $\zeta_j$. Next we will prove that $\Phi_\pm(x,z)$ cannot be singular at $\mu_j$ unless $\mu_j=E_k$ for some $k$. We will discuss three cases: $\nu_j=\pm1$ and $\zeta_{2j}=\zeta_{2j-1}=\mu_j$.

Suppose that $\nu_j=1$, which occurs only when $j=k$ for some $k$. From the definition of $\nu_j$ and Lemma 2.5, we have that $\tilde{y}_{11}(L,\mu_j)=\rho^{-1}(\mu_j)$, and $\rho^{-1}(z)-\tilde{y}_{11}(L,z)$ is holomorphic in a neighbourhood of $\mu_j$. For $z\in{\mathbb C}^+$, we have

(4.10)\begin{equation} \Phi_+ = ib(z)\big(\,\psi^-f^-\,,\,\psi^+f^+\,\big)\,\mathrm{e}^{\text{i} zx\sigma_3}. \end{equation}

For the 1st column, the zeros of $\rho^{-1}(z)-\tilde{y}_{11}(L,z)$ at $\mu_j$ can cancel the zeros of $\tilde{y}_{12}(L,z)$ at $\mu_j$. For the 2nd column, the zeros of $f^+$ cancel the singularities of $\psi^+$. Therefore, $\Phi_+$ is nonsingular at $\mu_j$. We can get the same conclusion for $z\in{\mathbb C}^-$ using the same method.

Suppose that $\nu_j=-1$, which occurs only when $j=k$ for some $k$. We also have $\rho(\mu_j)=\tilde{y}_{11}(L,\mu_j)$. So $\rho(z)-\tilde{y}_{11}(L,z)$ is holomorphic in $\mu_j$’s neighbourhood, and $\mu_j$ are the zeros of $\rho(z)-\tilde{y}_{11}(L,z)$. Using the same method as $\nu_j=1$, we see that $\Phi_\pm$ are nonsingular at $\mu_j$ when $\nu_j=-1$.

Suppose that $\zeta_{2j-1}=\zeta_{2j}$. In this case, $\rho(\mu_j)=\rho^{-1}(\mu_j)=\pm1$. Therefore, $\mu_j$ are the zeros of both $\rho(z)-\tilde{y}_{11}(L,z)$ and $\rho^{-1}(z)-\tilde{y}_{11}(L,z)$. The zeros of $\rho(z)-\tilde{y}_{11}(L,z)$ and $\rho^{-1}(z)-\tilde{y}_{11}(L,z)$ at $\mu_j$ can cancel the zeros of $\tilde{y}_{12}(L,z)$ at $\mu_j$. Moreover, the square root zeros of $\sqrt{f^0}$ at $\mu_j$ cancel the square root of $\sqrt[4]{\Delta^2-1}$ at $\zeta_{2j}=\zeta_{2j-1}$. So $\Phi_\pm$ are nonsingular at $\mu_j$.

Next we will prove that $\Phi_\pm(x,z)$ have at worst quartic root singularities on $\{E_k\}$. Suppose that $\nu_{k}=\pm1$, then the only singular contribution to $\Phi_\pm$ at $z\to E_{2k}$ or $E_{2k-1}$ is from the boundary values of $1/{\sqrt[4]{\Delta^2-1}}$ for $z\in\mathbb{R}$. Suppose that $\nu_{k}=0$, then $\mu_j=E_{2j}$ or $E_{2j-1}$. If $\mu_j=E_{2j}$, then the only singular contribution to $\Phi_\pm$ near $E_{2j-1}$ is from the boundary values of $1/{\sqrt[4]{\Delta^2-1}}$ for $z\in\mathbb{R}$. If $\mu_j=E_{2j-1}$, then the only singular contribution to $\Phi_\pm$ near $E_{2j}$ is from the boundary values of ${1}/{\sqrt[4]{\Delta^2-1}}$ for $z\in\mathbb{R}$.

Suppose that either $\mu_{j}=E_{2j}$ or $\mu_j=E_{2j-1}$. By Lemma 2.5, we have $\rho^{\pm1}(\mu_j)-\tilde{y}_{11}(L,x)=0$, since $\rho(\mu_j)=\pm1$ at $\mu_j$. As the singular behaviours of $\psi^\pm(x,z)$ are from $(\rho^{\pm1}(z)-\tilde{y}_{11}(L,z))/\tilde{y}_{12}(L,z)$, and the numerator of $\psi^\pm(x,z)$ comprises only holomorphic functions and the square root of a holomorphic function, the zeros of the numerator at $\mu_j$ must have an order of at least ${\textstyle\frac12}$. Since $\tilde{y}_{12}(L,z)$ has a simple zero at $\mu_j$, $\psi_\pm^\pm(x,z)$ must then have at worst square root singularities at $\mu_j$. The boundary values of ${1}/{\sqrt[4]{\Delta^2-1}}$ for $z\in\mathbb{R}$ have a quartic root singularity at $\mu_j$ and the boundary values of $\sqrt{f^0}$ have a square root zero at $\mu_j$. Hence, $\Phi$ has at worst a quartic root singularity at $\mu_j$.

Condition 3. Recalling the definition of $\rho(z)$ in Section 2, we have that the discrepancy between the boundary values of $\sqrt{\Delta^2-1}$ on the branch cut  $\Sigma(\mathcal{L})$ is

(4.11)\begin{equation} \rho_+(z)=\rho_-^{-1}(z),\qquad z\in\Sigma(\mathcal{L}). \end{equation}

Therefore,

(4.12)\begin{equation} \Psi_+(x,z)=\Psi_-(x,z)\,,\quad z\in\Sigma(\mathcal{L})\,, \end{equation}

(4.13)\begin{equation} \Psi_+(x,z)=\Psi_-(x,z)\sigma_1\,,\quad z\in\mathbb{R}\setminus\Sigma(\mathcal{L})\,, \end{equation}

where $\sigma_1$ is the 1st Pauli matrix (cf. Appendix A.1). As a result, we see that the jump matrix $V(x,z)$ takes the form

(4.14)\begin{equation} V(x,z) = \begin{cases}\mathrm{e}^{-\text{i} zx\sigma_3}(B_-)^{-1}B_+\mathrm{e}^{\text{i} zx\sigma_3}, &z\in\Sigma(\mathcal{L})\,, \\ \mathrm{e}^{-\text{i} zx\sigma_3}(B_-)^{-1}\sigma_1B_+\mathrm{e}^{\text{i} zx\sigma_3}, &z\in\mathbb{R}\setminus\Sigma(\mathcal{L})\,. \end{cases} \end{equation}

Next, we show that (4.14) reduces to (4.5). To do this, we need to compute the product $(B_-)^{-1}B_+$. Since the functions $f^\pm(z)$ appearing in (4.1a) are holomorphic, it is sufficient to compute the jump of the scalar factor $\sqrt{f^0(z)}/\sqrt[4]{\Delta^2(z)-1}$. To this end, recall that one can equivalently express this ratio as (4.2a). It is convenient to consider separately the numerator and denominator of the right hand side of (4.2a) and to compute their jumps individually. We therefore define

(4.15a)\begin{equation} p(z) = \left(\prod_{k=g^-\atop\nu_{k}=0 }^{g^+}\bigg(1 - \frac{z}{\gamma_k}\bigg)\,\mathrm{e}^{z/\gamma_k}\right)^{\frac{1}{2}}\,, \end{equation}

(4.15b)\begin{equation} h(z) = \left(\prod_{k=g^- }^{g^+}{\frac{1}{E_{2k-1}E_{2k}}(E_{2k-1}-z)(E_{2k}-z)\mathrm{e}^{z\big(\frac{1}{E_{2k-1}}+\frac{1}{E_{2k}}\big)}} \right)^{\frac{1}{4}}\,, \end{equation}

so that $\sqrt{f^0(z)}/\sqrt[4]{\Delta^2(z)-1} = \mathrm{e}^{{\text{i}\pi}/{4}} p(z)/(\sqrt{2}h(z))$ by (4.2a). We choose the branch cut for the square root in (4.15a) consistently with the definition of the counting function $m(z)$ in (4.6). Namely, if there are no $\gamma_k$ with $\nu_k=0$, $p(z)\equiv1$ and there is no branch cut. If there is at least one $\gamma_k$ with $\nu_k=0$, recall that $\gamma_*$ is the value of $\gamma_k$ closest to zero with $\nu_k=0$. Then we take the branch cut for $p(z)$ to be the subset of the real $z$-axis for which $m(z)$ is odd. With this choice, we have

(4.16a)\begin{equation} \frac{p_-(z)}{p_+(z)} = (-1)^{m(z)}. \qquad z\in\mathbb{R}, \end{equation}

Moreover, the choice of branch cut for the fourth root given after Definition 4.1c implies

(4.16b)\begin{equation} \frac{h_-(z)}{h_+(z)} = \begin{cases} (-1)^{n-1}\text{i} , &z\in [E_{2n-2},E_{2n-1}]\subset\Sigma(\mathcal{L}) \,,\\ (-1)^n , &z\in [E_{2n-1},E_{2n}]\subset\mathbb{R}\setminus\Sigma(\mathcal{L})\,. \end{cases} \end{equation}

Inserting (4.16a) and (4.16b) in (4.14), one now obtains (4.5).

