# Spectral theory for self-adjoint Dirac operators with
periodic potentials and inverse scattering transform
for the defocusing nonlinear Schrödinger equation
with periodic boundary conditions

Gino Biondini and Zechuan Zhang

###### Abstract

We formulate the inverse spectral theory for a self-adjoint one-dimensional
Dirac operator associated periodic potentials via a Riemann-Hilbert problem
approach. We also use the resulting formalism to solve the initial value
problem for the nonlinear Schrödinger equation. We establish a uniqueness
theorem for the solutions of the Riemann-Hilbert problem, which provides a new
method for obtaining the potential from the spectral data. Two additional,
scalar Riemann-Hilbert problems are also formulated that provide conditions
for the periodicity in space and time of the solution generated by arbitrary
sets of spectral data. The formalism applies for both finite-genus and
infinite-genus potentials. Importantly, the formalism shows that only a single
set of Dirichlet eigenvalues is needed in order to uniquely reconstruct the
potential of the Dirac operator and the corresponding solution of the
defocusing NLS equation, in contrast with the representation of the solution
of the NLS equation via the finite-genus formalism, in which two different
sets of Dirichlet eigenvalues are used.

###### Contents

  1. 1 Introduction and outline
  2. 2 Direct spectral theory for the periodic self-adjoint Zakharov-Shabat problem
  3. 3 Modified Bloch-Floquet solutions, their asymptotics and infinite product expansions
  4. 4 Inverse spectral theory for the periodic self-adjoint Zakharov-Shabat problem
  5. 5 Initial value problem for the defocusing NLS equation with periodic BC
  6. 6 Periodicity conditions in space and time
  7. 7 Baker-Akhiezer functions and finite-gap potentials
  8. 8 Concluding remarks
  9. Appendix

##  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 for the defocusing nonlinear
Schrödinger (NLS) equation with periodic boundary conditions. More precisely,
we study the eigenvalue problem

$\mathcal{L}\,v=z\,v\,,$ (1.1)

where the matrix-valued differential operator

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

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

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

the asterisk denoting complex conjugation, and the “potential”
$q:{\mathbb{R}}\to{\mathbb{C}}$ is a function with minimal period $L$, i.e.,

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

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

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

i.e., the set of complex numbers $z$ such that (1.1) has at least one bounded
nonzero 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 $\mathcal{L}$ [75]. 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 nonlinear evolution equations, since it comprises the
first half of the Lax pair for the defocusing NLS equation, namely the partial
differential equation (PDE)

$\text{i}q_{t}+q_{xx}-2|q|^{2}q=0\,,$ (1.6)

with $q=q(x,t)$, and where subscripts $x$ and $t$ denoting partial
differentiation. Indeed, Zakharov and Shabat [81, 82] showed that (1.6) is the
compatibility condition of the following overdetermined linear system of
ordinary differential equations (ODEs)

$\displaystyle v_{x}=(-\text{i}z\sigma_{3}+Q(x,t))\,v\,,$ (1.7a)
$\displaystyle v_{t}=(-2\text{i}z^{2}\sigma_{3}+H(x,t,z))\,v\,,$ (1.7b)

where $v=v(x,t,z)$, with $H(x,t,z)=2zQ-\text{i}\sigma_{3}(Q^{2}-Q_{x})$ and
$Q(x,t)$ has the same form as in (1.3). For this reason, (1.7a) is referred to
as the Zakharov-Shabat scattering problem, $z\in{\mathbb{C}}$ as the
scattering (or spectral) parameter, and $v$ as the scattering eigenfunction.
It is trivial to see (1.7a) is equivalent to the eigenvalue problem (1.1).
Together, (1.7a) and (1.7b) comprise the Lax pair of the NLS equation (1.6).

In 1967, Gardner, Greene, Kruskal and Miura pioneered the use of direct and
inverse spectral methods to solve the initial value problem (IVP) for the
Korteweg-de Vries equation [32], whose scattering problem is the time-
independent Schrödinger equation. Then, using a similar approach, in [81, 82]
Zakharov and Shabat showed how the Lax pair (1.7) 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 Zakharov-Shabat 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 nonlinear PDEs [2]. Various
significant further developments of the IST then followed (e.g., see [1, 3, 4,
7, 18, 29, 54, 72]), 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) [1, 6, 17, 43, 72, 83], which
in turn made it possible to formulate powerful asymptotic techniques such as
the Deift-Zhou method [20], 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 dependending on the particular PDE
under consideration), a natural next step was the study of IVP with periodic
boundary conditions. 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 boundary
conditions has of course a long and distinguished history, going back to the
early 1970s, related to the so-called finite-genus formalism, e.g., see [8,
24, 45, 46, 47, 52, 58, 61, 62, 63, 64, 68, 71, 72]. The theory was also
extended to the case of infinite genus in [52, 66, 67], 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 Zakharov-Shabat operators and in
particular their finite-gap formalism also have a long history [35, 36, 37,
44, 55, 60, 65, 69]. In particular, it is well known that, for the self-
adjoint Zakharov-Shabat spectral problem (1.1) with periodic potentials,
$\Sigma(\mathcal{L})$ is purely continuous [15, 25, 42, 75] (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.
Anothed 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 [70]. 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 Riemann-Hilbert problems in
order to study a variety of physically interesting problems. In this work we
generalize the approach of [70] to the self-adjoint Dirac operator (1.2) with
periodic potentials and we use the resulting formalism to solve the initial
value problem 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 [31, 60, 69]).

Specifically, the outline of this work is as follows. In section 2 we begin by
recalling some well-known results from Bloch-Floquet thoery, 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 of
$\tilde{y}_{12}(L,z)$, $\Delta(z)-1$ and $\Delta(z)+1$. All these results are
used to formulate the inverse problem as a suitable matrix Riemann-Hilbert
problem 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 a Riemann–Hilbert problem characterization for the solution of
the initial value problem 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 section 4
and section 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 RHP 4.1 and RHP 5.1 to a uniqueness
results 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 Zakharov-Shabat
problem

In this section we begin formulating the direct spectral theory for the
defocusing ZS spectral problem by briefly recalling some well-known results
from Bloch-Floquet theory, and by definining the spectral data that will allow
us to uniquely reconstruct the potential in section 4. 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.

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 [12, 25, 30] states that
any fundamental matrix solution $Y(x)$ of (2.1) of the system of linear
homogeneous ODEs

$y_{x}=A(x)\,y$ (2.1)

can be written in the Floquet normal form

$Y(x)=W(x)\,\mathrm{e}^{Rx}\,,$ (2.2)

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

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

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

$\psi(x+L,z)=\rho\,\psi(x,z)\,,$ (2.4)

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

$Y(x+L,z)=Y(x,z)M(z)\,,$ (2.5)

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

$M(z)=Y(L,z)\,.$ (2.6)

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

$Y(x,z)=\mathrm{e}^{-\text{i}z\sigma_{3}x}+\int_{0}^{x}\mathrm{e}^{-\text{i}z\sigma_{3}(x-x^{\prime})}Q(x^{\prime})Y(x^{\prime},z)\,\mathrm{d}x^{\prime}\,,$
(2.7)

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 eigenvalues of $M(z)$, i.e., the
Floquet multipliers, are given by roots of the quadratic equation

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

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

$\Delta:=\Delta(z)={\textstyle\frac{1}{2}}\mathop{\rm tr}\nolimits M(z)\,,$
(2.9)

which is also is an entire function of $z$ [31, 60, 69]. 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 $\Delta^{2}(z)>1$, 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
$\Delta^{2}(z)<1$, 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 as follows:

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

For $z\in\mathbb{C}$, denote the roots of (2.8) as

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

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 $|\rho_{1,2}(z)|\neq 1$.

######  Remark 2.1.

Let $\rho(z)$ be the root of (2.8) that is analytic for
$z\notin\Sigma(\mathcal{L})$ and such that $|\rho(z)|<1$ for
$z\in{\mathbb{C}}\setminus\Sigma(\mathcal{L})$. Setting $\rho(z)=\rho_{1}(z)$
$\forall z\in\mathbb{C}$ uniquely defines the complex square root
$\sqrt{\Delta^{2}(z)-1}$ so that:

  1. (i)

its branch cut coincides with $\Sigma(\mathcal{L})$,

  2. (ii)

$\sqrt{\Delta^{2}(z)-1}>0$ for all
$z\in\mathbb{R}\setminus\Sigma(\mathcal{L})$.

  3. (iii)

$\sqrt{\Delta^{2}(z)-1}$ is continuous from above for all
$z\in\Sigma(\mathcal{L})$.

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
$\Sigma(\mathcal{L})=\cup_{j\in\mathbb{Z}}[\zeta_{2j},\zeta_{2j+1}]$. On the
other hand, 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 [60, 69],
we therefore define the Dirichlet spectrum associated with (1.7a) as follows:

$\Sigma_{{\mathrm{Dir}}}(x_{o}):=\\{\mu\in{\mathbb{C}}:\exists\leavevmode\nobreak\
v\not\equiv 0\in H^{1}([x_{o},x_{o}+L],{\mathbb{C}}^{2})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
\mathcal{L}\,v=\mu\,v\leavevmode\nobreak\ \wedge\leavevmode\nobreak\ v\in{\rm
BC}_{{\rm Dir},x_{o}}\\}\,,$ (2.12)

where “BC${}_{{\rm Dir},x_{o}}$” denote the following Dirichlet boundary
conditions (BC) with base point $x_{o}$:

$v_{1}(x_{o})+v_{2}(x_{o})=v_{1}(x_{o}+L)+v_{2}(x_{o}+L)=0\,,$ (2.13)

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:

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

where

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

(The above transformation could be trivially made unitary by including a
factor $1/\sqrt{2}$ in (2.15), but the present normalization is more useful
for our purposes.) Then $\tilde{Y}(x,z)$ is the fundamental solution of the
following modified scattering problem

$\tilde{y}_{x}=U(-\text{i}z\sigma_{3}+Q)U^{-1}\,\tilde{y}\,.$ (2.16)

For convenience, also let

$\tilde{M}(z)=\tilde{Y}(L,z)\,,$ (2.17)

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

$\Sigma_{{\mathrm{Dir}}}(0)=\left\\{\mu\in{\mathbb{C}}:\tilde{y}_{12}(L,\mu)=0\right\\}.$
(2.18)

###### Proof.

Any vector solution of (1.1) can be expressed as
$v(x,z)=c_{1}y_{1}(x,z)+c_{2}y_{2}(x,z)$ for suitable constants $c_{1,2}$.
Evaluating this expression at $x=0$ and imposing $v_{1}(0)+v_{2}(0)=0$, we
obtain $c_{2}=-c_{1}$. Then, evaluating it at $x=L$ and imposing
$v_{1}(L)+v_{2}(L)=0$, we obtain $c_{2}(M_{11}+M_{21}-M_{12}-M_{22})=0$. Since
$c_{2}$ cannot be zero, we obtain that the Dirichlet boundary conditions
(2.13) are equivalent to the condition $M_{11}+M_{21}-M_{12}-M_{22}=0$, i.e.,
$\tilde{y}_{12}(L,z)=0$ by (2.14). ∎

Figure 1: Schematic representation of the behavior 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.

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 [23, 40, 41, 69, 60] (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 nondegenerate band edges $E_{j}$ and degenerate band edges
$\hat{\zeta}_{j}$, at which one has $\Delta^{\prime}(E_{j})\neq 0$ and
$\Delta^{\prime}(\hat{\zeta}_{j})=0$, respectively. Each $\zeta_{j}$ is either
a simple or a double root of 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 unstable band. All the
$\mu_{j}$ are simple roots of $\tilde{y}_{12}(L,z)$.

  3. 3.

$\Delta^{\prime}(z)\neq 0$ for all $z\in\mathbb{R}$ such that
$-1<\Delta(z)<1$.

  4. 4.

The periodic, antiperiodic and Dirichlet eigenvalues $\zeta_{j}$ and $\mu_{j}$
have the following asymptotic behavior as $|j|\to\infty$:

$\zeta_{j}=\frac{\pi j}{L}+o(1),\qquad\mu_{j}=\frac{\pi j}{L}+o(1).$ (2.19)

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

$\cdots<\zeta_{-2j}\leq\zeta_{-2j+1}<\cdots<{\zeta}_{-1}\leq{\zeta}_{0}<{\zeta}_{1}\leq\cdots<\zeta_{2j}\leq\zeta_{2j+1}<\cdots$
(2.20)

(cf. Fig. 1). We also number the Dirichlet eigenvalues so that
$\zeta_{2j-1}\leq\mu_{j}\leq\zeta_{2j}$. If $z=0$ lies a in spectral band
instead of a spectral gap, we set can simply set $\zeta_{-1}$ to be the first
point of the main spectrum that is greater to or equal than zero. The Lax
spectrum can still be expressed as above, and the whole formalism remains
unchanged. (Moreover, if $\zeta_{-1}>0$, 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 (3.10b) and (3.10c) in section 4.

######  Lemma 2.4.

The vector-valued fundamental solution $\tilde{y}_{2}(x,\mu_{j})$ is also a
Bloch-Floquet solution of (2.16) with Floquet multiplier
$\tilde{y}_{22}(L,\mu_{j})=1/\tilde{y}_{11}(L,\mu_{j})=\rho(\mu_{j})$ or
$1/\rho(\mu_{j})$.

###### Proof.

The Dirichlet eigenfunction $\tilde{y}_{2}(x,\mu_{j})$ solves the modified ZS
problem with spectral parameter $z=\mu_{j}$, and so does
$\tilde{y}_{2}(x+L,\mu_{j})$. So we have
$\tilde{y}_{2}(x+L,\mu_{j})=a\tilde{y}_{1}(x,\mu_{j})+b\tilde{y}_{2}(x,\mu_{j})$
for some constants $a$ and $b$. Evaluating this expression at $x=0$, we
conclude $a=0$ and $b=\tilde{y}_{22}(L,\mu_{j})$. Therefore, by Floquet’s
theorem, $\tilde{y}_{2}(x,\mu_{j})$ is a Bloch-Floquet solution with
multiplier $\tilde{y}_{22}(L,\mu_{j})=\rho(\mu_{j})$ or
$\tilde{y}_{22}(L,\mu_{j})=\rho(\mu_{j})^{-1}$. Since
$\det\tilde{Y}(L,\mu_{j})=1$, we also have
$\tilde{y}_{22}(L,\mu_{j})=1/\tilde{y}_{11}(L,\mu_{j})$. ∎

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
$q(x)\equiv 0$ from consideration.