Condition 4. Using the asymptotic behaviour of $\psi^\pm(x,z)$ in Proposition 3.6, we have

(4.17)\begin{equation} \Psi(x,z)= (U + O(1/z))\,\mathrm{e}^{-\text{i} zx\sigma_3},\qquad z\to\infty,~z\in\mathbb{C}. \end{equation}

Therefore,

(4.18)\begin{equation} U^{-1}\Psi(x,z)\mathrm{e}^{\text{i} zx\sigma_3}=I+O(1/z),\qquad z\to\infty \end{equation}

for $z\in\mathbb{C}$, implying (4.9).

Condition 5. Let $y_{ij}$, $\phi_{ij}$ and $\psi_{ij}$ denote the $(i,j)$th entries of the matrices $Y$, $\Phi$ and $\Psi$, respectively. We will prove that

(4.19)\begin{equation} \phi_{ij}\in\mathcal{A}_2\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup\overline{D}_{k}\right)\right),\qquad \psi_{ij}\in\mathcal{A}_2\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup\overline{D}_{k}\right)\right), \end{equation}

where the function class $\mathcal{A}_p(\Omega)$ is defined in Appendix A.6. In Appendix A.5, we show that the entire functions $y_{ij}(x,z)$ have order at most  $1$. Thus, by Proposition A.11, we have

(4.20)\begin{equation} \tilde{y}_{ij}(x,z)\in\mathcal{A}_2(\mathbb{C}). \end{equation}

Next, since

(4.21)\begin{equation} |\rho^{\pm1}(z)|\leq|\Delta(z)|+\sqrt{|\Delta^2(z)-1|}\leq1+|\Delta(z)|, \end{equation}

and $\Delta(z)$ is an entire function of order $1$, we also have

(4.22)\begin{equation} \rho^{\pm1}(z)\in\mathcal{A}_2(\mathbb{C}\setminus\mathbb{R}). \end{equation}

Applying Lemma A.17 to (3.11a), we establish that

(4.23)\begin{equation} \tilde{y}_{12}(L,z)^{-1}\in\mathcal{A}_2\left(\mathbb{C}\setminus\bigcup_{j\in\mathbb{Z}}\overline{D}_j\right). \end{equation}

Therefore, using Propositions A.10 and A.11, we conclude that

(4.24)\begin{equation} \psi_{ij}\in\mathcal{A}_2(\mathcal{D}). \end{equation}

Next, note that the smallest numbers $a_\pm$ and $a_0$ so that

(4.25)\begin{equation} \sum_{\nu_j=\pm1}\mu_j^{-a_\pm},\sum_{\nu_j=0}\mu_j^{-a_0} \end{equation}

converge are such that $a_\pm$, $a_0\leq 1$. So $f^\pm$ and $f^0$ are entire functions with orders at most $1$. Hence,

(4.26)\begin{equation} f^\pm,f^0\in\mathcal{A}_2(\mathbb{C}). \end{equation}

Applying Lemma A.17 to (3.11b) and (3.11c), we conclude that

(4.27)\begin{equation} \frac{1}{\Delta^2-1}\in\mathcal{A}_2\left(\mathbb{C}\setminus\bigcup_{j}\overline{D}_j\right). \end{equation}

Combining both (4.26) and (4.27), we have

(4.28)\begin{equation} g^\pm(z):=\frac{f^0(z)f^\pm(z)}{\Delta^2(z)-1}\in\mathcal{A}_2\left(\mathbb{C}\setminus\bigcup_{j}\overline{D}_j\right). \end{equation}

If $\mu_j\notin\{E_k\}$, then we can easily prove that $g^\pm(z)$ is analytic at $\mu_j$, which coincides with the centre of $D_j$. Therefore, $g^\pm(z)$ are analytic in such a $D_j$. Finally, using Lemma A.17, we have

(4.29)\begin{equation} g^\pm(z)\in\mathcal{A}_2\left(\mathbb{C}\setminus\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right). \end{equation}

Recalling the definition (4.4) of $\Phi(x,z)$, this completes the proof of condition 5.

Proof. We first prove that the particular solution to RHP 4.1 given by (4.4) has unit determinant. We have

(4.30)\begin{equation} \det\Phi(x,z)=\det B(z)\det\Psi(x,z) = \mp \frac{\tilde{y}_{12}(L,z)}{2\sqrt{\Delta^2-1}}\det\Psi_\pm\,, \qquad z\in\mathbb{C}^\pm\,. \end{equation}

Moreover,

(4.31)\begin{equation} \det\Psi(x,z) = \det\Psi(0,z) = \pm \frac{\rho(z)-\rho^{-1}(z)}{\tilde{y}_{12}(L,z)}= \mp \frac{2\sqrt{\Delta^2-1}}{\tilde{y}_{12}(L,z)}\,,\qquad z\in\mathbb{C}^\pm\,. \end{equation}

Thus, we conclude that $\det\Phi(x,z)=1$ for all $x$ and $z$. Comparing $\det\Phi\equiv1$ with the asymptotics of $\Phi(x,z)$ in (4.9), we have

(4.32)\begin{equation} \det\left(U\,B(z)\right)=1+O(1/z)\,, \end{equation}

as $z\to\infty$ for $z\in\mathbb{C}$.

Next we show that all solutions of the RHP 4.1 have unit determinant. Let $\tilde{\Phi}(x,z)$ be an arbitrary solution to RHP 4.1. Define $d_\Phi(x,z) =\det\tilde{\Phi}(x,z)$, which is holomorphic in $\mathbb{C}\setminus\mathbb{R}$. As $\det V(x,z)=1$, we can derive that for $z\in\mathbb{R}\setminus\{E_k\}$

(4.33)\begin{equation} d_{\Phi,+}(x,z)=\det\tilde{\Phi}_+(x,z)=\det(V(x,z)\tilde{\Phi}_-(x,z))=\det\tilde{\Phi}_-(x,z)=d_{\Phi,-}(x,z)\,. \end{equation}

As $\tilde{\Phi}$ also satisfies condition 2 in RHP 4.1, $d_\Phi(x,z)$ has at worst radical root singularities at $E_k$. Nevertheless, isolated singularities of a holomorphic function must have at least order 1. Thus, $d_\Phi$ can be extended to an entire function.

To get $d_\Phi\equiv1$, we still need to prove that $d_\Phi$ is bounded. First, we prove that $d_\Phi\in\mathcal{A}_2(\mathbb{C})$. From condition 5 of RHP 4.1, we have

(4.34)\begin{equation} \phi_{ij},\tilde{\phi}_{ij}\in\mathcal{A}_2\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right)\right)\,. \end{equation}

Hence, we can conclude that

(4.35)\begin{equation} d_\Phi\in\mathcal{A}_2\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right)\right)\,. \end{equation}

Since $d_{\Phi}$ is an entire function, by continuity, we have

(4.36)\begin{equation} d_\Phi\in\mathcal{A}_2\left(\mathbb{C}\setminus\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right)\,. \end{equation}

By Proposition A.14, we have that $d_\Phi\in\mathcal{A}_2(\mathbb{C})$ . Comparison of (4.9) with (4.32) implies that

(4.37)\begin{equation} d_\Phi(x,z)=1+O(1/z),\qquad z\to\infty\,. \end{equation}

But since $d_\Phi(x,z)$ is entire, the asymptotic behaviour in (4.37) implies that $d_\Phi(x,z)$ is bounded for all $z\in\mathbb{C}$. So, by Liouville’s theorem, $d_\Phi(x,z)$ is constant and therefore $d_\Phi(x,z)\equiv1~\forall z\in\mathbb{C}$.

Proof. For fixed $x\in\mathbb{R}$, let $\Phi$ and $\tilde{\Phi}$ be the solutions of RHP 4.1. Define

(4.38)\begin{equation} R(x,z):=\Phi(x,z)\tilde{\Phi}^{-1}(x,z)\,. \end{equation}

To establish the uniqueness of the solution to RHP 4.1, it suffices to prove that $\det R(x,z)\equiv 1$.

As $\det\tilde{\Phi}=1$, we can express $\tilde{\Phi}^{-1}$ as

(4.39)\begin{equation} \tilde{\Phi}^{-1}(x,z)=\begin{pmatrix}\tilde{\phi}_{22}(x,z) &-\tilde{\phi}_{12}(x,z)\\-\tilde{\phi}_{21}(x,z)&\tilde{\phi}_{11}(x,z)\end{pmatrix}\,, \end{equation}

where $\tilde{\phi}_{ij}(x,z)$ is the $ij$-element of $\tilde{\Phi}(x,z)$. It follows from condition 3 of RHP 4.1 that $R(x,z)$ has no jump, so it can be extended to a holomorphic function on $\mathbb{C}\setminus\{E_k\}$. Additionally, since the elements of $\Phi$ and $\tilde{\Phi}$ have at worst quartic root singularities at $E_k$, $R(x,z)$ has at worst square root singularities at $E_k$. Nevertheless, isolated singularities of a holomorphic function must have at least order 1. So $R(x,z)$ is an entire function of $z$.

Next, we will show that $r_{ij}\in\mathcal{A}_2(\mathbb{C})$. Since $\Phi$ and $\tilde{\Phi}$ solve RHP 4.1, we have that

(4.40)\begin{equation} \phi_{ij},\tilde{\phi}_{ij}\in\mathcal{A}_2\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right)\right)\,. \end{equation}

Hence, we conclude that

(4.41)\begin{equation} r_{ij}\in\mathcal{A}_2\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right)\right)\,. \end{equation}

Since $R(x,z)$ is an entire function, by continuity, we have

(4.42)\begin{equation} r_{ij}\in\mathcal{A}_2\left(\mathbb{C}\setminus\bigcup_{k=g^-}^{g^+}\overline{D}_{k}\right)\,. \end{equation}

By Proposition A.14, we can derive that $r_{ij}\in\mathcal{A}_2(\mathbb{C})$.

The asymptotic behaviour of both of the matrices $\Phi$ and $\tilde{\Phi}$ as $z\to\infty$ is given by (4.9). Therefore we can calculate the asymptotic behaviour of $\tilde{\Phi}^{-1}(x,z)$ as

(4.43)\begin{equation} \tilde{\Phi}^{-1}(x,z)=B^{-1}(z)(I+O(1/z))U^{-1}\,,\qquad z\to\infty\,. \end{equation}

We can then easily get the asymptotic behaviour of $R(x,z)$ as

(4.44)\begin{equation} R(x,z) = U\,(I+O(1/z))\,U^{-1} = I+O(1/z)\,,\qquad z\to\infty\,. \end{equation}

Again, since $R(x,z)$ is entire, the asymptotic behaviour (4.44) indicates that $r_{ij}$s are bounded for $z\in\mathbb{C}$. Using Liouville’s theorem, we have that $R(x,z)$ is constant. The asymptotic behaviour of $R(x,z)$ in (4.44) implies that $R(x,z)\equiv I~\forall z\in\mathbb{C}$. Hence, the solution of RHP 4.1 is unique.

Corollary 4.8. The potential matrix $Q(x)$ of the Dirac equation (1.1) is obtained from any solution $\Phi(x,z)$ of RHP 4.1 by

(4.45)\begin{equation} Q(x) = \lim_{z\to\infty} \text{i} z [\sigma_3,U^{-1}\Phi(x,z)B^{-1}(z)]\,, \end{equation}

with $U$ as in (2.15).

5. IVP for the defocusing NLS equation with periodic BC

We now show how the results of the previous sections can be used to solve the IVP for the defocusing NLS equation with periodic BC,

(5.1)\begin{equation} q(x+L,t)=q(x,t)\,. \end{equation}

Specifically, we construct a Riemann–Hilbert characterization of periodic solutions of the defocusing NLS equation with infinite-gap initial conditions. To this end, recall that the defocusing NLS equation (1.6) is the compatibility condition (or zero curvature condition) of the Lax pair (1.7a and 1.7b). An equivalent formulation of the zero-curvature condition is the Lax equation

(5.2a)\begin{equation} \mathcal{L}_t=[\mathcal{A},\mathcal{L}]\,, \end{equation}

with $\mathcal{L}$ as in (1.2) and

(5.2b)\begin{equation} \mathcal{A}=2\text{i}\sigma_3\partial_x^2+2\text{i} Q\sigma_3\partial_x-\text{i}(Q^2-Q_x)\sigma_3\,. \end{equation}

It is well known that, if the potential $q$ of (1.1) evolves according to the NLS equation (1.6), the Lax spectrum $\Sigma(\mathcal{L})$ of the ZS problem is invariant in time (which is easily proven by showing that the Floquet discriminant $\Delta(z)$ is time-independent). On the other hand, the movable Dirichlet eigenvalues are time dependent, and so are the Bloch–Floquet eigenfunctions. Thus, in order to construct an effective time-dependent RHP, one must determine the time evolution of the Dirichlet eigenvalues and of the Bloch–Floquet eigenfunctions. We turn to these tasks next.

Proposition 5.1. For all $n\in\mathbb{Z}$, the Dirichlet eigenvalues $\mu_n(t)$ satisfy the ODE

(5.3a)\begin{equation} \frac{\partial{\mu_{n}(t)}}{\partial{t}}=\nu_n(t)c_1(\mu_{n}(t))\frac{\rho(\mu_n(t))-\rho^{-1}(\mu_n(t))}{\tilde{y}'_{12}(L,t,\mu_{n}(t))}. \end{equation}

where

(5.3b)\begin{equation} c_1(z)=-2z^2-\text{i} z(q^*(0,t)-q(0,t))-|q(0,t)|^2-\frac{1}{2}(q_x(0,t)+q_x^*(0,t))\,. \end{equation}

Proof. Recall that the Dirichlet eigenvalues are poles of the modified Bloch–Floquet solutions, which, in turn, are defined via (2.14). The transformation (2.14) yields the modified Lax equation

(5.4a)\begin{equation} \tilde{\mathcal{L}}_t=[\tilde{\mathcal{A}},\tilde{\mathcal{L}}]\,, \end{equation}

with

(5.4b)\begin{equation} \tilde{\mathcal{L}}= U\mathcal{L} U^{-1}\,, \qquad \tilde{\mathcal{A}} = U\mathcal{A}U^{-1}\,. \end{equation}

Recall that $\tilde{y}_2$ is a Bloch–Floquet solution of $\tilde{\mathcal{L}}$, i.e., $\tilde{\mathcal{L}}\tilde{y}_2=z\tilde{y}_2$. Differentiating both sides with respect to $t$, we have $\tilde{\mathcal{L}}_t\tilde{y}_2+\tilde{\mathcal{L}}\tilde{y}_{2,t}=z\tilde{y}_{2,t}$, which, combined with the modified Lax equation (5.4a) implies

(5.5)\begin{equation} \tilde{\mathcal{L}}(\tilde{y}_{2,t}-\tilde{\mathcal{A}}\tilde{y}_2)=z(\tilde{y}_{2,t}-\tilde{\mathcal{A}}\tilde{y}_2)\,. \end{equation}

Therefore, for fixed $t$ and $z$, there exist constants $c_1$ and $c_2$ such that

(5.6)\begin{equation} \tilde{y}_{2,t}-\tilde{\mathcal{A}}\tilde{y}_2=c_1\tilde{y}_1+c_2\tilde{y}_2\,. \end{equation}

The normalization condition of $\tilde{Y}(x,z)$ at $x=0$ gives that $\tilde{y}_{12,t}(0,t,z)=\tilde{y}_{22,t}(0,t,z)=0$, so we obtain

(5.7)\begin{equation} (c_1,c_2)^T = -\tilde{\mathcal{A}}\tilde{y}_{2}(0,t,z)\,. \end{equation}

Moreover, the modified equation (2.16) also implies

(5.8)\begin{equation} \tilde{y}_{2,xx}(x,t,z)=-\text{i} zU\sigma_3U^{-1}\tilde{y}_{2,x}(x,t,z)+UQ(x,t)U^{-1}\tilde{y}_{2,x}(x,t,z)+UQ_x(x,t)U^{-1}\tilde{y}_2(x,t,z)\,, \end{equation}

and therefore

(5.9)\begin{equation} \tilde{\mathcal{A}}\tilde{y}_2(x,t,z) = -2\text{i} z^2U\sigma_3U^{-1}\tilde{y}_2(x,t,z)+2zUQU^{-1}\tilde{y}_2(x,t,z)-\text{i} U(Q^2+Q_x)\sigma_3U^{-1}\tilde{y}_2(x,t,z)\,. \end{equation}

Evaluation at $x=0$ yields

(5.10a)\begin{equation} c_1=-2z^2-\text{i} z(q^*(0,t)-q(0,t))-|q(0,t)|^2-{\textstyle\frac12}(q_x(0,t)+q_x^*(0,t))\,, \end{equation}

(5.10b)\begin{equation} c_2=z(q^*(0,t)+q(0,t))-\frac{\text{i}}{2}(q_x(0,t)-q_x^*(0,t))\,. \end{equation}

Plugging (5.9), (5.10a) and (5.10b) into (5.6) and evaluating (5.6) at $x=L$ and $z=\mu_n(t)$, we get

(5.11)\begin{align} \tilde{y}_{12,t}(L,t,\mu_n(t)) &= \big(2\mu_n(t)^2+\text{i}\mu_n(t)(q^*(0,t)-q(0,t))+|q(0,t)|^2+{\textstyle\frac12}(q_x(0,t)+q_x^*(0,t))\big)\cr & \qquad\big(\tilde{y}_{22}(L,t,\mu_n(t))-\tilde{y}_{11}(L,t,\mu_n(t))\big)\,. \end{align}

At the same time, differentiating the expansion (3.11a) in $t$ we have

(5.12)\begin{equation} \tilde{y}_{12,t}(L,t,z)=\mathrm{e}^{A_1z+B_1}\sum_{n\in\mathbb{Z}}\left[\frac{z\mu_{n,t}(t)}{\mu_n^2(t)}\mathrm{e}^{\frac{z}{\mu_n(t)}}\frac{z}{\mu_n(t)}\prod_{j\neq n}\left(1-\frac{z}{\mu_j(t)}\right)\mathrm{e}^{\frac{z}{\mu_j(t)}}\right]\,, \end{equation}

whereas differentiating (3.11a) in $z$ we have

(5.13)\begin{equation} \tilde{y}'_{12}(L,t,z)=A_1\mathrm{e}^{A_1z+B_1}\prod_{j\in\mathbb{Z}}\left(1-\frac{z}{\mu_j(t)}\right)\mathrm{e}^{\frac{z}{\mu_j(t)}}+\mathrm{e}^{A_1z+B_1}\sum_{n\in\mathbb{Z}}\left[-\frac{z}{\mu_n^2(t)}\mathrm{e}^{\frac{z}{\mu_n(t)}}\prod_{j\neq n}\left(1-\frac{z}{\mu_j(t)}\right)\mathrm{e}^{\frac{z}{\mu_j(t)}}\right]\,. \end{equation}

Evaluating (5.12) and (5.13) at $z=\mu_n(t)$ gives $\tilde{y}_{12,t}(L,t,\mu_n(t))=-\mu_{n,t}(t)\tilde{y}'_{12}(L,t,\mu_{n}(t))$. Therefore, we can express $\mu_{n,t}(t)$ as:

\begin{equation*} \mu_{n,t}(t) = -\frac{\tilde{y}_{12,t}(L,t,\mu_n(t))}{\tilde{y}'_{12}(L,t,\mu_{n}(t))} = \frac{c_1(\mu_{n}(t))(\tilde{y}_{22}(L,t,\mu_n(t))-\tilde{y}_{11}(L,t,\mu_n(t)))}{\tilde{y}'_{12}(L,t,\mu_{n}(t))}\,, \nonumber \end{equation*}

which yields (5.3a) upon substituting $c_1$ via (5.3b).