######  Definition 2.5.

(Non-degenerate gaps) Fix $g_{-}$ and $g_{+}$ with $g_{-}<g_{+}$ so that
$\\{E_{2g_{-}-1},\dots,E_{2g_{+}}\\}$ is the subsequence of
periodic/antiperiodic eigenvalues $\\{{\zeta}_{j}\\}_{j\in\mathbb{Z}}$ for
which the spectrum $\Sigma(\mathcal{L})$ is the disjoint union

$\Sigma(\mathcal{L})=(-\infty,E_{2g_{-}-1}]\cup[E_{2g_{+}},\infty)\bigcup_{j=g_{-}}^{g_{+}-1}[E_{2j},E_{2j+1}],$
(2.21)

where each of the numbers $g_{\pm}$ is either finite or countably infinite.
The finite-length
intervals$(E_{2g_{-}-1},E_{2g_{-}}),\dots,(E_{2g_{+}-1},E_{2g_{+}})$ comprise
the spectral gaps of the potential.

######  Remark 2.6.

When $g_{-}$ and $g_{+}$ are finite, their value denotes the smallest and
largest values of $j$ for which the corresponding spectral gaps are non-empty.
In other words, $\\{E_{2g_{-}-1},\dots,E_{2g_{+}}\\}$ is the subset of the
periodic/antiperiodic eigenvalues $\\{\zeta_{j}\\}_{j\in\mathbb{Z}}$ that are
associated to open spectral gaps. Note that the set of open spectral gaps
might not necessarily be consecutive.

If both $g_{\pm}$ are finite, the total number of gaps is $g_{+}-g_{-}+1$, and
the genus of the potential (and the associated solution of the defocusing NLS
equation) is $g=g_{+}-g_{-}$. (That is, 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 the
indices $g_{\pm}$ up to an identical arbitrary additive constant. We used the
freedom to choose this constant when definining $\zeta_{0}$ above.

######  Definition 2.7.

For each Dirichlet eigenvalue $\mu_{k}$, we define $\sigma_{k}=-\mathop{\rm
sgn}\nolimits(\log|\tilde{y}_{22}(L,\mu_{k})|)$.

######  Definition 2.8.

(Fixed and movable Dirichlet eigenvalues) We denote by
$\\{\gamma_{g_{-}},\dots,\gamma_{g_{+}}\\}$ the subsequence of Dirichlet
eigenvalues that lie in the non-degenerate spectral gaps, numbered so that
${E}_{2k-1}\leq\gamma_{k}\leq{E}_{2k}$. These Dirichlet eigenvalues are
movable, i.e., they depend on the base point $x_{o}$ (here chosen to be zero)
and also on time (cf. section 5). The remaining eigenvalues are independent of
the base point, and are called the fixed Dirichlet eigenvalues.

######  Definition 2.9.

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

$S(q):=\\{{E}_{2k-1},{E}_{2k},\gamma_{k},\sigma_{k}\\}_{k=g_{-},\dots,g_{+}}.$
(2.22)

######  Remark 2.10.

Lemma 2.4 implies that, for all $k\in\mathbb{Z}$: (i) If $\sigma_{k}=\pm 1$,
$\tilde{y}_{2}(x,\mu_{k})$ is a Bloch-Floquet solution associated with the
Floquet multiplier $\rho^{\pm 1}(\mu_{k})$, respectively. (ii) If
$\sigma_{k}=0$, $\mu_{k}$ lies on one of the edges of a (degenerate or non-
degenerate) spectral gap. Thus, $\sigma_{k}=0$ for all fixed Dirichlet
eigenvalues, whereas $\sigma_{k}$ can equal $\pm 1$ or 0 for the movable
Dirichlet eigenvalues.

######  Remark 2.11.

In section 4 we will show that the spectral data (2.22) defined by the main
spectrum and the single set of Dirichlet eigenvalues of (1.7a) enables the
unique reconstruction of the potential $q$. The result is in contrast with the
finite-genus formalism (e.g., see [60, 69]), for which two sets of Dirichlet
eigenvalues are used. (See Appendix A.7 for further discussion.)

##  3 Modified Bloch-Floquet solutions, their asymptotics and infinite product
expansions

Some additional results are needed before we can begin to formulate the
inverse spectral theory. Specifically, as usual one needs knowledge of the
asymptotic behavior 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 $Y(x,z)$ have the following asymptotic behavior as
$z\to\infty$:

$Y(x,z)=\begin{pmatrix}1-\frac{1}{2\text{i}z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}+O(1/z^{2})&\frac{1}{2\text{i}z}(q(x)-q(0)\,\mathrm{e}^{-2\text{i}zx})+O(1/z^{2})\\\
-\frac{1}{2\text{i}z}(q^{*}(x)-q^{*}(0)\mathrm{e}^{2\text{i}zx})+O(1/z^{2})&1+\frac{1}{2\text{i}z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}+O(1/z^{2})\end{pmatrix}\,\mathrm{e}^{\text{i}zx\sigma_{3}}\,.$
(3.1)

######  Remark 3.2.

The off-diagonal terms of $Y(x,z)$ at $O(1/z)$ contain the value of the
potential $q(x)$ at $x=0$ (as well as its complex conjugate). As a result,
$Y(x,z)$ is not sufficient for the formulation of an inverse spectral theory.
As with Hill’s equation [70], one uses the fundamental solution to define
suitable Bloch-Floquet eigenfunctions instead. In the case of the Zakharov-
Shabat problem (1.7a), however, a further step needed is the similarity
transformation from $Y(x,z)$ to $\tilde{Y}(x,z)$ introduced in (2.14).

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 behavior of $Y(x,z)$ in Lemma 3.1
implies the following:

######  Proposition 3.3.

The Floquet discriminant $\Delta(z)$ has the following asymptotic behavior as
$z\to\infty$:

$\displaystyle\Delta(z)={\textstyle\frac{1}{2}}\mathrm{e}^{\mp\text{i}zL}\left(1\mp\frac{1}{2\text{i}z}\int_{0}^{L}|q(x)|^{2}\mathrm{d}x+O\left(\frac{1}{z^{2}}\right)\right),\qquad
z\in{\mathbb{C}}^{\pm}\,.$ (3.2)

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

$\psi^{\pm}(x+L,z)=\rho^{\pm
1}(z)\psi^{\pm}(x,z)\,,\qquad\psi_{1}^{\pm}(0,z)=1.$ (3.3)

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

$\Psi(x,z)=\big{(}\,\psi^{\mp}\,,\,\psi^{\pm}\,\big{)}\,,\qquad
z\in\mathbb{C}^{\pm}\setminus\mathbb{R}\,.$ (3.4)

In Appendix A.2 and Appendix A.4 we prove the following:

######  Proposition 3.5.

The Bloch-Floquet solutions $\psi^{\pm}(x,z)$ defined by (3.3) are given by

$\psi^{\pm}(x,z)=\tilde{y}_{1}(x,z)+\frac{\rho^{\pm
1}(z)-\tilde{y}_{11}(L,z)}{\tilde{y}_{12}(L,z)}\tilde{y}_{2}(x,z),$ (3.5)

where $\tilde{Y}(x,z)=(\tilde{y}_{1},\tilde{y}_{2})$.

######  Proposition 3.6.

For all $x>0$, $\psi^{-}(x,z)$ has the following asymptotic behavior as
$z\to\infty$

$\displaystyle\psi^{-}(x,z)=\mathrm{e}^{-\text{i}zx}\left(\begin{array}[]{c}1-\frac{1}{2\text{i}z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}-\frac{q^{*}(x)}{2\text{i}z}+O(1/z^{2})\\\
-\text{i}+\frac{1}{2z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}-\frac{q^{*}(x)}{2z}+O(1/z^{2})\end{array}\right)\,,\quad
z\in{\mathbb{C}}^{+}\,,$ (3.6c)
$\displaystyle\psi^{-}(x,z)=\mathrm{e}^{\text{i}zx}\left(\begin{array}[]{c}1+\frac{1}{2\text{i}z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}+\frac{q(x)}{2\text{i}z}+O(1/z^{2})\\\
\text{i}+\frac{1}{2z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}-\frac{q(x)}{2z}+O(1/z^{2})\end{array}\right),\quad
z\in{\mathbb{C}}^{-}\,.$ (3.6f)

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

$\displaystyle\psi^{+}(x,z)$
$\displaystyle=\mathrm{e}^{\text{i}zx}\left(\begin{array}[]{c}1+\frac{1}{2\text{i}z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}+\frac{q(x)}{2\text{i}z}+O(1/z^{2})\\\
\text{i}+\frac{1}{2z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}-\frac{q(x)}{2z}+O(1/z^{2})\end{array}\right),\quad
z\in{\mathbb{C}}^{+}\,,$ (3.7c) $\displaystyle\psi^{+}(x,z)$
$\displaystyle=\mathrm{e}^{-\text{i}zx}\left(\begin{array}[]{c}1-\frac{1}{2\text{i}z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}-\frac{q^{*}(x)}{2\text{i}z}+O(1/z^{2})\\\
-\text{i}+\frac{1}{2z}\int_{0}^{x}|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}-\frac{q^{*}(x)}{2z}+O(1/z^{2})\end{array}\right)\,,\quad
z\in{\mathbb{C}}^{-}\,.$ (3.7f)

######  Remark 3.7.

The introduction of the transformation (2.14) is needed to ensure that the
$O(1/z)$ terms in (3.6) and (3.7) only contain $q(x)$ and not $q(0)$. This
will make it possible to use the Bloch-Floquet eigenfunctions $\psi^{\pm}$ to
define a suitable Riemann-Hilbert problem in section 4 (see RHP 4.1), and
would have not be possible if one defined $\psi^{\pm}$ using $Y(x,z)$ instead
of $\tilde{Y}(x,z)$ (see Appendix A.7 for details). The second components of
$\psi^{\pm}$ are the Weyl-Titchmarsh $m$-functions [76, 77]. Note that
$\psi_{\pm}(x,z)$ are undefined at points $z$ for which
$\tilde{y}_{12}(L,z)=0$, due to the normalization $\psi_{\pm}(0,z)=1$, which
cannot be satisfied if the first component of the eigenvector of the monodromy
matrix vanishes.

It is straightforward to see the asymptotic behavior in Proposition 3.6,
together with the definitions (2.14) and (3.5), yields:

######  Corollary 3.8.

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

$Q(x)=\lim_{z\to\infty}iz[\sigma_{3},U^{\dagger}\Psi(x,z)\mathrm{e}^{\text{i}zx\sigma_{3}}]\,.$
(3.8)

Proposition 3.6 immediately gives the asymptotic behavior 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 (3.6) and (3.7) we also
have, as $z\to\infty$:

$\displaystyle\psi_{2}^{-}(0,z)=\begin{cases}\displaystyle-\text{i}-\frac{q^{*}(0)}{z}-\frac{q^{2}(0)+q_{x}(0)}{2\text{i}z^{2}}+O(1/z^{3})\,,&z\in{\mathbb{C}}^{+}\,,\\\\[4.30554pt]
\displaystyle\text{i}-\frac{q(0)}{z}+\frac{q^{2}(0)+q_{x}(0)}{2\text{i}z^{2}}+O(1/z^{3})\,,&z\in{\mathbb{C}}^{-}\,,\end{cases}$
(3.9a)
$\displaystyle\psi_{2}^{+}(0,z)=\begin{cases}\displaystyle\text{i}-\frac{q(0)}{z}+\frac{q^{2}(0)+q_{x}(0)}{2\text{i}z^{2}}+O(1/z^{3})\,,&z\in{\mathbb{C}}^{+}\,,\\\\[4.30554pt]
\displaystyle-\text{i}-\frac{q^{*}(0)}{z}-\frac{q^{2}(0)+q_{x}(0)}{2\text{i}z^{2}}+O(1/z^{3})\,,&z\in{\mathbb{C}}^{-}\,.\end{cases}$
(3.9b)

######  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 [5, 57]),
allows us to write $\Delta(z)\pm 1$ and $\tilde{y}_{12}(L,z)$ as the following
infinite products: if $\mu_{0}\neq 0$ and $\zeta_{0}\neq 0$,

$\displaystyle\tilde{y}_{12}(L,z)=\mathrm{e}^{A_{1}z+B_{1}}\prod_{j\in\mathbb{Z}}\left(1-\frac{z}{\mu_{j}}\right)\mathrm{e}^{\frac{z}{\mu_{j}}}\,,$
(3.10a)
$\displaystyle\Delta(z)-1=\mathrm{e}^{A_{2}z+B_{2}}\prod_{j\in{\mathbb{Z}}\atop
j\text{
even}}\left(1-\frac{z}{\zeta_{2j-1}}\right)\left(1-\frac{z}{\zeta_{2j}}\right)\mathrm{e}^{z\big{(}\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big{)}}\,,$
(3.10b)
$\displaystyle\Delta(z)+1=\mathrm{e}^{A_{3}z+B_{3}}\prod_{j\in{\mathbb{Z}}\atop
j\text{
odd}}\left(1-\frac{z}{\zeta_{2j-1}}\right)\left(1-\frac{z}{\zeta_{2j}}\right)\mathrm{e}^{z\big{(}\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big{)}}\,.$
(3.10c)

If $\mu_{0}=0$, (3.10a) is replaced by

$\tilde{y}_{12}(L,z)=z\mathrm{e}^{A_{1}z+B_{1}}\prod_{j\neq
0}\bigg{(}1-\frac{z}{\mu_{j}}\bigg{)}\,\mathrm{e}^{\frac{z}{\mu_{j}}}.$
(3.11a) Similarly, if $\zeta_{0}=0$, (3.10b) is replaced respectively by
$\displaystyle\Delta(z)-1=z\bigg{(}1-\frac{z}{\zeta_{-1}}\bigg{)}\,\mathrm{e}^{A_{2}z+B_{2}+\frac{z}{\zeta_{-1}}}\prod_{j\neq
0\atop j\text{
even}}\left(1-\frac{z}{\zeta_{2j-1}}\right)\left(1-\frac{z}{\zeta_{2j}}\right)\mathrm{e}^{z\big{(}\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big{)}}\,,$
(3.11b)

while the factorization of $\Delta(z)+1$ remains unchanged. The presence of
the exponential terms in (3.10) is a significant difference between the
spectral theory for the ZS operator compared to that for Hill’s equation,
whose eigenfunctions have order of growth 1/2. A corresponding difference will
appear in the definition of the jump matrix for the RHP in section 4.