Proposition 5.3. The time-dependent Bloch–Floquet solutions $\psi^\pm(x,t,z)$ satisfy the ODEs

(5.14)\begin{equation} \psi^\pm_t(x,t,z)+\alpha^\pm(t,z)\psi^\pm(x,t,z)=\tilde{\mathcal{A}}\psi^\pm(x,t,z) \end{equation}

where

(5.15)\begin{equation} \begin{split} \alpha^\pm(t,z)=&z(q^*(0,t)+q(0,t))-\frac{\text{i}}{2}(q_x(0,t)-q^*_x(0,t))\\ &+\Big(2z^2+\text{i} z\big(q^*(0,t)-q(0,t)\big)+|q(0,t)|^2+{\textstyle\frac12} q_x(0,t)+{\textstyle\frac12} q_x^*(0,t)\Big) \psi^\pm_2(0,t,z)\,. \end{split} \end{equation}

As a result, for each $s \gt 0$, as $z\rightarrow\infty$, $z\in\mathbb{C}$, we have

(5.16a)\begin{equation} \alpha^\pm=\pm2\text{i} z^2+O(1/z),\qquad z\in\mathbb{C}^+, \end{equation}

(5.16b)\begin{equation} \alpha^\pm=\mp2\text{i} z^2+O(1/z),\qquad z\in\mathbb{C}^-. \end{equation}

Furthermore, if $\nu_n(t)=1$ then $\alpha^+(t,z)$ has a simple pole at $\mu_{n}(t)$ with residue $-\mu_{n,t}(t)$ and if $\nu_{n}=-1$ then $\alpha^-(t,z)$ has a simple pole at $\mu_{n}(t)$ with residue $-\mu_{n,t}(t)$.

Proof. For each $t$, we can factor the normalized Bloch–Floquet solutions $\psi^\pm(x,t,z)$ as

(5.17)\begin{equation} \psi^+(x,t,z)=p^+(x,t,z)\rho^{x/L}(z),\qquad \psi^-(x,t,z)=p^-(x,t,z)\rho^{-x/L}(z)\,. \end{equation}

Differentiating $(\tilde{\mathcal{L}}-z)\psi^+=0$ with respect to $t$ and using the modified Lax equation imply that

(5.18)\begin{equation} \tilde{\mathcal{L}}_t\psi^++\tilde{\mathcal{L}}\psi^+_t-z\psi^+_t=[\tilde{\mathcal{A}},\tilde{\mathcal{L}}]\psi^++\tilde{\mathcal{L}}\psi^+_t-z\psi_t^+=(\tilde{\mathcal{L}}-z)(\psi^+_t-\tilde{\mathcal{A}}\psi^+)=0. \end{equation}

So $\psi_t^+-\tilde{\mathcal{A}}\psi^+$ solves (2.16) for all $t \gt 0$. Therefore, it can be written as a linear combination of $\psi^\pm$:

(5.19)\begin{equation} p^+_t\rho^{x/L}(z) - \tilde{\mathcal{A}}\,p^+\rho^{x/L}(z) = - \alpha^+(t,z)p^+\rho^{x/L}(z) - \beta^+(t,z)p^-\rho^{-x/L}(z), \end{equation}

where $\alpha^+(t,z)$ and $\beta^+(t,z)$ are some undetermined functions independent of $x$. Recall that $|\rho(z)| \lt 1~\forall z\in\mathbb{C}\setminus\Sigma(\mathcal{L})$. Evaluating the LHS and RHS of the above equation as $x\to\infty$, every term decays exponentially except the last one. Therefore, in order for the equation to hold, one needs $\beta^+(t,z)=0$. Hence $\psi^+$ solves (5.14) with the plus sign. An analogous argument, but now taking the limit as $x\to-\infty$, shows that $\psi^-$ solves (5.14) with the minus sign, for some $\alpha^-(t,z)$.

Next we derive (5.15), (5.16a) and (5.16b). To do so, we evaluate the first component of (5.14) at $x=0$. Since $\psi^\pm_1(0,t,z)=1$ for all time, we have $\psi^\pm_{1,t}(0,t,z)=0$. Therefore (5.14) yields

(5.20)\begin{equation} \alpha^\pm(t,z) = (\mathcal{A}\psi^\pm)_1(0,t,z)\,. \end{equation}

From (5.9) we have that

(5.21)\begin{equation} \tilde{\mathcal{A}}v = -2\text{i} z^2U\sigma_3U^{-1}v + 2zUQU^{-1}v-\text{i} U(Q^2+Q_x)\sigma_3U^{-1}v\,. \end{equation}

Explicitly, the first component of the above equation is

(5.22)\begin{equation} (\tilde{\mathcal{A}}v)_1 = 2z^2\,v_2 + z\,\big[(q^* + q)\,v_1 + \text{i}(q^*-q)v_2 \big] \\ - \frac{i}{2} \big[ (q^*_x - q_x)\,v_1 + \text{i}(2|q|^2 + q^*_x + q_x ) v_2 \big]\,. \end{equation}

Letting $v(x,t,z) = \psi^\pm(x,t,z)$, evaluating (5.22) at $x=0$ recalling that (3.6) evaluated at $x=0$ yields $\psi^\pm_1(0,t,z)=1$ and $\psi^\pm_2(0,t,z) = (\rho^{\pm1}-\tilde{y}_{11}(L,t,z))/\tilde{y}_{12}(L,t,z)$, (5.20) then yields (5.15).

We now introduce scalar factors that will be used to define modified time-dependent Bloch–Floquet solutions. Let $e^\pm(t,z)$ be solutions of the differential equation $e^\pm_t(t,z)=\alpha^\pm(t,z)e^\pm(t,z)$ with the initial condition $e^\pm(0,z)=1$, i.e.,

(5.23)\begin{equation} e^\pm(t,z)=\exp\left(\int_0^t\alpha^\pm(\tau,z)\mathrm{d}\tau\right)\,. \end{equation}

Define

(5.24)\begin{equation} \check{\psi}^\pm(x,t,z)=\psi^\pm(x,t,z)e^\pm(t,z)\,. \end{equation}

Then $\check{\psi}^\pm(x,t,z)$ satisfy the system of $\tilde{\mathcal{L}}\check\psi^\pm = z\check\psi^\pm$ and $\check{\psi}^\pm_t=\tilde{\mathcal{A}}\check{\psi}^\pm$, for which the modified Lax equation (5.4a) is the compatibility condition.

Proof. Note that since $\alpha^\pm$ are holomorphic for $z\in{\mathbb C}\setminus\mathbb{R}$ for all $t$, it is obvious that $e^\pm$ are holomorphic for $z\in{\mathbb C}\setminus\mathbb{R}$. In the process of proving Condition 2, we will prove that $e^\pm$ can be extended to meromorphic functions for $z\in{\mathbb C}\setminus\Sigma(\mathcal{L})$, thus also proving Condition 1.

Condition 2. Suppose that at $t=0$, $\mu_n\in(\zeta_{2n-1},\zeta_{2n})$ and $\nu_n=1$. For $t$ small enough, we have $\mu_n(t)\in(\zeta_{2n-1},\zeta_{2n})$, then from Proposition 5.3, $\mu_n$ is a simple pole of $\alpha^+$, so for $z$ near $\mu_n(t)$, $e^+$ could be written as

(5.26)\begin{equation} e^+(t,z)=\exp\left(-\int_{0}^t\frac{\mu_{n\tau}(\tau)\mathrm{d}\tau}{z-\mu_n(\tau)}\right)e^+_{\rm reg}(t,z)\,, \end{equation}

where $e^+_{\rm reg}(t,z)$ is holomorphic and non-zero in an open set that encompasses the image of $\mu_n(t)$ for some small time window. We can rewrite the above function in terms of $\mu_n$

(5.27)\begin{equation} -\int_{0}^t\frac{\mu_{n\tau}(\tau)\mathrm{d}\tau}{z-\mu_n(\tau)}=\int_{\mu_n(0)}^{\mu_n(t)}\frac{\mathrm{d}\mu_n}{\mu_n-z}=\log(\mu_n(t)-z)-\log(\mu_n(0)-z)\,, \end{equation}

which implies

(5.28)\begin{equation} e^+(t,z)=\frac{z-\mu_n(t)}{z-\mu_n(0)}e^+_{\rm reg}(t,z)\,. \end{equation}

The proof for $e^-$ is entirely analogous. While the Dirichlet eigenvalues $\mu_n(t)$ can also be equal to one of the endpoints $\zeta_{2n-1}$ or $\zeta_{2n}$ of the gaps. Here we apply a time translation so that $\mu_n(0)=\zeta_{2n}$. Without loss of generality, we assume that $\mu_n(t)$ is approaching $\zeta_{2n}$ and that $\mu_n(t)$ is a pole of $\alpha^+$. Then a local coordinate $w$ can be defined by

(5.29)\begin{equation} w(z)=\begin{cases} \rho(z)-\rho^{-1}(z)\,,&z\in{\mathbb C}^+,\quad z~\text{near}~\zeta_{2n}\,, \\ \rho^{-1}(z)-\rho(z)\,,&z\in{\mathbb C}^-,\quad z~\text{near}~\zeta_{2n}\,. \end{cases} \end{equation}

This is an analytic invertible transformation from a neighbourhood of $w = 0$ to the $z$ plane. In the $w$ plane, the single function

(5.30)\begin{equation} \alpha(t,w) = \alpha^\pm(t,z(w))\,,\qquad z(w)\in{\mathbb C}^\pm\,, \end{equation}

is meromorphic in a neighbourhood of $w=0$, with a single simple pole at $w(\mu_n)$. Then

(5.31)\begin{equation} e(t,w) = e^\pm(t,z(w))\,,\qquad z(w)\in{\mathbb C}^\pm\,, \end{equation}

is related to $\alpha$ by

(5.32)\begin{equation} e(t,w)=\exp\bigg(\int_0^t\alpha(\tau,w)\mathrm{d} \tau\bigg). \end{equation}

The derivative of $w(z)$ is

(5.33)\begin{equation} w'(z)= \pm\frac{\rho'(z)}{\rho(z)}\Delta(z)\,,\quad z\in{\mathbb C}^\pm, \end{equation}

which implies that

(5.34)\begin{equation} w_t(\mu_n(t))=\nu_n(t)\frac{\rho'(z)}{\rho(z)}\Delta(z)|_{z=z(w(\mu_n(t)))}\mu_{n,t}(t). \end{equation}

Then by differentiating (2.8), we get $\rho'(z)(\rho(z)-\Delta(z))=\Delta'(z)\rho(z)$, which gives the logarithmic derivative of $\rho(z)$

(5.35)\begin{equation} \frac{\rho'(z)}{\rho(z)}=-\frac{\Delta'(z)}{\sqrt{\Delta^2(z)-1}}. \end{equation}

Using (5.3) and (5.35) in (5.34) indicates

(5.36)\begin{equation} w_t(\mu_n(t))=\frac{\Delta(z)\Delta'(z)}{\tilde{y}'_{12}(L,t,z)}\bigg|_{z=z(w(\mu_n(t))}c_1(\mu_n(t)), \end{equation}

where $c_1(z)$ is defined in (5.3b).

Since $\mu_n(t)$ is a pole of $\alpha$, a straightforward computation gives that

(5.37)\begin{equation} \alpha(w)=\frac{-w_t(\mu_n(t))}{w-w(\mu_n(t))}+\text{ reg. near }\zeta_{2n}. \end{equation}

Now we compute the local behaviour of $e(w)$ for $w$ near $0$ as

(5.38)\begin{equation} e(w)=\exp\left(-\int_0^t\frac{-w_t(\mu_n(\tau))}{w-w(\mu_n(\tau))}\mathrm{d}\tau\right)e_{\rm reg}(w)=\frac{w-w(\mu_n(t))}{w-w(\mu_n(0))}e_{\rm reg}(w), \end{equation}

where $e_{\rm reg}$ is holomorphic and non-zero near $w=0$. So we complete the proof of Condition 2.

Condition 3. Condition 3 follows from considering the boundary behaviour of $\rho(z)$.

Condition 4. By (4.11), $\rho_+(z)=\rho_-^{-1}(z)$ for $z\in\Sigma(L)$, while $\tilde y_{11}(L,t,z)$ and $\tilde y_{12}(L,t,z)$ are entire functions of $z$. Hence, $\psi_2^\pm(0,t,z)=\big(\rho^{\pm1}(z)-\tilde y_{11}(L,t,z)\big)/\tilde y_{12}(L,t,z)$ satisfy $\psi^\pm_{2,+}(0,t,z)=\psi^\mp_{2,-}(0,t,z)$ for $z$ in the interior of $\Sigma(L)$, and (5.15) then gives $\alpha^\pm_+(t,z)=\alpha^\mp_-(t,z)$ there. Integrating in $t$ and using (5.23) yields $e^\pm_+(t,z)=e^\mp_-(t,z)$ for $z\in\Sigma(L)$.

Condition 5. By Proposition 5.3, we have that as $z\rightarrow\infty$, for $z\in\overline{\mathbb{C}^+}$, $\pm{\alpha^\pm(t,z)}/{(2\text{i} z^2)}$ are continuous function of $z$ converging to $1$, while for $z\in\overline{\mathbb{C}^-}$, $\mp{\alpha^\pm(t,z)}/{(2\text{i} z^2)}$ are continuous function of $z$ converging to $1$. By the dominated convergence theorem, as $z\to\infty$ we have

(5.39)\begin{equation} \int_0^t \alpha^+(\tau,z)\,\mathrm{d} \tau= \pm 2\text{i} z^2t + O(1/z)\,,\quad z\in{\mathbb C}^\pm\,, \end{equation}

which yields the asymptotic behaviour of $e^+(t,z)$ as $z\rightarrow\infty$. The same procedure gives us the asymptotic behaviour of $e^-(t,z)$ as $z\rightarrow\infty$.

Condition 6. From the asymptotic behaviours of $\alpha^\pm$ (5.16a and 5.16b), a subalgebraic bound of the form $|\alpha^\pm(t,z)|\leq\mathop{\rm max}\nolimits\{Cz^2,C'\}$ always holds for $z\in\mathbb{C}\setminus\mathcal{D}$. Here we denote the upper half region of $\mathcal{D}$ by $\mathcal{D}^+$. So the domain $\mathcal{D}^+$ consists of the upper half plane with excised half-domes centred on the real line whose heights are bounded above by $R/n$. We then divide $\mathcal{D}^+$ into two parts and denote the first part by $\mathcal{D}^+_U=\{z\in\mathbb{C}^+:\mathop{\rm Im}\nolimits{z} \gt \mathop{\rm max}\nolimits\{R,\log2/L\}\}$. From (3.1), we have that

(5.40)\begin{equation} \tilde{y}_{12}(L,z)=\sin(zL)+O(1/z)\,. \end{equation}

Using the lower bound $|\sin(zL)|\geq{\textstyle\frac12}\left(|\mathrm{e}^{-\text{i} zL}|-{\textstyle\frac12}\right)\geq\frac{|\mathrm{e}^{-\text{i} zL}|}{4}$ we have, for $|z|$ large enough,

(5.41)\begin{equation} |\tilde{y}_{12}(L,z)|\geq C|\mathrm{e}^{-\text{i} zL}|. \end{equation}

From (A.29) and the fact that $\tilde{y}_{11}(L,z)$ has growth order 1, we have the bound

(5.42)\begin{equation} |\rho^\pm(z)-\tilde{y}_{11}(L,z)|\leq C|\mathrm{e}^{-\text{i} zL}|. \end{equation}

These together with (5.15) give us

(5.43)\begin{equation} |\alpha^\pm(t,z)|\leq\mathop{\rm max}\nolimits\{C|z|^2,C'\} \end{equation}

for $z\in\mathcal{D}^+_U$. We use $\mathcal{D}^+_L$ to denote the remaining part of $\mathcal{D}^+$. For $z$ large enough, one of the excised discs will overlap. At this point the non-straight pieces of boundary are deformed semicircles $C_n$ labelled by $n$ with the closest point to $\mu_n$ a distance away from $\mu_n$ bounded below by $R/n$ for some constant $R$, and furthest point from $\mu_n$ a distance away from $\mu_n$ bounded above by $R'/n$ for some constant $R'$. As $z\rightarrow\infty$, (5.40) gives us that

(5.44)\begin{equation} \tilde{y}'_{12}(L,z)=L\cos(zL)+O(1/z). \end{equation}

The values $z=\mu_n$ are a distance $O(n^{-1})\sim\mu_n^{-1}$ away from the values of $z$ where $\cos(zL)=\pm1$, so for large $n$

(5.45)\begin{equation} \tilde{y}'_{12}(L,\mu_n)=L+O(\mu_n^{-1}). \end{equation}

Hence, on $C_n$, the functions $\tilde{y}_{12}(L,z)$ are approximated by

(5.46)\begin{equation} \tilde{y}_{12}(L,z)=(L+O(\mu_n^{-1}))(z-\mu_n)+O(z-\mu_n)\geq\frac{RL}{n} \end{equation}

for $n$ large enough. As $z\sim\frac{n\pi}{L}$ for $z\in C_n$ for $n$ large enough, we have $\tilde{y}_{12}(L,z)^{-1}=O(z)$ for $z\in C_n$ for $n$ large enough. Finally, we have $|\tilde{y}_{12}(L,z)^{-1}|\leq\mathop{\rm max}\nolimits\{C|z|, C'\}$ for $z\in\mathcal{D}^+_L$. This completes the proof of condition 6.