##  4 Inverse spectral theory for the periodic self-adjoint Zakharov-Shabat
problem

Using the results of section 3, we now construct a 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 $2\times 2$ matrix-valued function $B(z)$ be defined as

$B(z)=\begin{cases}\displaystyle\frac{i\sqrt{f^{0}(z)}}{\sqrt[4]{\Delta^{2}(z)-1}}\mathop{\rm
diag}\nolimits(f^{-},f^{+}),&z\in\mathbb{C}^{+},\\\\[8.61108pt]
\displaystyle\frac{\sqrt{f^{0}(z)}}{\sqrt[4]{\Delta^{2}(z)-1}}\mathop{\rm
diag}\nolimits(f^{+},f^{-}),&z\in\mathbb{C}^{-},\end{cases}$ (4.1)

where $f^{0}(z)$ and $f^{\pm}(z)$ are defined as follows. If $\mu_{0}\neq 0$,

$\displaystyle
f^{0}(z)={\textstyle\frac{1}{2}}\mathrm{e}^{A_{1}z+B_{1}}\prod_{\sigma_{j}=0}\left(1-\frac{z}{\mu_{j}}\right)\mathrm{e}^{\frac{z}{\mu_{j}}}\,,\quad
f^{+}(z)=\prod_{\sigma_{j}=1}\left(1-\frac{z}{\mu_{j}}\right)\mathrm{e}^{\frac{z}{\mu_{j}}}\,,\quad
f^{-}(z)=\prod_{\sigma_{j}=-1}\left(1-\frac{z}{\mu_{j}}\right)\mathrm{e}^{\frac{z}{\mu_{j}}}\,,$

with $A_{1}$ and $B_{1}$ as in (3.10). Otherwise, if $\mu_{0}=0$ and
$\sigma_{0}=-1$, $f^{-}$ is replaced by

$f^{-}(z)=z\prod_{\sigma_{j}=-1\atop j\neq
0}\left(1-\frac{z}{\mu_{j}}\right)\mathrm{e}^{\frac{z}{\mu_{j}}}\,.$ (4.3)

Finally, if $\mu_{0}=0$ and $\sigma_{0}=1$, or if $\sigma_{0}=0$, $f^{\pm}$
and $f^{0}$ are the same as above except that the factor $z$ appears
respectively in $f^{+}$ or $f^{0}$ instead of $f^{-}$.

Note that the functions $f^{0}(z)$, $f^{\pm}(z)$ defined by the infinite
products in (LABEL:e:f+-0def) are entire. As usual, if any of the products in
(LABEL:e:f+-0def) 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 $f^{0}(z)\leq 0$, 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:
$g=2$.

######  Proposition 4.2.

The matrix $B(z)$ is completely determined by the spectral data $S(q)$ defined
in (2.22). Explicitly, if $\mu_{0}\neq 0$, we have

$\displaystyle\frac{\sqrt{2f^{0}(z)}}{\sqrt[4]{\Delta^{2}(z)-1}}=\mathrm{e}^{\frac{\text{i}\pi}{4}}\left(\prod_{k=g^{-}\atop\sigma_{k}=0}^{g^{+}}\frac{\big{(}1-\frac{z}{\gamma_{k}}\big{)}^{2}\mathrm{e}^{\frac{2z}{\gamma_{k}}}}{\big{(}1-\frac{z}{E_{2k-1}}\big{)}\big{(}1-\frac{z}{E_{2k}}\big{)}\mathrm{e}^{z\big{(}\frac{1}{E_{2k-1}}+\frac{1}{E_{2k}}\big{)}}}\prod_{k=g^{-},\atop\sigma_{k}\neq
0}^{g^{+}}\frac{1}{\big{(}1-\frac{z}{E_{2k-1}}\big{)}\big{(}1-\frac{z}{E_{2k}}\big{)}\mathrm{e}^{z\big{(}\frac{1}{E_{2k-1}}+\frac{1}{E_{2k}}\big{)}}}\right)^{\frac{1}{4}}\,,$
(4.4a) $\displaystyle
f^{+}(z)=\prod_{k=g^{-},\atop\sigma_{k}=1}^{g^{+}}\left(1-\frac{z}{\gamma_{k}}\right)\mathrm{e}^{\frac{z}{\gamma_{k}}}\,,\qquad
f^{-}(z)=\prod_{k=g^{-}\atop\sigma_{k}=-1}^{g^{+}}\left(1-\frac{z}{\gamma_{k}}\right)\mathrm{e}^{\frac{z}{\gamma_{k}}}\,.$
(4.4b)

The corresponding expressions if $\mu_{0}=0$ are similar.

###### Proof.

Using the infinite products (3.10b) and (3.10c), we have

$\displaystyle\sqrt{f^{0}(z)}={\textstyle\frac{1}{\sqrt{2}}}\mathrm{e}^{(A_{1}z+B_{1})/2}\prod_{j\neq
0\atop\sigma_{j}=0}\sqrt{1-\frac{z}{\mu_{j}}}\mathrm{e}^{\frac{z}{2\mu_{j}}},$
(4.5a)
$\displaystyle\sqrt[4]{\Delta^{2}(z)-1}=\mathrm{e}^{[(A_{2}+A_{3})z+B_{2}+B_{3}]/4}\prod_{j\in\mathbb{Z}}\sqrt[4]{1-\frac{z}{\zeta_{2j}}}\sqrt[4]{1-\frac{z}{\zeta_{2j-1}}}\mathrm{e}^{\frac{z}{4}\big{(}\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big{)}}\,.$
(4.5b)

Now recall that $\zeta_{2j-1}\leq\mu_{j}\leq\zeta_{2j}$, and that
$\sigma_{j}=0$ iff $\mu_{j}$ coincides with $\zeta_{2j}$ or $\zeta_{2j-1}$.
Thus, when $\zeta_{2j}=\zeta_{2j-1}$ (i.e., for all degenerate gaps), we have
$\mu_{j}=\zeta_{2j-1}=\zeta_{2j}$, and

$\sqrt{1-\frac{z}{\mu_{j}}}\mathrm{e}^{\frac{z}{2\mu_{j}}}\Bigg{/}\left(\sqrt[4]{1-\frac{z}{\zeta_{2j}}}\sqrt[4]{1-\frac{z}{\zeta_{2j-1}}}\mathrm{e}^{\frac{z}{4}\big{(}\frac{1}{\zeta_{2j-1}}+\frac{1}{\zeta_{2j}}\big{)}}\right)=1\,.$

Consequently, the scalar ratio appearing in (4.1) can be written as (4.4a).
Equation (4.4a) will also be useful in what follows. Note that the products
for $f^{\pm}(z)$ in (LABEL:e:f+-0def) only involve Dirichlet eigenvalues
located in non-degenerate gaps, and can therefore be rewritten as (4.4b).

To complete the proof, it remains to show that
$\exp\big{(}\frac{1}{4}[(2A_{1}-A_{2}-A_{3})z+2B_{1}-B_{2}-B_{3}]\big{)}=\mathrm{e}^{\frac{\text{i}\pi}{4}}$.
To see this, note that, by (3.10a) we have
$f^{0}(z)f^{+}(z)f^{-}(z)={\textstyle\frac{1}{2}}\tilde{y}_{12}(L,z)$. From
(3.1), the asymptotic behaviors of both $\Delta(z)$ and $\tilde{y}_{12}(L,z)$
are:

$\displaystyle\Delta(z)=\cos Lz+\frac{\sin
Lz}{2z}\int_{0}^{L}|q(x)|^{2}\mathrm{d}x+O(1/z^{2}),$ (4.6a)
$\displaystyle\tilde{y}_{12}(L,z)=-\sin
Lz\left(1-\frac{\text{i}}{2z}(q(0)-q^{*}(0))\right)-\frac{\cos
Lz}{2z}\int_{0}^{L}|q(x)|^{2}\mathrm{d}x+O(1/z^{2}).$ (4.6b)

These expression yields

$\frac{\tilde{y}^{2}_{12}(L,z)}{\Delta^{2}(z)-1}=-1+O(1/z),$ (4.7)

Then, recalling (3.10) and using Liouville’s theorem, we have that indeed
$\exp\big{(}\frac{1}{4}[(2A_{1}-A_{2}-A_{3})z+2B_{1}-B_{2}-B_{3}]\big{)}=\mathrm{e}^{\frac{\text{i}\pi}{4}}$,
which completes the proof. ∎

######  Definition 4.3.

Let the sectionally meromorphic $2\times 2$ matrix $\Phi(x,z)$ be defined as

$\Phi(x,z)=\Psi(x,z)B(z)\mathrm{e}^{\text{i}zx\sigma_{3}}\,,\quad
z\in\mathbb{C}\,.$ (4.8)

Also, let

$V(x,z)=\begin{cases}(-1)^{n+m(z)}\begin{pmatrix}{f^{-}(z)}/{f^{+}(z)}&0\\\
0&{f^{+}(z)}/{f^{-}(z)}\end{pmatrix}\,,&z\in[E_{2n-2},E_{2n-1}]\,,\\\
(-1)^{n+m(z)}\text{i}\mathrm{e}^{-2\text{i}zx\sigma_{3}}\sigma_{1}\,,&z\in[E_{2n-1},E_{2n}]\,.\end{cases}$
(4.9)

The counting function $m(z)$ in (4.9) is defined as follows: If there are no
$\gamma_{k}$ with $\sigma_{k}=0$, let $m(z)=0$. Alternatively, if there is at
least one $\gamma_{k}$ with $\sigma_{k}=0$, let $\gamma_{*}$ be the value of
$\gamma_{k}$ closest to zero with $\sigma_{k}=0$, and let

$m(z)=\begin{cases}\big{|}\\{\gamma_{k}:\gamma_{*}\leq\gamma_{k}\leq
z,\sigma_{k}=0\\}\big{|}\,,&z\geq\gamma_{*},\\\\[1.72218pt]
\big{|}\\{\gamma_{k}:z<\gamma_{k}<\gamma_{*},\sigma_{k}=0\\}\big{|}\,,&z<\gamma_{*}\,.\end{cases}$
(4.10)

In other words, for $z\geq\gamma_{*}$, $m(z)$ counts the number of Dirichlet
eigenvalues $\gamma_{k}$ smaller than or equal to $z$ with $\sigma_{k}=0$,
whereas for $z<\gamma_{*}$, $m(z)$ counts the number of Dirichlet eigenvalues
$\gamma_{k}$ larger than or equal to $z$ with $\sigma_{k}=0$.

######  Definition 4.4.

Let $R=\mathop{\rm max}\nolimits_{g^{-}\leq k\leq
g^{+}}\left\\{E_{2k}-E_{2k-1}\right\\}$. Define $D_{j}$ to be the open disc of
radius $R$ centered at $c_{j}=\frac{1}{2}(\zeta_{2j}+\zeta_{2j-1})$, and

$\mathcal{D}=\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^{-}}^{g^{+}}\bar{D}_{k}\right).$
(4.11)

######  Theorem 4.5.

The $2\times 2$ matrix-valued function $\Phi(x,z)$ defined by (4.8) solves the
following Riemann-Hilbert problem:

######  RHP 4.1.

Find a $2\times 2$ matrix-valued function $\Phi(x,z)$ such that

  1. 1.

$\Phi(x,z)$ is a holomorphic function for of $z$ for
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$.

  2. 2.

$\Phi^{\pm}(x,z)$ are continuous functions of $z$ for
$z\in{\mathbb{R}}\setminus\\{E_{k}\\}$, and have at worst quartic root
singularities on $\\{E_{k}\\}$.

  3. 3.

$\Phi^{\pm}(x,z)$ satisfies the jump relation

$\Phi^{+}(x,z)=\Phi^{-}(x,z)V(x,z),\qquad z\in\mathbb{R}\,,$ (4.12)

where $V(x,z)$ is given by (4.9).

  4. 4.

As $z\to\infty$ with $z\in\mathbb{C}$, $\Phi(x,z)$ has the following
asymptotic behavior

$\Phi(x,z)=\tilde{U}\,(I+O(1/z))B(z),$ (4.13)

with $\tilde{U}$ as in (2.15) and $B(z)$ as in (4.1).

  5. 5.

There exist positive constants $c$ and $M$ such that $|\phi_{ij}(x,z)|\leq
M\mathrm{e}^{c|z|^{2}}$ for all $z\in\mathcal{D}$.

###### Proof.

We prove each condition separately.

Condition 1. It follows from (3.5) that $\psi^{\pm}$ are meromorphic functions
of $z$ for $z\in\mathbb{C}\setminus\Sigma(\mathcal{L})$, so that $\Psi(x,z)$
is meromorphic in $\mathbb{C}\setminus\Sigma(\mathcal{L})$. From (4.1), 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 (3.5), we can
derive directly that $\Psi_{\pm}(x,z)$ can only be singular at $\mu_{j}$. From
(4.1), 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 dicuss
three cases: $\sigma_{j}=\pm 1$ and $\zeta_{2j}=\zeta_{2j-1}=\mu_{j}$.

Suppose that $\sigma_{j}=1$, which is only possible when $j=k$ for some $k$.
By looking at the definition of $\sigma_{j}$, we can derive that
$\tilde{y}_{11}(L,\mu_{j})=\rho(\mu_{j})^{-1}$. So that
$\rho^{-1}(z)-\tilde{y}_{11}(L,z)$ is holomorphic in $\mu_{j}$’s neighborhood,
and $\mu_{j}$ are the zeros of $\rho^{-1}(z)-\tilde{y}_{11}(L,z)$. For
$z\in{\mathbb{C}}^{+}$, we have

$\Phi_{+}=\frac{i\sqrt{f^{0}}}{\sqrt[4]{\Delta^{2}-1}}\big{(}\,\psi^{-}f^{-}\,,\,\psi^{+}f^{+}\,\big{)}\,\mathrm{e}^{\text{i}zx\sigma_{3}}.$
(4.14)

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 $\psi^{+}$. So
that $\Phi_{+}$ is nonsingular $\mu_{j}$. We can get the same conclusion for
$z\in{\mathbb{C}}^{-}$ using the same method.

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

Suppose that $\zeta_{2j-1}=\zeta_{2j}$. In this case,
$\rho(\mu_{j})=\rho^{-1}(\mu_{j})=\pm 1$. 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 that $\Phi_{\pm}$ are nonsigular at
$\mu_{j}$.

Next we will prove that $\Phi_{\pm}(x,z)$ have at worst quartic root
singularities on $\\{E_{k}\\}$. Suppose that $\sigma_{k}=\pm 1$, 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 $\sigma_{k}=0$, 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 either $\mu_{j}=E_{2j}$ or $\mu_{j}=E_{2j-1}$, it follows from Lemma
2.4 that $\rho^{\pm 1}(\mu_{j})-\tilde{y}_{11}(L,x)=0$ since
$\rho(\mu_{j})=\pm 1$ at $\mu_{j}$. As the singular behaviors of
$\psi^{\pm}(x,z)$ are from $(\rho^{\pm
1}(z)-\tilde{y}_{11}(L,z))/\tilde{y}_{12}(L,z)$, and the numerator involves
only holomorphic functions and the square root of a holomorphic function, the
zeros of the numerator at $\mu_{j}$ must have order at least
${\textstyle\frac{1}{2}}$. Since $\tilde{y}_{22}(L,z)$ 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. By considering the discrepancy in the boundary values of
$\sqrt{\Delta^{2}-1}$ on the branch cuts $\Sigma(\mathcal{L})$ we find that

$\rho_{+}(z)=\rho_{-}^{-1}(z),\hskip
14.22636pt\rho_{+}^{-1}(z)=\rho_{-}(z),\hskip
14.22636ptz\in\Sigma(\mathcal{L}).$ (4.15)

Therefore,

$\displaystyle\Psi_{+}(x,z)=\Psi_{-}(x,z)\,,\quad z\in\Sigma(\mathcal{L})\,,$
(4.16) $\displaystyle\Psi_{+}(x,z)=\Psi_{-}(x,z)\sigma_{1}\,,\quad
z\in\mathbb{R}\setminus\Sigma(\mathcal{L})\,,$ (4.17)

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

$V(x,z)=\begin{cases}\mathrm{e}^{-\text{i}zx\sigma_{3}}(B_{-})^{-1}B_{+}\mathrm{e}^{\text{i}zx\sigma_{3}},&z\in\Sigma(\mathcal{L})\,,\\\
\mathrm{e}^{-\text{i}zx\sigma_{3}}(B_{-})^{-1}\sigma_{1}B_{+}\mathrm{e}^{\text{i}zx\sigma_{3}},&z\in\mathbb{R}\setminus\Sigma(\mathcal{L})\,,\end{cases}$
(4.18)

Next, we show that (4.18) reduces to (4.9). To do this, we need to compute the
product $(B_{-})^{-1}B_{+}$. Since the functions $f^{\pm}(z)$ appearing in
(4.1) 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 (4.4a). It is convenient to
consider separately the numerator and denominator of the right hand side of
(4.4a) and to compute their jumps individually. We therefore define

$\displaystyle
p(z)=\left(\prod_{k=g^{-}\atop\sigma_{k}=0}^{g^{+}}\bigg{(}1-\frac{z}{\gamma_{k}}\bigg{)}\,\mathrm{e}^{z/\gamma_{k}}\right)^{\frac{1}{2}}\,,$
(4.19a) $\displaystyle
h(z)=\left(\prod_{k=g^{-}\atop\sigma_{k}=0}^{g^{+}}{\frac{1}{E_{2k-1}E_{2k}}(E_{2k-1}-z)(E_{2k}-z)\mathrm{e}^{z\big{(}\frac{1}{E_{2k-1}}+\frac{1}{E_{2k}}\big{)}}}\right)^{\frac{1}{4}}\,,$
(4.19b)

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 (4.4a). We choose the branch cut for the square root in (4.19a)
consistently with the definition of the counting function $m(z)$ in (4.10).
Namely, if there are no $\gamma_{k}$ with $\sigma_{k}=0$, $p(z)\equiv 1$ and
there is no branch cut. If there is at least one $\gamma_{k}$ with
$\sigma_{k}=0$, recall that $\gamma_{*}$ is the value of $\gamma_{k}$ closest
to zero with $\sigma_{k}=0$. 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

$\displaystyle\frac{p_{-}(z)}{p_{+}(z)}=(-1)^{m(z)},\qquad z\in\mathbb{R},$
(4.20a)
$\displaystyle\frac{h_{-}(z)}{h_{+}(z)}=\begin{cases}(-1)^{n-1}\text{i},&z\in[E_{2n-2},E_{2n-1}]\subset\Sigma(\mathcal{L})\,,\\\
(-1)^{n},&z\in[E_{2n-1},E_{2n}]\subset\mathbb{R}\setminus\Sigma(\mathcal{L})\,.\end{cases}$
(4.20b)

Inserting (4.20) in (4.18), one now obtains (4.9).

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

$\Psi(x,z)=(\tilde{U}+O(1/z))\,\mathrm{e}^{-\text{i}zx\sigma_{3}},\qquad
z\to\infty,\leavevmode\nobreak\ z\in\mathbb{C}.$ (4.21)

Therefore,

$\tilde{U}^{-1}\Psi(x,z)\mathrm{e}^{\text{i}zx\sigma_{3}}=I+O(1/z),\qquad
z\to\infty$ (4.22)

for $z\in\mathbb{C}$, implying (4.13).

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

$\phi_{ij}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup\overline{D}_{k}\right)\right),\hskip
14.22636pt\psi_{ij}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup\overline{D}_{k}\right)\right).$
(4.23)

In appendix A.5 we show that the entire functions $y_{ij}(x,z)$ have order at
most $1$. Thus, by Proposition A.10, we have

$\tilde{y}_{ij}(x,z)\in\mathcal{A}_{2}(\mathbb{C}).$ (4.24)

Next, since

$\rho^{\pm 1}(z)\leq|\Delta(z)|+\sqrt{|\Delta^{2}(z)-1|}\leq 1+|\Delta(z)|,$
(4.25)

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

$\rho^{\pm 1}(z)\in\mathcal{A}_{2}(\mathbb{C}\setminus\mathbb{R}).$ (4.26)

Applying Lemma A.16 to (3.10a), we can conclude that

$\tilde{y}_{12}(L,z)^{-1}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\bigcup_{j\in\mathbb{Z}}\overline{D}_{j}\right).$
(4.27)

Therefore, using Proposition A.9 and A.10, we can establish that

$\psi_{ij}\in\mathcal{A}_{2}(\mathcal{D}).$ (4.28)

The smallest numbers $a_{\pm}$ and $a_{0}$ so that

$\sum_{\sigma_{j}=\pm 1}\mu_{j}^{-a_{\pm}},\hskip
5.69046pt\sum_{\sigma_{j}=0}\mu_{j}^{-a_{0}}$ (4.29)

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,

$f^{\pm},f^{0}\in\mathcal{A}_{2}(\mathbb{C}).$ (4.30)

Applying Lemma A.16 to (3.10b) and (3.10c), we can conclude that

$\frac{1}{\Delta^{2}-1}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\bigcup_{j}\overline{D}_{j}\right).$
(4.31)

Combining both (4.30) and (4.31), we have

$g^{\pm}(z):=\frac{f^{0}(z)f^{\pm}(z)}{\Delta^{2}(z)-1}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\bigcup_{j}\overline{D}_{j}\right).$
(4.32)

If $\mu_{j}\notin\\{E_{k}\\}$, then we can easily prove that $g^{\pm}(z)$ is
analytic at $\mu_{j}$, which coincides with the center of $D_{j}$. Therefore,
$g^{\pm}(z)$ are analytic in such a $D_{j}$. Finally, using Lemma A.16, we
have

$g^{\pm}(z)\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right).$
(4.33)

Recalling the definition (4.8) of $\Phi(x,z)$, this completes the proof of
condition 5. ∎

######  Lemma 4.6.

If $\Phi(x,z)$ solves the RHP 4.1, then $\det\Phi(x,z)\equiv 1$.

###### Proof.

We first prove that the particular solution to RHP 4.1 given by (4.8) has unit
determinant. We have

$\det\Phi(x,z)=\det
B(z)\det\Psi(x,z)=\begin{cases}\displaystyle-\frac{\tilde{y}_{12}(L,z)}{2\sqrt{\Delta^{2}-1}}\det\Psi_{+}\,,&z\in\overline{\mathbb{C}^{+}}\,,\\\\[4.30554pt]
\displaystyle\frac{\tilde{y}_{12}(L,z)}{2\sqrt{\Delta^{2}-1}}\det\Psi_{-}\,,&z\in\overline{\mathbb{C}^{-}}\,.\end{cases}$
(4.34)

We also have

$\det\Psi(x,z)=\det\Psi(0,z)=\begin{cases}\displaystyle\frac{\rho(z)-\rho^{-1}(z)}{\tilde{y}_{12}(L,z)}=-\frac{2\sqrt{\Delta^{2}-1}}{\tilde{y}_{12}(L,z)}\,,&z\in\overline{\mathbb{C}^{+}}\,,\\\
\displaystyle\frac{\rho^{-1}(z)-\rho(z)}{\tilde{y}_{12}(L,z)}=\frac{2\sqrt{\Delta^{2}-1}}{\tilde{y}_{12}(L,z)}\,,&z\in\overline{\mathbb{C}^{-}}\,.\end{cases}$
(4.35)

Thus, we conclude that $\det\Phi(x,z)=1$ for all $x$ and $z$. Comparing
$\det\Phi\equiv 1$ with the asymptotics of $\Phi(x,z)$ in (4.13), we have

$\det\left(\tilde{U}\,B(z)\right)=1+O(1/z)\,,$ (4.36)

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
$f(x,z)=\det\tilde{\Phi}(x,z)$, which is holomorphic in
$\mathbb{C}\setminus\mathbb{R}$. As $\det V(x,z)=1$, we can derive that for
$z\in\mathbb{R}\setminus\\{E_{k}\\}$

$f_{+}(x,z)=\det\tilde{\Phi}_{+}(x,z)=\det(V(x,z)\tilde{\Phi}_{-}(x,z))=\det\tilde{\Phi}_{-}(x,z)=f_{-}(x,z)\,.$
(4.37)

As $\tilde{\Phi}$ also satisfies the condition 2 in RHP 4.1, $f(x,z)$ has at
worst square root singularities at $E_{k}$. However, isolated singularities of
a holomorphic function have at least order 1. So that $f$ can be extended to
an entire function.

To get $f\equiv 1$, we still need to prove that $f$ is bounded. First, we
prove that $f\in\mathcal{A}_{2}(\mathbb{C})$. From condition 5 of RHP 4.1, we
have

$\phi_{ij},\tilde{\phi}_{ij}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right)\right)\,.$
(4.38)

Hence, we can conclude that

$f\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right)\right)\,.$
(4.39)

Since $f$ is an entire function, by continuity, we have

$f\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right)\,.$
(4.40)

According to Proposition A.13, we can derive that
$f\in\mathcal{A}_{2}(\mathbb{C})$ . Comparison of (4.13) with (4.36) implies
that

$f(x,z)=1+O(1/z),$ (4.41)

as $z\to\infty$ and $z\in\mathbb{C}$. Again since $f(x,z)$ is entire, the
asymptotic behavior of $f(x,z)$ in (4.41) indicates that $f(x,z)$ is bounded
for all $z\in\mathbb{C}$. By Liouville’s theorem, $f(x,z)$ is constant and
therefore $f(x,z)\equiv 1$. ∎

######  Theorem 4.7.

For any fixed $x\in\mathbb{R}$, the solution of RHP 4.1 is unique.

###### Proof.

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

$R(x,z):=\Phi(x,z)\tilde{\Phi}^{-1}(x,z)\,.$ (4.42)

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

$\tilde{\Phi}^{-1}(x,z)=\begin{pmatrix}\tilde{\phi}_{22}(x,z)&-\tilde{\phi}_{12}(x,z)\\\
-\tilde{\phi}_{21}(x,z)&\tilde{\phi}_{11}(x,z)\end{pmatrix}\,.$ (4.43)

The function $R(x,z)$ has no jump, so that it can be extended to a holomorphic
function on $\mathbb{C}\setminus\\{E_{k}\\}$. Since the entries of $\Phi$ and
$\tilde{\Phi}$ have at worst quartic root singularities at $E_{k}$, $R(x,z)$
has at worst square root singularities at $E_{k}$. However, isolated
singularities of a holomorphic function has at least order 1. Thus $R(x,z)$ 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

$\phi_{ij},\tilde{\phi}_{ij}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right)\right)\,.$
(4.44)

Hence, we can conclude that

$r_{ij}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\left(\mathbb{R}\cup\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right)\right)\,.$
(4.45)

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

$r_{ij}\in\mathcal{A}_{2}\left(\mathbb{C}\setminus\bigcup_{k=g^{-}}^{g^{+}}\overline{D}_{k}\right)\,.$
(4.46)

By Proposition A.13, we can derive that
$r_{ij}\in\mathcal{A}_{2}(\mathbb{C})$.

The asymptotic behavior of both of the matrices $\Phi$ and $\tilde{\Phi}$ as
$z\to\infty$ is given by (4.13). Therefore we can calculate the asymptotic
behavior of $\tilde{\Phi}^{-1}(x,z)$ as

$\tilde{\Phi}^{-1}(x,z)=B^{-1}(z)(I+O(1/z))\tilde{U}^{-1}\,,\qquad
z\to\infty\,.$ (4.47)

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

$R(x,z)=\tilde{U}\,(I+O(1/z))\,\tilde{U}^{-1}=I+O(1/z)\,,\qquad z\to\infty\,.$
(4.48)

Again, since$R(x,z)$ is entire, the asymptotic behavior (4.48) indicates that
$r_{ij}$ are bounded for $z\in\mathbb{C}$. Using Liouville’s theorem, we have
that $R(x,z)$ is constant. The asymptotic behavior of $R(x,z)$ in (4.48)
implies that

$R(x,z)\equiv I\,.$ (4.49)

Hence, the solution of RHP 4.1 is unique. ∎

######  Corollary 4.8.

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

$Q(x)=\lim_{z\to\infty}\text{i}z[\sigma_{3},U^{\dagger}\Phi(x,z)B^{-1}(z)]\,,$
(4.50)

with $U$ as in (2.15).

###### Proof.

The reconstruction of $Q(x)$ in (4.50) follows from (4.8) and Corollary 3.8. ∎

##  5 Initial value problem for the defocusing NLS equation with periodic BC

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

$q(x+L,t)=q(x,t)\,.$ (5.1)

Specifically, we construct a Riemann-Hilbert characterization 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 (1.7), whose first half is the
ZS problem (1.7a). An equivalent formulation of the zero-curvature condition
is the Lax equation

$\mathcal{L}_{t}=[\mathcal{A},\mathcal{L}]\,,$ (5.2)

with $\mathcal{L}$ as in (1.2) and

$\mathcal{A}=2\text{i}\sigma_{3}\partial_{x}^{2}+2\text{i}Q\sigma_{3}\partial_{x}-\text{i}(Q^{2}-Q_{x})\sigma_{3}\,.$
(5.3)

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 can be easily proven by showing that the Floquet
discriminant $\Delta(z)$ is time-independent). On the other hand, the movable
Dirichlet eigenvalues depend on time. Thus, in order to construct an effective
time-dependent Riemann-Hilbert problem, one needs to obtain evolution
equations that determine the time evolution of the Dirichlet eigenvalues and
Bloch-Floquet eigenfunctions. We turn to this task next.

######  Proposition 5.1.

The Dirichlet eigenvalues satisfy the following ODE

${\frac{\partial\mu_{n}(t)}{\partial
t}}=\sigma_{n}(t)c_{1}(\mu_{n}(t))\frac{\rho(\mu_{n}(t))-\rho^{-1}(\mu_{n}(t))}{\tilde{y}^{\prime}_{12}(L,t,\mu_{n}(t))},\qquad
n\in\mathbb{Z}\,,$ (5.4a) where
$c_{1}(z)=-2z^{2}-\text{i}z(q^{*}(0,t)-q(0,t))-|q(0,t)|^{2}-\frac{1}{2}(q_{x}(0,t)+q_{x}^{*}(0,t))\,.$
(5.4b)

######  Remark 5.2.

As a special case, (5.4a) shows that the fixed Dirichlet eigenvalues, for
which $\sigma_{n}=0$, are indeed time-independent, as was mentioned in section
2.