Definition 5.5. Let $\check{V}(x,t,z)$ be defined by

(5.47)\begin{equation} \check{V}(x,t,z) = \begin{cases} (-1)^{n + m(z)} \begin{pmatrix}{f^-(z)}/{f^+(z)} & 0 \\ 0 & {f^+(z)}/{f^-(z)}\end{pmatrix}, &z\in[E_{2n-2},E_{2n-1}],\\ (-1)^{n + m(z)}\text{i}\mathrm{e}^{-2\text{i} zx\sigma_3-4\text{i} z^2t\sigma_3}\sigma_1, &z\in[E_{2n-1},E_{2n}], \end{cases} \end{equation}

with $m(z)$ still defined by (4.6), and let the matrix-valued time-dependent Bloch–Floquet solution $\check\Psi(x,t,z)$ be defined as in (3.5) but with $\psi_j^\pm$ replaced by $\check\psi_j^\pm$, as defined in (5.24). Finally, let $\check{\Phi}(x,t,z)$ be given by

(5.48)\begin{equation} \check{\Phi}(x,t,z)=\check{\Psi}(x,t,z)B(z)\mathrm{e}^{\text{i}\sigma_3zx+2\text{i}\sigma_3z^2t}\,. \end{equation}

For fixed $x,t\in\mathbb{R}$, the solution of RHP 5.1 is unique, as can be shown using the same arguments as for Theorem 4.7. Moreover, the potential matrix $Q(x,t)$ of the Dirac operator (1.2) is given in terms of the unique solution $\check\Phi(x,t,z)$ of RHP 5.1 by

(5.50)\begin{equation} Q(x,t)=\lim_{z\to\infty}\text{i} z [\sigma_3,U^\dagger\check\Phi(x,t,z)B(z)^{-1}], \end{equation}

with $U$ as in (2.15).

6. Periodicity conditions in space and time

It is well known that, even in the finite-genus case, and even for Hill’s equation, the potential generated by a generic set of spectral data is not periodic in general, but only quasi-periodic. A natural question is therefore whether it is possible to identify a subset of the spectral data which guarantees that the associated potential is in fact periodic. In this section, assuming existence of the solutions of the RHP 4.1, and following [Reference McLaughlin and Nabelek73], we prove that it is possible to define a scalar RHP for which the existence of a solution implies the potential is periodic. Furthermore, we also prove there is an analogous scalar RHP for which the existence of a solution implies temporal periodicity.

To identify the desired RHP, note first that, if the potential is periodic, the Bloch–Floquet theory of the ZS problem yields the existence of the Floquet multiplier $\rho(z)$. Next, note that the function equal to $\rho(z)$ for $z\in\mathbb{C}^+$ and to $\rho(z)^{-1}$ for $z\in\mathbb{C}^-$ satisfies the scalar RHP defined by Conditions (i)–(iii) in Theorem 6.1. Therefore, the latter is the desired RHP, since the existence of solutions for it guarantees the periodicity of the potential, and hence the existence of the Floquet multiplier $\rho(z)$. Next we make these considerations more precise.

Consider a ‘candidate spectral data’, namely a sequence $\{E_{2n-1},E_{2n},\gamma_n,\nu_n\}_{n=g_-,\dots,g_+}$ of periodic and antiperiodic eigenvalues $E_n$, Dirichlet eigenvalues $\gamma_n$ and signs $\nu_n$, with $g_- \lt g_+$ either finite or infinite, satisfying the following conditions:

• • $E_{2g_--1} \lt \dots \lt E_{n-1} \lt E_n \lt E_{n+1} \lt \dots \lt E_{2g_+}$;

• • $\forall n = g_-,\dots,g_+$, $\gamma_n\in[E_{2n-1},E_{2n}]$ and $\nu_n\in\{-1,0,1\}$;

• • $R=\mathop{\rm max}\nolimits_k|E_{2k}-E_{2k-1}| \lt \infty$;

• • If $g_-$ and $g_+$ are finite, there exists $N$, $C$ such that $|E_n| \gt Cn^2$ for all $|n| \gt N$;

• • If either $g_-$ or $g_+$ are infinite, $\exists N \gt 0$ such that the discs $D_n$ of radius $R$ centred at $(E_{2n}+E_{2n-1})/2$ are disjoint for $|n|\geq N$.

Here and below, $\mathcal D$ denotes the domain introduced in Definition 4.4, and $\mathcal A_2(\mathcal D)$ denotes the Phragmén–Lindelöf growth class of functions on $\mathcal D$, as defined in Appendix A.6.

Proof of Theorem 6.1

Suppose that $r_1(z)$ exists and that $\Phi(x,t,z)$ solves the RHP 5.1, and define the function

(6.1)\begin{equation} \tilde{\Phi}(x,t,z)=\Phi(x+L_1,t,z)r_1(z)^{\sigma_3}\mathrm{e}^{-\text{i} z\sigma_3 L_1}\,. \end{equation}

We next prove that $\tilde{\Phi}$ also solves RHP 5.1. Conditions 1 and 2 are obvious. Regarding condition 3 (the jump), note that on the spectral band $[E_{2n-2},E_{2n-1}]$, the jump relation for $\tilde{\Phi}$ is

(6.2)\begin{equation} \begin{aligned} \tilde{\Phi}_+(x,t,z) = \Phi_+(x+L_1,t,z)r_1(z)^{\sigma_3}\mathrm{e}^{-\text{i} z\sigma_3 L_1} \\ = \Phi_-(x+L_1,t,z)(-1)^{n+m(z)}\begin{pmatrix} f^-/f^+ & 0 \\ 0 & f^+/f^- \end{pmatrix}r_1^{\sigma_3}\mathrm{e}^{-\text{i} z\sigma_3 L_1} \\ = \tilde{\Phi}_-(x,t,z) (-1)^{n+m(z)}\begin{pmatrix} f^-/f^+ & 0 \\ 0 & f^+/f^- \end{pmatrix}. \end{aligned} \end{equation}

On the other hand, on $[E_{2n-1},E_{2k}]$, the jump can be derived by

(6.3)\begin{equation} \begin{aligned} \tilde{\Phi}_+(x,t,z) = \Phi_+(x+L_1,t,z)r_{1+}(z)^{\sigma_3}\mathrm{e}^{-\text{i} z\sigma_3 L_1} \\ = \Phi_-(x+L_1,t,z)(-1)^{n+m(z)}\text{i}\mathrm{e}^{-2\text{i} zx\sigma_3}\sigma_1r_{1-}^{-\sigma_3}\mathrm{e}^{-\text{i} z\sigma_3 L_1} \\ = \tilde{\Phi}_-(x,t,z) (-1)^{n+m(z)}\text{i}\mathrm{e}^{-2\text{i} zx\sigma_3}\sigma_1. \end{aligned} \end{equation}

Therefore, $\tilde{\Phi}$ satisfies condition 3. Condition 4 follows from the asymptotic behaviour of $r_1(z)$. Condition 5 comes from the fact that $\mathcal{A}_2(\mathcal{D})$ is an algebra. Theorem 5.6 and the reconstruction formula (5.50) imply that

(6.4)\begin{equation} \Phi(x,t,z)=\Phi(x+L_1,t,z)r_1(z)^{\sigma_3}\mathrm{e}^{-\text{i} zL_1\color{blue}{\sigma_3}}, \end{equation}

which indicates $q(x+L_1,t)=q(x,t)$.

In the finite-gap case, the existence of solutions of the scalar RHP for $r_1(z)$ formulated in Theorem 6.1 is equivalent to the usual commensurability condition for the periods of the normalized quasi-momentum differential, see Section 7. Indeed, the situation is analogous to that of the finite-gap solutions of the KdV equation, as discussed in [Reference McLaughlin and Nabelek73]. If the corresponding periods are commensurate, the Floquet multiplier can be constructed as an exponential of an Abelian integral and satisfies the scalar RHP conditions above. Conversely, if the periods are not commensurate, the associated finite-gap solution is only quasi-periodic, and the scalar RHP cannot have a solution giving a fixed spatial period. Thus, Theorem 6.1 gives the scalar Riemann–Hilbert formulation of the standard commensurability condition.

7. Baker–Akhiezer functions and finite-gap potentials

Up to this point, we have made no use of the spectral curve underlying the periodic potentials of the Dirac operator. In this section, we show how the formalism presented above can also be interpreted in terms of Baker–Akhiezer functions. Then we show how, when $g_-$ and $g_+$ are both finite, the formalism can be used to recover the finite-genus potentials of the Dirac operator and therefore the corresponding finite-gap solutions of the defocusing NLS equation. Specifically, we provide a corollary to Theorems 4.5 and 5.6 that allows us to interpret the method we have presented in terms of Baker–Akhiezer functions on a Riemann surface of possibly infinite genus.

Let $\mathcal{X}\subset\mathbb{C}\times\mathbb{C}$ be the curve defined by

(7.1)\begin{equation} w^2=P(z)=-L^2(E_{-1}-z)(E_0-z)\prod_{k=g_-}^{g_+}(z-E_{2k-1})(z-E_{2k}). \end{equation}

Similarly to [Reference McLaughlin and Nabelek73], this curve is diffeomorphic via a holomorphic map to the desingularization of the curve defined by $w^2=\Delta^2(z)-1$ by two-point blowups at the degenerate Dirichlet eigenvalues. As in [Reference McLaughlin and Nabelek73], if $g=\infty$, we choose not to compactify  $\mathcal{X}$ as a topological space as the compactification would not be smooth at infinity (because of the accumulation of ‘holes’ at infinity). The projection $\pi((z,w))\rightarrow z$ onto the $z$ plane has two inverses under composition:

(7.2)\begin{equation} \pi_\pm^{-1}(z)=(z,\pm\sqrt{P(z)}), \end{equation}

which hold only at branch points.