###### Proof of Proposition 5.1.

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

$\tilde{\mathcal{L}}_{t}=[\tilde{\mathcal{A}},\tilde{\mathcal{L}}]\,,$ (5.5)

where

$\displaystyle\tilde{\mathcal{L}}=U\mathcal{L}U^{-1}\,,\qquad\tilde{\mathcal{A}}=U\tilde{\mathcal{A}}U^{-1}\,.$
(5.6)

From Lemma 2.4, we know that $\tilde{y}_{2}$ is a Bloch-Floquet solution.
Therefore, $\tilde{y}_{2}$ satisfies
$\tilde{\mathcal{L}}\tilde{y}_{2}=z\tilde{y}_{2}$. Differentiating both sides
with respect to $t$, we can obtain

$\tilde{\mathcal{L}}_{t}\tilde{y}_{2}+\tilde{\mathcal{L}}\tilde{y}_{2,t}=z\tilde{y}_{2,t}\,.$
(5.7)

Combining (5.7) with the modified Lax equation (5.5) implies

$\tilde{\mathcal{L}}_{t}\tilde{y}_{2}=z\tilde{y}_{2,t}-\tilde{\mathcal{L}}\tilde{y}_{2,t}=\tilde{\mathcal{A}}\tilde{\mathcal{L}}\tilde{y}_{2}-\tilde{\mathcal{L}}\tilde{\mathcal{A}}\tilde{y}_{2}=z\tilde{\mathcal{A}}\tilde{y}_{2}-\tilde{\mathcal{L}}\tilde{\mathcal{A}}\tilde{y}_{2}\,.$
(5.8)

Hence we have

$\tilde{\mathcal{L}}(\tilde{y}_{2,t}-\tilde{\mathcal{A}}\tilde{y}_{2})=z(\tilde{y}_{2,t}-\tilde{\mathcal{A}}\tilde{y}_{2})\,.$
(5.9)

Therefore, for fixed $t$ and $z$, there exist constants $c_{1}$ and $c_{2}$
such that

$\tilde{y}_{2,t}-\tilde{\mathcal{A}}\tilde{y}_{2}=c_{1}\tilde{y}_{1}+c_{2}\tilde{y}_{2}\,.$
(5.10)

From the normalizations of $\tilde{y}_{1}$ and $\tilde{y}_{2}$ at $x=0$ we
find that $\tilde{y}_{12,t}(0,t,z)=\tilde{y}_{22,t}(0,t,z)=0$, so we can
derive that

$c_{1}=-(\tilde{\mathcal{A}}\tilde{y}_{2}(0,t,z))_{1},\hskip
14.22636ptc_{2}=-(\tilde{\mathcal{A}}\tilde{y}_{2}(0,t,z))_{2}\,.$ (5.11)

From the modified equation (2.16) it follows that

$\tilde{y}_{2,xx}(x,t,z)=-\text{i}zU\sigma_{3}U^{-1}\tilde{y}_{2,x}(x,t,z)+UQ(x,t)U^{-1}\tilde{y}_{2,x}(x,t,z)+UQ_{x}(x,t)U^{-1}\tilde{y}_{2}(x,t,z)\,,$
(5.12)

and therefore

$\displaystyle\tilde{\mathcal{A}}\tilde{y}_{2}(x,t,z)=-2\text{i}z^{2}U\sigma_{3}U^{-1}\tilde{y}_{2}(x,t,z)+2zUQU^{-1}\tilde{y}_{2}(x,t,z)-\text{i}U(Q^{2}+Q_{x})\sigma_{3}U^{-1}\tilde{y}_{2}(x,t,z)\,.$
(5.13)

Evaluation at $x=0$ gives

$\displaystyle
c_{1}=-2z^{2}-\text{i}z(q^{*}(0,t)-q(0,t))-|q(0,t)|^{2}-{\textstyle\frac{1}{2}}(q_{x}(0,t)+q_{x}^{*}(0,t))\,,$
(5.14a) $\displaystyle
c_{2}=z(q^{*}(0,t)+q(0,t))-\frac{\text{i}}{2}(q_{x}(0,t)-q_{x}^{*}(0,t))\,.$
(5.14b)

Plugging (5.13) and (5.14) into (5.10) and evaluating (5.10) at $x=L$ and
$z=\mu_{n}(t)$, we get

$\tilde{y}_{12,t}(L,t,\mu_{n}(t))=\big{(}2\mu_{n}(t)^{2}+\text{i}\mu_{n}(t)(q^{*}(0,t)-q(0,t))+|q(0,t)|^{2}+{\textstyle\frac{1}{2}}(q_{x}(0,t)+q_{x}^{*}(0,t))\big{)}\\\
\big{(}\tilde{y}_{22}(L,t,\mu_{n}(t))-\tilde{y}_{11}(L,t,\mu_{n}(t))\big{)}\,.$
(5.15)

Differentiating the expansion (3.10a) in $t$, we have

$\tilde{y}_{12,t}(L,t,z)=\mathrm{e}^{A_{1}z+B_{1}}\sum_{n\in\mathbb{Z}}\left[\frac{z\mu_{n,t}(t)}{\mu_{n}^{2}(t)}\mathrm{e}^{\frac{z}{\mu_{n}(t)}}\frac{z}{\mu_{n}(t)}\prod_{j\neq
n}\left(1-\frac{z}{\mu_{j}(t)}\right)\mathrm{e}^{\frac{z}{\mu_{j}(t)}}\right]\,.$
(5.16)

While differentiating (3.10a) in $z$ we have

$\tilde{y}^{\prime}_{12}(L,t,z)=A_{1}\mathrm{e}^{A_{1}z+B_{1}}\prod_{j\in\mathbb{Z}}\left(1-\frac{z}{\mu_{j}(t)}\right)\mathrm{e}^{\frac{z}{\mu_{j}(t)}}+\mathrm{e}^{A_{1}z+B_{1}}\sum_{n\in\mathbb{Z}}\left[-\frac{z}{\mu_{n}^{2}(t)}\mathrm{e}^{\frac{z}{\mu_{n}(t)}}\prod_{j\neq
n}\left(1-\frac{z}{\mu_{j}(t)}\right)\mathrm{e}^{\frac{z}{\mu_{j}(t)}}\right]\,.$
(5.17)

Evaluating (5.16) and (5.17) at $z=\mu_{n}(t)$ gives
$\tilde{y}_{12,t}(L,t,\mu_{n}(t))=-\mu_{n,t}(t)\tilde{y}^{\prime}_{12}(L,t,\mu_{n}(t))$.
Therefore, we can express $\mu_{n,t}(t)$ as:

$\mu_{n,t}(t)=-\frac{\tilde{y}_{12,t}(L,t,\mu_{n}(t))}{\tilde{y}^{\prime}_{12}(L,t,\mu_{n}(t))}=\frac{c_{1}(\mu_{n}(t))(\tilde{y}_{22}(L,t,\mu_{n}(t))-\tilde{y}_{11}(L,t,\mu_{n}(t)))}{\tilde{y}^{\prime}_{12}(L,t,\mu_{n}(t))}\,,$

which becomes (5.4a) upon substituting $c_{1}$ using (5.4b). ∎

######  Proposition 5.3.

The time-dependent Bloch-Floquet solutions $\psi^{\pm}(x,t,z)$ satisfy the
ODEs

$\psi^{\pm}_{t}(x,t,z)+\alpha^{\pm}(t,z)\psi^{\pm}(x,t,z)=\tilde{\mathcal{A}}\psi^{\pm}(x,t,z)$
(5.18)

where

$\displaystyle\begin{split}\alpha^{\pm}(t,z)=&z(q^{*}(0,t)+q(0,t))-\frac{\text{i}}{2}(q_{x}(0,t)-q^{*}_{x}(0,t))\\\
&+\Big{(}2z^{2}+\text{i}z\big{(}q^{*}(0,t)-q(0,t)\big{)}+|q(0,t)|^{2}+{\textstyle\frac{1}{2}}q_{x}(0,t)+{\textstyle\frac{1}{2}}q_{x}^{*}(0,t)\Big{)}\psi^{\pm}_{2}(0,t,z)\,.\end{split}$
(5.19)

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

$\displaystyle\alpha^{\pm}=\pm 2\text{i}z^{2}+O(1/z),\hskip
14.22636ptz\in\overline{\mathbb{C}^{+}};$ (5.20a)
$\displaystyle\alpha^{\pm}=\mp 2\text{i}z^{2}+O(1/z),\hskip
14.22636ptz\in\overline{\mathbb{C}^{-}}.$ (5.20b)

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

###### Proof.

At each time $t$, we can produce normalized Bloch-Floquet solutions
$\psi^{\pm}(x,t,z)$ that factor as

$\psi^{+}(x,t,z)=p^{+}(x,t,z)\rho(z)^{x/L},\hskip
14.22636pt\psi^{-}(x,t,z)=p^{-}(x,t,z)\rho(z)^{-x/L}\,.$ (5.21)

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

$\tilde{\mathcal{L}}_{t}\psi^{+}+\tilde{\mathcal{L}}\psi^{+}_{t}-z\psi^{+}_{t}=[\tilde{\mathcal{A}},\tilde{\mathcal{L}}]\psi^{+}+\tilde{\mathcal{L}}\psi^{+}_{t}-z\psi_{t}^{+}=(\tilde{\mathcal{L}}-z)(\psi^{+}_{t}-\tilde{\mathcal{A}}\psi^{+})=0.$
(5.22)

So $\psi_{t}^{+}-\tilde{\mathcal{A}}\psi^{+}$ is a solution to (2.16) for all
time. Therefore, it can be written as a linear combination of $\psi^{+}$ and
$\psi^{-}$:

$p^{+}_{t}\rho(z)^{x/L}-\tilde{\mathcal{A}}\,p^{+}\rho(z)^{x/L}=-\alpha^{+}(t,z)p^{+}\rho(z)^{x/L}-\beta^{+}(t,z)p^{-}\rho(z)^{-x/L},$
(5.23)

where $\alpha^{+}(t,z)$ and $\beta^{+}(t,z)$ are some undetermined functions
independent of $x$. Recall that, for all
$z\in\mathbb{C}\setminus\Sigma(\mathcal{L})$, we have $|\rho(z)|<1$.
Evaluating the LHS and RHS of the above equation in the limit $x\to\infty$,
every term in the above equation decays exponentially except the last one.
Therefore, in order for the equation to hold, one needs $\beta^{+}(t,z)=0$.
Hence $\psi^{+}$ solves (5.18) with the + sign. An analogous argument, but now
taking the limit as $x\to-\infty$, shows that $\psi^{-}$ solves (5.18) with
the minus sign, for some $\alpha^{-}(t,z)$.

Next we derive (5.19) and (5.20). To do so, we evaluate the first component of
(5.18) 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 (5.18) yields

$\alpha^{\pm}(t,z)=(\mathcal{A}\psi^{\pm})_{1}(0,t,z)\,.$ (5.24)

From (5.13) we have that

$\tilde{\mathcal{A}}v=-2\text{i}z^{2}U\sigma_{3}U^{-1}v+2zUQU^{-1}v-\text{i}U(Q^{2}+Q_{x})\sigma_{3}U^{-1}v\,.$
(5.25)

Explicitly, the first component of the above equation is

$(\tilde{\mathcal{A}}v)_{1}=2z^{2}\,v_{2}+z\,\big{[}(q^{*}+q)\,v_{1}+\text{i}(q^{*}-q)v_{2}\big{]}\\\
-{\textstyle\frac{\text{i}}{2}}\big{[}(q^{*}_{x}-q_{x})\,v_{1}+\text{i}(2|q|^{2}+q^{*}_{x}+q_{x})v_{2}\big{]}\,.$
(5.26)

Letting $v(x,t,z)=\psi^{\pm}(x,t,z)$, evaluating (5.26) at $x=0$ recalling
that (3.5) evaluated at $x=0$ yields $\psi^{\pm}_{1}(0,t,z)=1$ and
$\psi^{\pm}_{2}(0,t,z)=(\rho^{\pm
1}-\tilde{y}_{11}(L,t,z))/\tilde{y}_{12}(L,t,z)$, (5.24) then yields (5.19). ∎

We are now ready to define the 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.,

$e^{\pm}(t,z)=\exp\left(\int_{0}^{t}\alpha^{\pm}(\tau,z)\mathrm{d}\tau\right)\,.$
(5.27)

Define

$\check{\psi}^{\pm}(x,t,z)=\psi^{\pm}(x,t,z)e^{\pm}(t,z)\,.$ (5.28)

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 (5.5) is the compatibility condition.

######  Proposition 5.4.

The functions $e^{\pm}(t,z)$ satisfy the following properties:

  1. 1.

$e^{\pm}(t,z)$ are meromorphic functions in
$\mathbb{C}\setminus\Sigma(\mathcal{L})$.

  2. 2.

$e^{+}(t,z)$ has simple poles on $\gamma_{k}(0)$ as $\sigma_{k}(0)=1$and
simple zeros on $\gamma_{k}(t)=1$ as $\sigma_{k}(t)=1$; $e^{-}(t,z)$ has
simple poles on $\gamma_{k}(0)$ as $\sigma_{k}(0)=-1$ and simple zeros on
$\gamma_{k}(t)$ as $\sigma_{k}(t)=-1$.

  3. 3.

$e^{\pm}(t,z)$ both have square root singularities at $\gamma_{k}(0)$ when
$\sigma_{k}(0)=0$ and square root zeros at $\gamma_{k}(t)$ when
$\sigma_{k}(t)=0$.

  4. 4.

The boundary values of $e^{\pm}$ satisfy $e^{\pm}_{+}(t,z)=e^{\mp}_{-}(t,z)$
for $z\in\Sigma(\mathcal{L})$.

  5. 5.

For fixed $t$, $e^{\pm}$ have the following asymptotic behaviors as
$z\rightarrow\infty$:

$e^{\pm}(t,z)=\begin{cases}\mathrm{e}^{\pm
2\text{i}z^{2}t}(1+O(1/z)),&z\in\overline{\mathbb{C}^{+}}\\\ \mathrm{e}^{\mp
2\text{i}z^{2}t}(1+O(1/z)),&z\in\overline{\mathbb{C}^{-}}\end{cases}\,.$
(5.29)

  6. 6.

There exist positive constants $c$ and $M$ such that $|e^{\pm}(t,z)|\leq
M\mathrm{e}^{c|z|^{2}}$.

###### 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
$\sigma_{n}=1$. 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

$e^{+}(t,z)=\exp\left(-\int_{0}^{t}\frac{\mu_{n\tau}(\tau)\mathrm{d}\tau}{z-\mu_{n}(\tau)}\right)e^{+}_{\rm
reg}(t,z)\,,$ (5.30)

where $e^{+}_{\rm reg}(t,z)$ is holomorphic and nonzero 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 $\mu_{n}$

$-\int_{0}^{t}\frac{\mu_{n\tau}(\tau)\mathrm{d}\tau}{z-\mu_{n}(\tau)}=\int_{\mu_{n}(0)}^{\mu_{n}(t)}\frac{\mathrm{d}\mu_{n}}{\mu_{n}-z}=\log(\mu_{n}(t)-z)-\log(\mu_{n}(0)-z)\,,$
(5.31)

which implies

$e^{+}(t,z)=\frac{z-\mu_{n}(t)}{z-\mu_{n}(0)}e^{+}_{\rm reg}(t,z)\,.$ (5.32)

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

$w(z)=\begin{cases}\rho(z)-\rho^{-1}(z)\,,&z\in{\mathbb{C}}^{+},\quad
z\leavevmode\nobreak\ \text{near}\leavevmode\nobreak\ \zeta_{2n}\,,\\\
\rho^{-1}(z)-\rho(z)\,,&z\in{\mathbb{C}}^{-},\quad z\leavevmode\nobreak\
\text{near}\leavevmode\nobreak\ \zeta_{2n}\,.\end{cases}$ (5.33)

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

$\alpha(t,w)=\alpha^{\pm}(t,z(w))\,,\qquad z(w)\in{\mathbb{C}}^{\pm}\,,$
(5.34)

is meromorphic in a neighborhood of $w=0$, with a single simple pole at
$w(\mu_{n})$. Then

$e(t,w)=e^{\pm}(t,z(w))\,,\qquad z(w)\in{\mathbb{C}}^{\pm}\,,$ (5.35)

is related to $\alpha$ by

$e(t,w)=\exp\bigg{(}\int_{0}^{t}\alpha(\tau,w)\mathrm{d}\tau\bigg{)}.$ (5.36)

The derivatives $w(z)$ is

$w^{\prime}(z)=\pm\frac{\rho^{\prime}(z)}{\rho(z)}\Delta(z)\,,\quad
z\in{\mathbb{C}}^{\pm},$ (5.37)

which implies that

$w_{t}(\mu_{n}(t))=\sigma_{n}(t)\frac{\rho^{\prime}(z)}{\rho(z)}\Delta(z)|_{z=z(w(\mu_{n}(t)))}\mu_{n,t}(t).$
(5.38)

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

$\frac{\rho^{\prime}(z)}{\rho(z)}=-\frac{\Delta^{\prime}(z)}{\sqrt{\Delta^{2}(z)-1}}.$
(5.39)

Using (5.4a) and (5.39) in (5.38) indicates

$w_{t}(\mu_{n}(t))=\frac{\Delta(z)\Delta^{\prime}(z)}{\tilde{y}^{\prime}_{12}(L,t,z)}\bigg{|}_{z=z(w(\mu_{n}(t))}c_{1}(\mu_{n}(t)),$
(5.40)

where $c_{1}(z)$ is defined in (5.4b).

Since $\mu_{n}(t)$ is a pole of $\alpha$, a straightforward computation gives
that

$\alpha(w)=\frac{-w_{t}(\mu_{n}(t))}{w-w(\mu_{n}(t))}+\text{ reg. near
}\zeta_{2n}.$ (5.41)

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

$e(w)=\exp\left(-\int_{0}^{t}\frac{-w_{t}(\mu_{n}(\tau))}{w-w(\mu_{n}(\tau))}\mathrm{d}\tau\right)e_{\rm
reg}(w)=\frac{w-w(\mu_{n}(t))}{w-w(\mu_{n}(0))}e_{\rm reg}(w),$ (5.42)

where $e_{\rm reg}$ is analytic and nonzero near $w=0$. So we complete the
proof of Condition 2.

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

Condition 4. 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

$\int_{0}^{t}\alpha^{+}(\tau,z)\,\mathrm{d}\tau=\pm
2\text{i}z^{2}t+O(1/z)\,,\quad z\in\overline{{\mathbb{C}}^{\pm}}\,,$ (5.43)

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

Condition 5. From the asymptotic behaviors of $\alpha^{\pm}$ (5.20), a
subalgebraic bound of the form $|\alpha^{\pm}(t,z)|\leq\mathop{\rm
max}\nolimits\\{Cz^{2},C^{\prime}\\}$ 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 centered 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
$\mathcal{D}^{+}_{U}=\\{z\in\mathbb{C}^{+}:\mathop{\rm
Im}\nolimits{z}>\mathop{\rm max}\nolimits\\{R,\log 2/L\\}\\}$. From (3.1), we
have that

$\tilde{y}_{12}(L,z)=\sin(zL)+O(1/z)\,.$ (5.44)

Using the lower bound
$|\sin(zL)|\geq{\textstyle\frac{1}{2}}\left(|\mathrm{e}^{-\text{i}zL}|-{\textstyle\frac{1}{2}}\right)\geq\frac{|\mathrm{e}^{-\text{i}zL}|}{4}$
we have, for $|z|$ large enough,

$|\tilde{y}_{12}(L,z)|\geq C|\mathrm{e}^{-\text{i}zL}|.$ (5.45)

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

$|\rho^{\pm}(z)-\tilde{y}_{11}(L,z)|\leq C|\mathrm{e}^{-\text{i}zL}|.$ (5.46)

These together with (5.19) give us

$|\alpha^{\pm}(t,z)|\leq\mathop{\rm max}\nolimits\\{C|z|^{2},C^{\prime}\\}$
(5.47)

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 $C_{n}$ labeled 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 $R^{\prime}/n$ for some constant $R^{\prime}$. As
$z\rightarrow\infty$, (5.44) gives us that

$\tilde{y}^{\prime}_{12}(L,z)=L\cos(zL)+O(1/z).$ (5.48)

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

$\tilde{y}^{\prime}_{12}(L,\mu_{n})=L+O(\mu_{n}^{-1}).$ (5.49)

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

$\tilde{y}_{12}(L,z)=(L+O(\mu_{n}^{-1}))(z-\mu_{n})+O(z-\mu_{n})\geq\frac{RL}{n}$
(5.50)

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 $|\tilde{y}_{12}(L,z)^{-1}|\leq\mathop{\rm
max}\nolimits\\{C|z|,C^{\prime}\\}$ for $z\in\mathcal{D}^{+}_{L}$. This
completes the proof of condition 5. ∎

######  Definition 5.5.

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

$\check{V}(x,t,z)=\begin{cases}(-1)^{n+m(z)}\begin{pmatrix}{f^{-}(z)}/{f^{+}(z)}&0\\\
0&{f^{+}(z)}/{f^{-}(z)}\end{pmatrix},&z\in[E_{2n-2},E_{2n-1}],\\\
(-1)^{n+m(z)}\text{i}\mathrm{e}^{-2\text{i}zx\sigma_{3}-4\text{i}z^{2}t\sigma_{3}}\sigma_{1},&z\in[E_{2n-1},E_{2n}],\end{cases}$
(5.51)

with $m(z)$ still defined by (4.10), and let the matrix-valued time-dependent
Bloch-Floquet solution $\check{\Psi}(x,t,z)$ be defined as in (3.4) but with
$\psi_{j}^{\pm}$ replaced by $\check{\psi}_{j}^{\pm}$, as defined in (5.28).
Finally, let $\check{\Phi}(x,t,z)$ be given by

$\check{\Phi}(x,t,z)=\check{\Psi}(x,t,z)B(z)\mathrm{e}^{\text{i}\sigma_{3}zx+2\text{i}\sigma_{3}z^{2}t}\,.$
(5.52)

######  Theorem 5.6.

Suppose that $q(x,t)$ is the solution to the defocusing NLS equation (1.6)
with smooth initial data $q(x,0)=q_{0}(x)$ and let $\Sigma(q_{0})$ be the
spectral data for corresponding Dirac operator. There exists a solution
$\check{\Phi}$ to the following Riemann-Hilbert problem, constructed by
(5.52), which is uniquely determined by this following Riemann-Hilbert
problem:

######  RHP 5.1.

Find a $2\times 2$ matrix-valued function $\check{\Phi}(x,t,z)$ such that

  1. 1.

$\check{\Phi}(x,t,z)$ is a holomorphic function of $z$ for
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$.

  2. 2.

$\check{\Phi}^{\pm}(x,t,z)$ are continuous functions of $z$ for
$z\in{\mathbb{R}}\setminus\\{E_{k}\\}$, and have at worst quartic root
singularities on $\\{E_{k}\\}$.

  3. 3.

$\check{\Phi}^{\pm}(x,t,z)$ satisfy the jump relation
$\check{\Phi}^{+}(x,t,z)=\check{\Phi}^{-}(x,t,z)\check{V}(x,t,z)$, with
$\check{V}(x,t,z)$ given by (5.51).

  4. 4.

As $z\to\infty$ with $z\in\mathbb{C}$, $\check{\Phi}(x,t,z)$ has the following
asymptotic behavior

$\check{\Phi}(x,t,z)=\tilde{U}\,(I+O(1/z))B(0,z).$ (5.53)

  5. 5.

There exist positive constants $c$ and $M$ such that
$|\check{\phi}_{ij}(x,t,z)|\leq M\mathrm{e}^{c|z|^{2}}$ for all
$z\in\mathcal{D}$.

For fixed $x,t\in\mathbb{R}$, the solution of RHP 5.1 is unique. Moreover, the
potential matrix $Q(x,t)$ of the Dirac operator (1.2) is given in terms any
solution $\check{\Phi}(x,t,z)$ of RHP 5.1 by

$Q(x,t)=\lim_{z\to\infty}\text{i}z[\sigma_{3},U^{\dagger}\hat{\Phi}(x,t,z)B(0,z)^{-1}],$
(5.54)

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 [70], 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 $\rho(z)$ satisfies the scalar RHP
defined by Conditions (i)–(iv) in Theorem 6.1 below. Therefore, the latter is
the desired RHP, since the existence of solutions for it guarantees the
existence of the Floquet multiplier $\rho(z)$. Next we make these
considerations more precise.

Consider a “candidate spectral data”, namely a a sequence
$\\{E_{2n-1},E_{2n},\gamma_{n},\sigma_{n}\\}_{n=g_{-},\dots,g_{+}}$ of
periodic and antiperiodic eigenvalues $E_{n}$, Dirichlet eigenvalues
$\gamma_{n}$ and signs $\sigma_{n}$, with $g_{-}<g_{+}$ either finite or
infinite, satisfying the following conditions:

  * •

$E_{2g_{-}-1}<\dots<E_{n-1}<E_{n}<E_{n+1}<\dots<E_{2g_{+}}$;

  * •

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

  * •

$R=\mathop{\rm max}\nolimits_{k}|E_{2k}-E_{2k-1}|<\infty$;

  * •

If $g_{-}$ and $g_{+}$ are finite, there exists $N$, $C$ such that
$|E_{n}|>Cn^{2}$ for all $|n|>N$;

  * •

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

######  Theorem 6.1.

Let $\Phi(x,t,z)$ be the solution of RHP 5.1 defined from the candidate
spectral data above, which determines the potential $q(x,t)$ via (5.54). If
there exists a function $r_{1}(z)$ such that

  1. (i)

$r_{1}$ is holomorphic in
${\mathbb{C}}\setminus(\mathbb{R}\setminus\Sigma(\mathcal{L}))$ with
continuous boundary values $r_{1\pm}$ on
$\mathbb{R}\setminus\Sigma(\mathcal{L})$ from above and below;

  2. (ii)

$r_{1}$ satisfies the jump relation $r_{1+}(z)=r_{1-}^{-1}(z)$ for
$z\in\mathbb{R}\setminus\Sigma(\mathcal{L})$;

  3. (iii)

$r_{1}$ satisfies the asymptotic condition
$r_{1}(z)=\mathrm{e}^{\text{i}zL_{1}}(1+O(1/z))$ for $L_{1}>0$;

  4. (iv)

$r_{1}(z)\in\mathcal{A}_{2}(\mathcal{D})$,

then $q(x+L_{1},t)=q(x,t)$.

######  Theorem 6.2.

Let $\Phi(x,t,z)$ be defined as in Theorem 6.1. If there exists a function
$r_{2}(z)$ such that

  1. (i)

$r_{2}$ is holomorphic in
${\mathbb{C}}\setminus(\mathbb{R}\setminus\Sigma(\mathcal{L}))$ with
continuous boundary values $r_{2\pm}$ on
$\mathbb{R}\setminus\Sigma(\mathcal{L})$ from above and below;

  2. (ii)

$r_{2}$ satisfies the jump relation $r_{2+}(z)=r_{2-}^{-1}(z)$ for
$z\in\mathbb{R}\setminus\Sigma(\mathcal{L})$;

  3. (iii)

$r_{2}$ satisfies the asymptotic condition
$r_{2}(z)=\mathrm{e}^{2\text{i}z^{2}L_{2}}(1+O(1/z))$ for $L_{2}>0$;

  4. (iv)

$r_{2}(z)\in\mathcal{A}_{2}(\mathcal{D})$,

then $q(x,t+L_{2})=q(x,t)$.

###### Proof of Theorem 6.1.

Suppose that $r_{1}(z)$ exists and define the function

$\tilde{\Phi}(x,t,z)=\Phi(x+L_{1},t,z)r_{1}(z)^{\sigma_{3}}\mathrm{e}^{-\text{i}z\sigma_{3}L_{1}}\,.$
(6.1)

We will prove that $\tilde{\Phi}$ also solves RHP 5.1. Condition 1 and
condition 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

$\tilde{\Phi}_{+}(x,t,z)=\Phi_{+}(x+L_{1},t,z)r_{1}(z)^{\sigma_{3}}\mathrm{e}^{-\text{i}z\sigma_{3}L_{1}}\\\
=\Phi_{-}(x+L_{1},t,z)(-1)^{n+m(z)}\begin{pmatrix}f^{-}/f^{+}&0\\\
0&f^{+}/f^{-}\end{pmatrix}r_{1}^{\sigma_{3}}\mathrm{e}^{-\text{i}z\sigma_{3}L_{1}}\\\
=\tilde{\Phi}_{-}(x+L_{1},t,z)(-1)^{n+m(z)}\begin{pmatrix}f^{-}/f^{+}&0\\\
0&f^{+}/f^{-}\end{pmatrix}.$ (6.2)

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

$\tilde{\Phi}_{+}(x,t,z)=\Phi_{+}(x+L_{1},t,z)r_{1+}(z)^{\sigma_{3}}\mathrm{e}^{-\text{i}z\sigma_{3}L_{1}}\\\
=\Phi_{-}(x+L_{1},t,z)(-1)^{n+m(z)}\text{i}\mathrm{e}^{-2\text{i}zx\sigma_{3}}\sigma_{1}r_{1-}^{-\sigma_{3}}\mathrm{e}^{-\text{i}z\sigma_{3}L_{1}}\\\
=c\tilde{\Phi}_{-}(x+L_{1},t,z)(-1)^{n+m(z)}\text{i}\mathrm{e}^{-2\text{i}zx\sigma_{3}}\sigma_{1}.$
(6.3)

Therefore, $\tilde{\Phi}$ satisfies condition 3. Condition 4 follows from the
asymptotic behavior 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 (5.54) imply that

$\Phi(x,t,z)=\Phi(x+L_{1},t,z)r_{1}(z)^{\sigma_{3}}\mathrm{e}^{-\text{i}zL_{1}},$
(6.4)

which indicates $q(x+L_{1},t)=q(x,t)$. ∎

###### Proof of Theorem 6.2.

To prove the statement about temporal periodicity, suppose that $r_{2}$
exists, and introduce $\tilde{\Phi}$ as
$\tilde{\Phi}(x,t,z)=\Phi(x,t+L_{2},z)r_{2}(z)^{\sigma_{3}}\mathrm{e}^{-2\text{i}z^{2}L_{2}}$.
Then the proof proceeds in a manner entirely analogous to the above. ∎

##  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

$w^{2}=P(z)=-L^{2}(E_{-1}-z)(E_{0}-z)\prod_{k=g_{-}}^{g_{+}}(z-E_{2k-1})(z-E_{2k})$
(7.1)

Similarly to [70], this curves 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 [70], 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:

$\displaystyle\pi_{\pm}^{-1}(z)=(z,\pm\sqrt{P(z)}),$ (7.2)

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
$p_{k}=(\mu_{n_{k}},\sigma_{n_{k}},P(\mu_{n_{k}}))$ for
$k=g_{-},g_{-}+1,...,g_{+}-1,g_{+}$ such that, as $z\rightarrow\infty$ with
$z\in{\mathbb{C}}\setminus{\mathbb{R}}$,

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

Moreover,
$\check{\psi}(x,t,\pi_{\pm}^{-1}(\,\cdot\,))\in\mathcal{A}_{2}(\mathcal{D})$.