Corollary 7.1. For every $x,t$ there is a unique meromorphic function $\check{\psi}(x,t,p)$ on $\mathcal{X}$ with only simple poles $p_k=(\mu_{n_k},\nu_{n_k},P(\mu_{n_k}))$ for $k=g_-,g_-+1,...,g_+-1,g_+$ such that, as $z\rightarrow\infty$ with $z\in{\mathbb C}\setminus{\mathbb R}$,

(7.3)\begin{equation} \check{\psi}(x,t,\pi_\pm ^{-1}(z))=\mathrm{e}^{\pm\text{i} (zx+2z^2t)}\left(\begin{pmatrix}1\\\mp\text{i}\end{pmatrix}+O(1/z)\right). \end{equation}

Moreover, $\check{\psi}(x,t,\pi_\pm^{-1}(\,\cdot\,))\in\mathcal{A}_2(\mathcal{D})$, where $\mathcal D$ and $\mathcal A_2(\mathcal D)$ are defined in Definition 4.4 and in Appendix A.6, respectively.

The function $\check\psi$ is known as the Baker–Akhiezer function for the NLS equation (e.g., see [Reference Belokolos, Bobenko, Enol’skii, Its and Matveev8, Reference Kotlyarov and Shepelsky56] for the finite-genus case).

In the finite gap case, let $\mathcal{X}$ be the Riemann surface of genus $g = g_+-g_-$ defined by the equation $w^2=P(z)$, where

(7.4)\begin{equation} P(z)=\prod_{j=2g_--1}^{2g_+}(z-E_{j}) \end{equation}

with cuts along the arcs $\Gamma_j=(E_{2j},E_{2j+1})$. The standard projection $\pi:\mathcal{X}\to \mathbb{CP}^1$ is defined by:

(7.5)\begin{equation} \pi(P)=z,\quad P=(w,z). \end{equation}

The projection $\pi$ defines $\mathcal{X}$ as a two-sheeted covering of $\mathbb{CP}^1$. There are two points $\infty^\pm\in\mathcal{X}$, with the property $\pi(\infty^\pm)=\infty\in\mathbb{CP}^1$.

Let us introduce a canonical basis $a_i$, $b_j$ of cycles satisfying $a_i\circ b_j=\delta_{ij}$, $a_i\circ a_j=0$ and $b_i\circ b_j=0$, where $\circ$ denotes minimal crossing number in the homology class of $a_i$ and $b_j$. Then we introduce the basis of Abelian differentials $\omega_i$ of the first kind on the Riemann surface $\mathcal{X}$, normalized such that

(7.6)\begin{equation} \int_{a_j}\omega_i=2\pi\text{i}\,\delta_{ij}. \end{equation}

The Abel map $\mathbf{A}: \mathcal{X}\rightarrow\mathbb{C}^{g}$ is defined as follows:

(7.7)\begin{equation} A_j(P)=\int_{P_{2g_-}}^P\omega_j. \end{equation}

The normalized holomorphic differentials define the $b$-period matrix as:

(7.8)\begin{equation} B_{jk}=\int_{b_k}\omega_j. \end{equation}

Associated with the matrix $B$ there is the Riemann theta function defined for $\textbf{u}\in{\mathbb C}^{g}$ by the Fourier series:

(7.9)\begin{equation} \Theta(\mathbf{p})=\sum_{\mathbf{m}\in\mathbb{Z}^{g}}\exp \{{\textstyle\frac12}\langle B\mathbf{m},\mathbf{m}\rangle+\langle\mathbf{p},\mathbf{m}\rangle\}, \end{equation}

where $\langle\mathbf{p},\mathbf{q}\rangle=p_1q_1+\dots+p_nq_n$ for $\mathbf{p}$, $\mathbf{q}\in{\mathbb C}^n$. The Abelian integrals $\Omega_1(P)$, $\Omega_2(P)$ and $\Omega_3(P)$, $P\in\mathcal{X}$ are fixed by the following conditions:

1. (i) $ \int_{a_i}\mathrm{d}\Omega_j=0.$

2. (ii) $\Omega_1(P)=\pm(z+O(1)),\quad P\to\infty^\pm$,

   $\Omega_2(P)=\pm(2z^2+O(1)),\quad P\to\infty^{\pm},$

   $\Omega_3(P)=\pm(\log z+O(1)),\quad P\to\infty^{\pm},\quad z=\pi(P).$

3. (iii) $\Omega_j(P)$ have no singularities at any points other than $\infty^{\pm}$.

Next we introduce an arbitrary divisor $\mathcal{D}$ with $\deg\mathcal{D}=g$ in general position, namely $\mathcal{D}=\sum\nolimits_{j=1}^g P_j$, with $\pi(P_j)\neq E_j$. Then following [Reference Belokolos, Bobenko, Enol’skii, Its and Matveev8], for all $\alpha\in\mathbb{C}$, we consider the vector-valued Baker–Akhiezer function $\psi(P,x,t)=(\psi_1,\psi_2)^T$ defined as follows:

(7.10a)\begin{equation} \psi_1(P)=\frac{\Theta(\int_{\infty^-}^P \boldsymbol{\omega}+\text{i} \mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D})\,\Theta(\mathbf{D})} {\Theta(\int_{\infty^-}^P\boldsymbol{\omega}-\mathbf{D})\,\Theta(\text{i}\mathbf{V} x+\text{i}\mathbf{W}t-\mathbf{D})} \exp\left\{\text{i} x\Omega_1(P)+\text{i} t\Omega_2(P)-{\textstyle\frac{\text{i}}{2}}E x+{\textstyle\frac{\text{i}}{2}}Nt\right\}, \end{equation}

(7.10b)\begin{equation} \psi_2(P)=\alpha\sqrt{\omega_0}\frac{\Theta(\int_{\infty^-}^P \boldsymbol{\omega}+\text{i} \mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D}-\mathbf{r})\,\Theta(\mathbf{D}-\mathbf{r})} {\Theta(\int_{\infty^-}^P\boldsymbol{\omega}-\mathbf{D})\,\Theta(\text{i}\mathbf{V} x+\text{i}\mathbf{W}t-\mathbf{D})} \exp\left\{\text{i} x\Omega_1(P)+\text{i} t\Omega_2(P)+{\textstyle\frac{\text{i}}{2}}E x-{\textstyle\frac{\text{i}}{2}}Nt+\Omega_3\right\}. \end{equation}

(7.11a)\begin{equation} \boldsymbol{\omega}=(\omega_1,\dots,\omega_{g}),\quad \mathbf{V}=(V_1,\dots,V_{g}),\quad \mathbf{W}=(W_1,\dots,W_{g}), \end{equation}

(7.11b)\begin{equation} \mathbf{r}=\int_{\infty^-}^{\infty^+}\boldsymbol{\omega},\quad\mathbf{D}=\sum_{j=1}^{g}\int_{\infty^-}^{P_j}\boldsymbol{\omega}+\mathbf{K}, \end{equation}

(7.11c)\begin{equation} V_j=\int_{b_j}\mathrm{d}\Omega_1,\quad W_j=\int_{b_j}\mathrm{d}\Omega_j,\quad K_j=\pi\text{i}+{\textstyle\frac12} B_{jj}-\frac{1}{2\pi\text{i}}\sum_{k\neq j}\int_{a_k}(\int_{\infty^-}^P\omega_j)\omega_k(P). \end{equation}

$E$

$N$

$\omega_0$

$\Omega_j(P)$

$\infty^{\pm}$

Finally, we fix some simply connected neighbourhood $U$ of the point $z=\infty$ which has no branch points. Then for each $z\in U$, $\pi^{-1}(z)$ identifies exactly two points denoted by $P^{\pm}\in\mathcal{X}$ so that $p^\pm\to\infty^\pm$ when $z\to\infty$. For all $z\in U$, we then define the matrix-valued function

(7.12)\begin{equation} \Psi(z,x,t)=(\psi(P^+),\psi(P^-)). \end{equation}

It is relatively straightforward to show that the above function satisfies all the properties of the finite-genus Baker–Akhiezer function [Reference Belokolos, Bobenko, Enol’skii, Its and Matveev8]. As a result, the solution to the NLS equation recovered from the Baker–Akhiezer function is given by

(7.13)\begin{equation} q(x,t)= 2\mathrm{e}^{-\text{i} Ex+\text{i} Nt} \frac{2\,\Theta(\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D}+\mathbf{r})\,\Theta(\mathbf{D})} {\alpha\,\Theta(\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D})\,\Theta(\mathbf{D}-\mathbf{r})}. \end{equation}

For any finite-gap potential, the spectral curve determines the normalized quasi-momentum Abelian differential $\Omega_1$ as defined above. Its $b$-periods determine the wavenumbers $V_1,\dots,V_g$ in the above theta-function representation of $q(x,t)$ via (7.11c). Thus finite-gap solutions are, in general, only quasi-periodic in  $x$. They become periodic in $x$ precisely when the corresponding wavenumbers are commensurate. The situation is entirely analogous to what happens for the finite-gap solutions of the KdV equation (e.g., see the discussion in [Reference McLaughlin and Nabelek73]). In the commensurate case, the scalar multiplier associated with the shift $x\mapsto x+T_1$ satisfies the scalar RHP in Theorem 6.1. Conversely, if the scalar problem in Theorem 6.1 is solvable, then Theorem 6.1 implies that the recovered potential is periodic in $x$. Hence, in the finite-gap case, the scalar Riemann–Hilbert condition in Theorem 6.1 is equivalent to the usual commensurability condition. In the non-commensurate case, the finite-gap solution is only quasi-periodic in $x$, and the scalar problem in Theorem 6.1 is not solvable. An analogous statement holds for Theorem 6.2, with the normalized quasi-energy differential $\Omega_2$ and a temporal period $T_2$.