The function $\check{\psi}$ is known as the Baker-Akhiezer function for the
NLS equation (e.g., see [8, 56] 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

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

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:

$\pi(P)=z,\quad P=(w,z).$ (7.5)

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 $\pi(\infty^{\pm})=\infty\in\mathbb{CP}^{1}$.

Here we 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

$\int_{a_{j}}\omega_{i}=2\pi\text{i}\,\delta_{ij}.$ (7.6)

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

$A_{j}(P)=\int_{P_{2g_{-}}}^{P}\omega_{j}.$ (7.7)

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

$B_{jk}=\int_{b_{k}}\omega_{j}.$ (7.8)

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

$\Theta(\mathbf{p})=\sum_{\mathbf{m}\in\mathbb{Z}^{g}}\exp\\{{\textstyle\frac{1}{2}}\langle
B\mathbf{m},\mathbf{m}\rangle+\langle\mathbf{p},\mathbf{m}\rangle\\},$ (7.9)

where $\langle\mathbf{p},\mathbf{q}\rangle=p_{1}q_{1}+\dots+p_{n}q_{n}$ 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:

  * •

$\int_{a_{i}}\mathrm{d}\Omega_{j}=0.$

  * •

$\Omega_{1}(P)=\pm(z+O(1)),\quad P\to\infty^{\pm}$,
$\Omega_{2}(P)=\pm(2z^{2}+O(1)),\quad P\to\infty^{\pm},$
$\Omega_{3}(P)=\pm(\log z+O(1)),\quad P\to\infty^{\pm},\quad z=\pi(P).$

  * •

$\Omega_{j}(P)$ 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 we consider the vector-valued Baker-Akhiezer
function $\psi(P,x)=(\psi_{1},\psi_{2})^{T}$ defined as follows:

$\displaystyle\psi_{1}(P)=\frac{\Theta(\int_{\infty^{-}}^{P}\boldsymbol{\omega}+\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D})\,\Theta(\mathbf{D})}{\Theta(\int_{\infty^{-}}^{P}\boldsymbol{\omega}-\mathbf{D})\,\Theta(\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D})}\exp\big{\\{}\text{i}x\Omega_{1}(P)+\text{i}t\Omega_{2}(P)-{\textstyle\frac{\text{i}}{2}}Ex+{\textstyle\frac{\text{i}}{2}}Nt\big{\\}},$
(7.10a)
$\displaystyle\psi_{2}(P)=\alpha\sqrt{\omega_{0}}\frac{\Theta(\int_{\infty^{-}}^{P}\boldsymbol{\omega}+\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D}-\mathbf{r})\,\Theta(\mathbf{D}-\mathbf{r})}{\Theta(\int_{\infty^{-}}^{P}\boldsymbol{\omega}-\mathbf{D})\,\Theta(\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D})}\exp\big{\\{}\text{i}x\Omega_{1}(P)+\text{i}t\Omega_{2}(P)+{\textstyle\frac{\text{i}}{2}}Ex-{\textstyle\frac{\text{i}}{2}}Nt+\Omega_{3}\big{\\}}.$
(7.10b)

The vector-valued parameters appearing in (7.10) are defined as follows:

$\displaystyle\boldsymbol{\omega}=(\omega_{1},\dots,\omega_{g}),\quad\mathbf{V}=(V_{1},\dots,V_{g}),\quad\mathbf{W}=(W_{1},\dots,W_{g}),$
$\displaystyle\mathbf{r}=\int_{\infty^{-}}^{\infty^{+}}\boldsymbol{\omega},\quad\mathbf{D}=\sum_{j=1}^{g}\int_{\infty^{-}}^{P_{j}}\boldsymbol{\omega}+\mathbf{K},$
$\displaystyle V_{j}=\int_{b_{j}}\mathrm{d}\Omega_{1},\quad
W_{j}=\int_{b_{j}}\mathrm{d}\Omega_{j},\quad
K_{j}=\pi\text{i}+{\textstyle\frac{1}{2}}B_{jj}-\frac{1}{2\pi\text{i}}\sum_{k\neq
j}\int_{a_{k}}(\int_{\infty^{-}}^{P}\omega_{j})\omega_{k}(P).$

The quantities $E$, $N$ and $\omega_{0}$ are determined by the second terms of
the asymptotic expansions of the integrals $\Omega_{j}(P)$ at the points
$\infty^{\pm}$.

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

$\Psi(z)=(\psi(P^{+}),\psi(P^{-})).$ (7.11)

The above function satisfies the propeties of the finite genus Baker-Akhiezer
function [8]. Finally, the solution to the NLS equation recovered from the
Baker-Akhiezer function is give by:

$q(x)=A\mathrm{e}^{-\text{i}Ex+\text{i}Nt}\frac{\Theta(\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D}+\mathbf{r})}{\Theta(\text{i}\mathbf{V}x+\text{i}\mathbf{W}t-\mathbf{D})},$
(7.12)

where $A={2\Theta(\mathbf{D})}/({\alpha\Theta(\mathbf{D}-\mathbf{r})})$.

##  8 Concluding remarks

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

The most striking result of this work is that a single 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 finite-gap
formalism, for which two sets of Dirichlet eigenvalues are used to reconstruct
the potential (e.g., see Theorem 2.2 in [69]). The result was illustrated by
computing explicitly the solution of the NLS in the case of genus zero. 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. We also note that, in the
finite-genus formalism, the dependence of the Dirichlet eigenvalues on the
base point (as governed by the Dubrovin equations) is highly nontrivial, and
the Dirichlet eigenvalues associated with all base points in the spatial
domain are needed in order to reconstruct the potential. With the Riemann-
Hilbert formalism, in contrast, only one base point is all that is needed, and
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 $t$. 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.

The formalism of this work generalizes the one developed in [70] for Hill’s
operator. Compared to [70], 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, for the Dirac problem, the bands 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 [70] is that,
in that case the spectral problem is an eigenvalue problem for a scalar
second-order differential operator. As a result, the asymptotic behavior of
various quantities contains square roots of the spectral parameter, which
complicates the analysis. In this respect, the Zakharov-Shabat problem is
simpler, since no square roots of $z$ appear in the analysis. On the other
hand, the Zakharov-Shabat problem is complicated significantly by the need to
introduce the modified Bloch-Floquet solutions.

The present work should also be compared with that in [16], where the IVP for
the focusing and defocusing NLS was studied using the unified transform
method. The latter 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 constrast, the present work uses the spectral theory of the ZS
problem, and is therefore much more analogous to the IVP on the line, and does
not use the values of the potential at the boundary of the domain.

The results of this work 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 [48, 53, 73, 74, 80, 84].

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 [52]. 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 nonzero), 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 inverse scattering transform reduces exactly to the direct and
inverse Fourier transform in the “linear limit”, i.e., the limit of
vanishingly small potentials [2]. 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 Zakharov-Shabat problem. In this respect, note that the analysis
of the focusing Zakharov-Shabat 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 [60, 69]. 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 [9,
10], the movable Dirichlet eigenvalues are not confined to curves in the
complex plane [65].

Another possible future extension of the present work concerns the periodic
problem for the Manakov system, i.e., the integrable two-component coupled
nonlinear Schrödinger 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 oustanding 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 [11, 19, 49,
51, 59, 78], which continue to receive considerable attention (e.g., see [27,
9, 10, 21, 22, 79] and references therein). Another possible application
concerns the emerging topics of soliton gases and the statistics of integrable
systems [13, 14, 26, 28, 33, 34, 38, 39]. We hope that the results of this
work and the present discussion will stimulate further work on all of these
topics.

Acknowledgment. We thank Marco Bertola, Dionyssis Mantzavinos, Ken McLaughlin,
Jeffrey Oregero, Barbara Prinari and Alex Tovbis for many insightful
conversations on topics related to the present work. This work work was
partially supported by the National Science Foundation under grant number
DMS-2009487.

## Appendix

###  A.1 Notation and standard definitions

We use the Pauli matrices defined as

$\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\,,\qquad\sigma_{1}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\,,\qquad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,.$ (A.1)

Throughout this work, subscripts $x$ and $t$ partial differentiation, the
superscript $T$ denotes matrix transpose, the asterisk complex conjugation,
and the dagger conjugate transpose. Also, we denote by
$f_{\mathrm{re}}={\textstyle\frac{1}{2}}(f+f^{*})$ and
$f_{\mathrm{im}}=-{\textstyle\frac{1}{2}}\text{i}(f-f^{*})$ the real and
imaginary parts of a complex quantity. For any $2\times 2$ matrix
$A=(a_{ij})$, we use the notation $a_{1}=(a_{11},a_{21})^{T}$ and
$a_{2}=(a_{12},a_{22})^{T}$ to denote its first and second columns,
respectively. The “half-plane residues” $\mathop{\rm
Res}\limits\nolimits^{\pm}$ used in section A.8 are defined as

$\mathop{\rm Res}\limits_{z=z_{o}}{\\!}^{\pm}f(z)=\lim_{{z\to
z_{o}^{\phantom{I}}}\atop{\mathop{\rm
Im}\nolimits(z-z_{o})\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{$\m@th\displaystyle\hfil#\hfil$\cr>\crcr<\crcr}
}}{\vbox{\offinterlineskip\halign{$\m@th\textstyle\hfil#\hfil$\cr>\crcr<\crcr}
}}{\vbox{\offinterlineskip\halign{$\m@th\scriptstyle\hfil#\hfil$\cr>\crcr<\crcr}
}}{\vbox{\offinterlineskip\halign{$\m@th\scriptscriptstyle\hfil#\hfil$\cr>\crcr<\crcr}
}}}0}}\\!\\![(z-z_{o})f(z)]\,.$ (A.2)

Finally, we take the signum function $\mathop{\rm sgn}\nolimits(z)$ to take
zero value when its argument is zero.

###  A.2 Basic properties of the spectrum

In this appendix we prove some basic properties of the Dirichlet eigenvalues,
largely following [60].

######  Proposition A.1.

All the Dirichlet eigenvalues $\mu_{j}$ are real.

###### Proof.

Since $\tilde{Y}$ a the fundamental solution of (2.16), its second column
$\tilde{y_{2}}$ also satisfies (2.16). Thus,

$\tilde{y}_{12,x}-z\tilde{y}_{22}=q_{\mathrm{re}}\tilde{y}_{12}+q_{\mathrm{im}}\tilde{y}_{22},\qquad\tilde{y}_{22,x}+z\tilde{y}_{12}=-q_{\mathrm{re}}\tilde{y}_{22}+q_{\mathrm{im}}\tilde{y}_{12}.$
(A.3)

Using (A.3), it is easy to show that

$(\tilde{y}_{12}\tilde{y}_{22}^{*}-\tilde{y}_{22}^{*}\tilde{y}_{12})_{x}+(z^{*}-z)(|\tilde{y}_{12}|^{2}+|\tilde{y}_{22}|^{2})=0;$
(A.4)

hence

$(z^{*}-z)\int_{0}^{L}(|\tilde{y}_{12}|^{2}+|\tilde{y}_{22}|^{2})\,\mathrm{d}x+\big{[}\tilde{y}_{12}\tilde{y}_{22}^{*}-\tilde{y}_{22}^{*}\tilde{y}_{12})\big{]}_{0}^{L}=0\,.$
(A.5)

Also, by definition we have
$\tilde{y}_{12}(0,\mu_{j})=\tilde{y}_{12}(L,\mu_{j})=0$. Evaluating (A.5) at
$z=\mu_{j}$, the second term in the above equation vanishes. Since the
integral is strictly positive, we conclude that $\mu_{j}=\mu_{j}^{*}$. ∎

The following result will be useful in the proof of Proposition A.5 later in
this section:

######  Proposition A.2.

For all $z\in\mathbb{R}$, the Floquet discriminant $\Delta(z)$ satisfies

$\Delta^{\prime}(z)={\textstyle\frac{1}{2}}\mathop{\rm
sgn}\nolimits[\tilde{y}_{21}(L,z)]\,\left\\{\int_{0}^{L}\big{[}(a\tilde{y}_{21}+b\tilde{y}_{21})^{2}+(a\tilde{y}_{11}+b\tilde{y}_{12})^{2}\big{]}\,\mathrm{d}x-\frac{\Delta^{2}-1}{|\tilde{y}_{21}(L,z)|}\int_{0}^{L}(\tilde{y}_{11}^{2}+\tilde{y}_{21}^{2})\,\mathrm{d}x\right\\},$
(A.6)

with

$a=\frac{\tilde{y}_{22}(L,z)-\tilde{y}_{11}(L,z)}{2\sqrt{|\tilde{y}_{21}(L,z)|}}\,,\qquad
b=-\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)\sqrt{|\tilde{y}_{21}(L,z)|}\,.$ (A.7)

As a result, $\mathop{\rm sgn}\nolimits(\Delta^{\prime}(z))=\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)$ for all $z\in\mathbb{R}$.