8. Concluding remarks

In summary, we formulated the direct and inverse transform for a one-dimensional self-adjoint Dirac operator with periodic BCs via Riemann–Hilbert techniques, and we used the formalism to solve the IVP for the defocusing NLS equation with periodic BCs.

The formalism of this work generalizes the one developed in [Reference McLaughlin and Nabelek73] for Hill’s operator. Compared to [Reference McLaughlin and Nabelek73], one of the main complications in the theory for Dirac operator is that the spectrum for Hill’s operator is bounded from below, which, even in the case of infinite genus, provides a natural starting point for the count of all sequences of eigenvalues and also for the definition of the branch points of all square roots and fourth roots. In contrast, the spectral bands of the Dirac operator extend to infinity in both directions along the real $z$-axis, which complicates the analysis. Another significant difference from the analysis of Hill’s equation in [Reference McLaughlin and Nabelek73] is that, in that case the spectral problem is an eigenvalue problem for a scalar second-order differential operator. As a result, the asymptotic behaviour of various quantities contains square roots of the spectral parameter, which complicates the analysis. In this respect, the ZS problem is simpler, since no square roots of $z$ appear in the analysis. On the other hand, the ZS problem is complicated significantly by the need to introduce the modified Bloch–Floquet solutions.

Recall that, in the finite-genus formalism, the Dirichlet eigenvalues associated with all base points in the spatial domain are needed in order to reconstruct the potential, and the dependence of the Dirichlet eigenvalues on the base point (as governed by the Dubrovin equations) is highly nontrivial and is linearized by the Abel map. With the Riemann–Hilbert formalism, in contrast, only one base point is all that is needed. Indeed, the correct spatial and temporal dependence of the solution is simply a consequence of the explicit, parametric dependence of the jump condition of the RHP on  $x$ and  $t$, precisely like in the IST formalism on the line.

The present work should also be compared with works based on the unified transform method. In particular, the IVP for the focusing and defocusing NLS was studied using the unified transform method in [Reference Deconinck, Fokas and Lenells16], while periodic finite-band solutions of the focusing NLS equation were studied by the Fokas method in [Reference Shepelsky, Karpenko, Bogdanov and Prilepsky79]. Recall that the unified transform method is based on simultaneous analysis of both parts of the Lax pair, but where the problem is posed on the finite interval. As a result, the formalism requires the use of unknown boundary values, which must then be eliminated. In contrast, the present work uses the spectral theory of the ZS problem, is therefore more analogous to the IVP on the line and does not use the value of the potential and its derivative at the boundary of the domain.

A result of this work is that only one set of Dirichlet eigenvalues is needed for the inverse problem, just like in the inverse spectral theory for the KdV equation. This is in contrast with the trace formulae obtained from the finite-gap formalism, in which two sets of Dirichlet eigenvalues are used to reconstruct the potential. The result was illustrated by computing explicitly the solution of the NLS in the case of genus zero in Appendix A.8. In Appendix A.7, we also showed that either of the two sets of Dirichlet eigenvalues used in the finite-genus formalism works equally well for the purposes of the present inverse spectral theory. The fact that only one set of Dirichlet eigenvalues is needed is consistent with the fact that, for the direct and inverse spectral theory on the line, each norming constant in the defocusing case is uniquely determined by one real degree of freedom. Similarly, in the theta function representation of the solution of the NLS equation, the real divisor also contains only one set of real degrees of freedom (cf. Section 7). Note also that, since the RHP 4.1 already admits a unique solution, one does not have the option of prescribing additional data (e.g., such as a second set of Dirichlet eigenvalues) while preserving the solvability of the RHP. This means that a bijective map must exist between the two sets of Dirichlet eigenvalues.

The results of this work also open a number of interesting problems for future study, which can be divided along three main classes. The first class of questions concerns the NLS equation, and a first question in this regard is whether the present results can be used to establish the well-posedness of the initial-value problem in some appropriate functional classes. In this regard, note that (as is usual) the IST was formulated under the assumption of existence and uniqueness of solutions of the IVP. On the other hand, one can now turn the perspective around and use the results of the present work to prove the well-posedness of the IVP in appropriate functional classes, similarly to what was done in [Reference Jenkins, Liu, Perry and Sulem50, Reference Kappeler and Topalov54, Reference Pelinovsky, Saalmann and Shimabukuro76, Reference Pelinovsky and Shimabukuro77, Reference Villarroel, Ablowitz and Prinari84, Reference Zhou88].

Another question is related to the fact that the NLS equation is an infinite-dimensional Hamiltonian system. It is well-known, when the IVP is posed on the line, the system is completely integrable, and the IST can be viewed as a canonical transformation to action-angle variables. For the KdV equation, action-angle variables with periodic BC are also known [Reference Kappeler and Pöschel53]. An obvious question is therefore what are the action-angle variables for the defocusing NLS equation with periodic BC in the infinite-genus case.

Yet another interesting question relates to possible existence of a suitable infinite period limit of the present formalism. In other words, the question is whether there is any way in which the formalism can reduce to the IST on the line with either zero or non-zero BC. The question is nontrivial, since the periodic problem and the problem on the line are fundamentally different. In particular, for the IST on the line, the value of the potential as $x\to\pm\infty$ must be prescribed (whether it be zero or non-zero), whereas in our case the value of $q(x)$ at $x= \pm L/2$ is not given. For the NLS equation, this difficulty is also compounded by the fact that the IST on the line differs quite a bit depending on whether zero or non-zero BC are given. (For the KdV equation, in contrast, the value of the potential at infinity can always be set without loss of generality thanks to the Galilean invariance of the equation.)

A last question in the first class relates to the ‘linearization’ of the direct and inverse transform. It is well known that, for potentials on the line, the IST reduces exactly to the direct and inverse Fourier transform in the ‘linear limit’, i.e., the limit of vanishingly small potentials [Reference Ablowitz, Kaup, Newell and Segur3]. A natural question is therefore whether there is any way in which the present formalism reduces to the solution of the periodic problem by Fourier series in the linear limit.

A second class of open questions concern whether the analysis of the present work can be generalized to other systems. For example, an obvious interesting question is whether the fact that a single set of Dirichlet eigenvalues suffices for the purposes of inverse spectral theory also carries through to the focusing ZS problem. In this respect, note that the analysis of the focusing ZS problem is significantly complicated by the fact that the corresponding Dirac operator is not self-adjoint, and as a result the eigenvalues are not relegated to the real $z$-axis [Reference Ma and Ablowitz63, Reference McLaughlin, Overman, Keller, McLaughlin and Papanicolaou72]. This difficulty is also compounded by the fact that, while the notion of spectral bands and gaps can still be introduced in a relatively straightforward way [Reference Biondini and Oregero9, Reference Biondini, Oregero and Tovbis10], the movable Dirichlet eigenvalues are not confined to curves in the complex plane [Reference McKean68].

Another possible future extension of the present work concerns the periodic problem for the Manakov system, i.e., the integrable two-component coupled NLS equation. Realizing such an extension will require overcoming several significant difficulties, which stem from the fact that the associated scattering problem is a three-component system. Therefore, even though in the defocusing case the spectral problem is still self-adjoint, the characterization of the spectrum is much more complicated and cannot be reduced to the study of a single entire function like with the KdV and NLS equations. At the same time, we remark that the periodic problem for the Manakov system is still completely open even with the finite-genus formalism. The reason for this state of affairs is similar: since the scattering problem is effectively third-order, the spectral curve associated with the finite-genus solutions is not hyperelliptic as in the case of the KdV and NLS equation. For this reason, the analysis of the periodic Manakov system remains as an outstanding open problem.

Finally, a third class of questions relates to the application of the present formalism to study concrete problems. One such possible application concerns the study of semiclassical limits and small dispersion problems [Reference Bronski11, Reference Deift, Venakides and Zhou19, Reference Jenkins and McLaughlin51, Reference Kamvissis, McLaughlin and Miller52, Reference Lax and Levermore60, Reference Tovbis and Venakides82], which continue to receive considerable attention (e.g., see [Reference Biondini and Oregero9, Reference Biondini, Oregero and Tovbis10, Reference Deng, Biondini and Trillo21, Reference Deng, Li, Biondini and Trillo22, Reference El and Hoefer27, Reference Trillo, Deng, Biondini, Klein, Clauss, Chabchoub and Onorato83] and references therein). Another possible application concerns the emerging topics of soliton gases and the statistics of integrable systems [Reference Congy, El and Roberti13, Reference Congy, El, Roberti, Tovbis, Randoux and Suret14, Reference El26, Reference El and Tovbis28, Reference Gelash and Agafontsev33, Reference Gelash, Agafontsev, Zakharov, El, Randoux and Suret34, Reference Girotti, Grava, Jenkins and McLaughlin38, Reference Girotti, Grava, Jenkins, McLaughlin and Minakov39]. We hope that the results of this work and the present discussion will stimulate further work on all of these topics.