###### Proof.

Let $W(x,z)=\tilde{Y}^{\prime}(x,z)=(w_{ij})$. Differentiating (A.3) with
respect to $z$ yields

$w_{1j,x}-zw_{2j}-q_{\mathrm{re}}w_{1j}-q_{\mathrm{im}}w_{2j}=\tilde{y}_{2j}\,,\qquad
w_{2j,x}+zw_{1j}+q_{\mathrm{re}}w_{2j}-q_{\mathrm{im}}w_{1j}=-\tilde{y}_{1j}\,.$
(A.8)

Comparing (A.8) with (A.3) shows that $(w_{1j},w_{2j})^{T}$ satisfies the same
ODE as $(\tilde{y}_{1j},\tilde{y}_{2j})^{T}$, except for the presence of a
“forcing” term on the right-hand side of (A.8). Taking $j=1$ in (A.8) and
expanding $(w_{11},w_{21})^{T}$ in terms of
$(\tilde{y}_{11},\tilde{y}_{21})^{T}$ and
$(\tilde{y}_{12},\tilde{y}_{22})^{T}$, we can write

$w_{11}=c_{11}\tilde{y}_{11}+c_{12}\tilde{y}_{12}\,,\qquad
w_{21}=c_{11}\tilde{y}_{21}+c_{12}\tilde{y}_{22}\,,$ (A.9)

where $c_{11}$ and $c_{12}$ are functions of $x$, and using variation of
parameters in the ODE we have

$c_{11}=\int_{0}^{x}(\tilde{y}_{21}\tilde{y}_{22}+\tilde{y}_{11}\tilde{y}_{12})(x^{\prime})\,\mathrm{d}x^{\prime}\,,\qquad
c_{12}=-\int_{0}^{x}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})(x^{\prime})\,\mathrm{d}x^{\prime}\,.$
(A.10)

Then

$\displaystyle
w_{11}(x)=\bigg{(}\int_{0}^{x}(\tilde{y}_{21}\tilde{y}_{22}+\tilde{y}_{11}\tilde{y}_{12})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{11}(x)-\bigg{(}\int_{0}^{x}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{12}(x)\,,$
(A.11a) $\displaystyle
w_{21}(x)=\bigg{(}\int_{0}^{x}(\tilde{y}_{21}\tilde{y}_{22}+\tilde{y}_{11}\tilde{y}_{12})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{21}(x)-\bigg{(}\int_{0}^{x}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})(x^{\prime})\mathrm{d}x^{\prime})\tilde{y}_{22}(x)\,,$
(A.11b) $\displaystyle
w_{12}(x)=\bigg{(}\int_{0}^{x}(\tilde{y}_{22}^{2}+\tilde{y}_{12}^{2})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{11}(x)-\bigg{(}\int_{0}^{x}(\tilde{y}_{12}\tilde{y}_{11}+\tilde{y}_{22}\tilde{y}_{21})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{12}(x)\,,$
(A.11c) $\displaystyle
w_{22}(x)=\bigg{(}\int_{0}^{x}(\tilde{y}_{22}^{2}+\tilde{y}_{12}^{2})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{21}(x)-\bigg{(}\int_{0}^{x}(\tilde{y}_{12}\tilde{y}_{11}+\tilde{y}_{22}\tilde{y}_{21})(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}\,\tilde{y}_{22}(x)\,.$
(A.11d)

Next, differentiating $\Delta(z)$ with respect to $z$, we have

$2\Delta^{\prime}(z)=w_{11}(L,z)+w_{22}(L,z)=\left(\int_{0}^{L}(\tilde{y}_{21}\tilde{y}_{22}+\tilde{y}_{11}\tilde{y}_{22})\mathrm{d}x\right)\tilde{y}_{11}(L,z)-\left(\int_{0}^{L}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})\mathrm{d}x\right)\tilde{y}_{12}(L,z)\\\
+\left(\int_{0}^{L}(\tilde{y}_{22}^{2}+\tilde{y}_{12}^{2})\mathrm{d}x\right)\tilde{y}_{21}(L,z)-\left(\int_{0}^{L}(\tilde{y}_{12}\tilde{y}_{11}+\tilde{y}_{22}\tilde{y}_{21})\mathrm{d}x\right)\tilde{y}_{22}(L,z)\,.$
(A.12)

Assuming for the moment that $\tilde{y}_{21}(L,z)\neq 0$, we have

$\tilde{y}_{12}(L,z)=\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)\frac{4(\Delta^{2}-1)-[\tilde{y}_{11}(L,z)-\tilde{y}_{22}(L,z)]^{2}}{4|\tilde{y}_{21}(L,z)|}\,.$
(A.13)

Next, to evaluate $\Delta^{\prime}(z)$ we write

$-\tilde{y}_{12}(L)\int_{0}^{L}\tilde{y}_{21}\mathrm{d}x=-\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)\frac{\Delta^{2}-1}{\tilde{y}_{21}(L,z)}\int_{0}^{L}\tilde{y}_{21}^{2}\mathrm{d}x+\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)\frac{[\tilde{y}_{11}(L,z)-\tilde{y}_{22}(L,z)]^{2}}{4|\tilde{y}_{21}(L,z)|}\int_{0}^{L}\tilde{y}_{21}^{2}\mathrm{d}x,$

and

$\tilde{y}_{21}(L)\int_{0}^{L}\tilde{y}_{22}^{2}\mathrm{d}x=\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)|\tilde{y}_{21}(L,z)|\int_{0}^{L}\tilde{y}_{22}^{2}\mathrm{d}x[\mathop{\rm
sgn}\nolimits\tilde{y}_{21}(L,z)]^{2}.$

Then, after some straightforward calculations, we finally obtain (A.6). ∎

######  Proposition A.3.

All the Dirichlet eigenvalues $\mu_{j}$s are simple roots of
$\tilde{y}_{12}(L,z)$.

###### Proof.

Since $\mu_{j}$ is a root of $\tilde{y}_{12}(L,z)$, we have
$\tilde{y}_{11}(L,z)\tilde{y}_{22}(L,z)=1$, which implies that
$\tilde{y}_{11}(L,z)\neq 0$. Therefore, from (A.11), we have $w_{12}(L,z)\neq
0$. Thus, we conclude that $\mu_{j}$ is a simple root of
$\tilde{y}_{12}(L,z)$. ∎

######  Proposition A.4.

All the main eigenvalues are simple or double roots of $\Delta^{2}(z)-1$.

###### Proof.

Here we only prove the result for the periodic band edges $\hat{\zeta}_{j}$.
The proof for the antiperiodic band edges follows similarly. First we claim
that $\tilde{y}_{21}(L,\hat{\zeta}_{j})=0$. At $z=\hat{\zeta}_{j}$, we also
have

$\displaystyle\tilde{y}_{11}(L,\hat{\zeta}_{j})+\tilde{y}_{22}(L,\hat{\zeta}_{j})=2,\qquad\tilde{y}_{12}(L,\hat{\zeta}_{j})=0.$
(A.14a)

Hence, we have

$\tilde{y}_{11}(L,\hat{\zeta}_{j})=\tilde{y}_{22}(L,\hat{\zeta}_{j})=1.$
(A.15)

So from (A.11), we obtain

$\displaystyle
w_{11}(L,\hat{\zeta}_{j})=\int_{0}^{L}(\tilde{y}_{21}\tilde{y}_{22}+\tilde{y}_{11}\tilde{y}_{12})(x)\mathrm{d}x,\qquad
w_{21}(L,\hat{\zeta}_{j})=-\int_{0}^{L}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})(x)\mathrm{d}x,$
$\displaystyle
w_{12}(L,\hat{\zeta}_{j})=\int_{0}^{L}(\tilde{y}_{22}^{2}+\tilde{y}_{12}^{2})(x)\mathrm{d}x,\qquad
w_{22}(L,\hat{\zeta}_{j})=-\int_{0}^{L}(\tilde{y}_{12}\tilde{y}_{11}+\tilde{y}_{22}\tilde{y}_{21})(x)\mathrm{d}x.$

From (A.12), we have

$2\frac{d^{2}\Delta}{d\zeta^{2}}=2\left\\{\left(\int_{0}^{L}(\tilde{y}_{21}\tilde{y}_{22}+\tilde{y}_{11}\tilde{y}_{22})(x)\mathrm{d}x\right)^{2}-\left(\int_{0}^{L}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})\mathrm{d}x\right)\left(\int_{0}^{L}(\tilde{y}_{22}^{2}+\tilde{y}_{21}^{2})\mathrm{d}x\right)\right\\}\leq
0\,,$ (A.16)

where the last inequality follows from the Cauchy-Schwarz inequality. The
equality sign occurs when $\tilde{y}_{1}$ and $\tilde{y}_{2}$ are linearly
dependent. This can never happen, however, since $\tilde{y}_{1}$ and
$\tilde{y}_{2}$ are linearly independent. Therefore, we have
$d^{2}\Delta/d\zeta^{2}<0$. ∎

######  Proposition A.5.

There is exactly one Dirichlet eigenvalue $\mu_{j}$ in each degenerate or non-
degenerate spectral gap.

###### Proof.

We first prove that there is at least one Dirichlet eigenvalue in each
spectral gap. At two adjacent band edges $\zeta_{j}$ and $\zeta_{j+1}$,
$\Delta^{\prime}(\zeta_{j})$ and $\Delta^{\prime}(\zeta_{j+1})$ have different
signs. From (A.6) this means that $\tilde{y}_{21}(L,\zeta_{j})$ and
$\tilde{y}_{21}(L,\zeta_{j+1})$ have different signs. By continuity, there
must be $\mu_{j}$ between $\zeta_{j}$ and $\zeta_{j+1}$, such that
$\tilde{y}_{21}(L,\mu_{j})=0$. We have thus proved the first part of the
proposition.

Next we prove that there is at most one Dirichlet eigenvalue in each spectral
gap. Since $\det\tilde{Y}(x;z)=1$, we have
$\tilde{y}_{11}(L,\mu_{j})\tilde{y}_{22}(L,\mu_{j})-\tilde{y}_{12}(L,\mu_{j})\tilde{y}_{21}(L,\mu_{j})=1$,
implying

$\tilde{y}_{22}(L,\mu_{j})=\frac{1}{\tilde{y}_{11}(L,\mu_{j})},\qquad
2\Delta(\mu_{j})=\tilde{y}_{11}(L,\mu_{j})+\frac{1}{\tilde{y}_{11}(L,\mu_{j})}.$

Therefore,

$\tilde{y}_{11}(L,\mu_{j})=\Delta(\mu_{j})+\sqrt{\Delta^{2}(\mu_{j})-1}.$
(A.17)

It is clear that $\tilde{y}_{11}(L,\mu_{j})>0$ if $\Delta(\mu_{j})>1$, and
$\tilde{y}_{11}(L,\mu_{j})<0$ if $\Delta(\mu_{j})<-1$. Then we have

$w_{21}(L,\mu_{j})=-\tilde{y}_{22}(L,\mu_{j})\int_{0}^{L}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})\mathrm{d}x=-\frac{1}{\tilde{y}_{11}(L,\mu_{j})}\int_{0}^{L}(\tilde{y}_{21}^{2}+\tilde{y}_{11}^{2})\mathrm{d}x\,.$
(A.18)

This implies that $w_{21}(L,\mu_{j})$ can only have a definite sign in each
unstable band, so that $\tilde{y}_{21}(L,\mu_{j})$ is monotonic in one band.
This means that there is at most one $\mu_{j}$ in each unstable band. ∎

###  A.3 Asymptotics of the modified fundamental matrix as $z\to\infty$

###### Proof of Lemma 3.1.

We want to derive the asymptotic expansion of $Y(x,z)$ as $z\to\infty$ up to
order $1/z^{2}$. Let

$\mu(x,z)=Y(x,z)\,\mathrm{e}^{\text{i}zx\sigma_{3}}\,,$ (A.19)

and expand $\mu(x,z)$ as

$\mu(x,z)=\sum_{n=0}^{\infty}\frac{\mu^{(n)}(x,z)}{z^{n}}\,.$ (A.20)

If $q\in L^{2}([0,L],\mathbb{C})$, integration by parts on the integral
equation (2.7) then yields the asymptotics of $\mu$ as $z\to\infty$ along the
real axis:

$\displaystyle\begin{aligned}
&\mu^{(0)}(x,z)=\mathbb{I}\,,\qquad\mu^{(1)}(x,z)=\frac{1}{2\text{i}z}\sigma_{3}\left(\bigg{(}Q(x)-\mathrm{e}^{-2\text{i}zx\sigma_{3}}Q(0)\bigg{)}-\int_{0}^{x}Q^{2}(x^{\prime})\mathrm{d}x^{\prime}\right)\,,\\\
&\begin{aligned}
\mu^{(2)}(x,z)=\frac{1}{4z^{2}}\Bigg{(}&Q_{x}(x)-\mathrm{e}^{-2\text{i}zx\sigma_{3}}Q_{x}(0)-\bigg{(}\mathrm{e}^{2\text{i}zx\sigma_{3}}Q(x)-Q(0)\bigg{)}Q(0)+\int_{0}^{x}Q(x^{\prime})Q_{x}(x^{\prime})\mathrm{d}x^{\prime}\\\
&\kern
80.00012pt-Q(x)\int_{0}^{x}Q^{2}(x^{\prime})\mathrm{d}x^{\prime}-{\textstyle\frac{1}{2}}\bigg{(}\int_{0}^{x}Q^{2}(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}^{2}-\mathrm{e}^{-2\text{i}zx\sigma_{3}}\bigg{(}Q^{2}(x^{\prime})\mathrm{d}x^{\prime}\bigg{)}Q(0)\Bigg{)}\,.\end{aligned}\end{aligned}$
(A.21a)

Combining these expressions then yields (3.1) for $z\in\mathbb{R}$.

When $z\not\in\mathbb{R}$, it is necessary to consider separately the first
and second column of $\mu(x,z)$. Nonetheless, one can rigorously prove [10]
that, if $q\in C^{2}([0,L])$, the asymptotic behavior of each of the columns
of $\mu(x,z)$ also holds as $z\to\infty$ for $z\in\overline{\mathbb{C}^{+}}$
(e.g., see Lemmas 6.1 and Lemma 6.2 in [10] for the first and second column of
$\mu$, respectively). For brevity we will not repeat the proof here.

Finally, the asymptotic behavior as $z\to\infty$ in the lower-half plane is
obtained by noting that, if $A(x,z)$ is any matrix solution of (1.7a), so is
$\sigma_{2}A^{*}(x,z^{*})$. Then, taking into account that $Y(0,z)=\mathbb{I}$
for all $z\in\mathbb{C}$, we have

$Y(x,z^{*})=\sigma_{2}Y^{*}(x,z)\sigma_{2}\,,$ (A.22)

which yields the desired behavior as $z\to\infty$ in
$\overline{\mathbb{C}^{-}}$. ∎

Next, combining (A.19), (A.20) and (A.21), and recalling the definition (2.14)
of $\tilde{Y}(x,z)$, tedious but straightforward calculations then yield the
following asymptotic behavior for the entries of $\tilde{Y}(x,z)$ as
$z\to\infty$ along the real axis:

$\displaystyle\begin{aligned}
\tilde{y}_{11}(x,z)=\frac{1}{2}\mathrm{e}^{-\text{i}zx}\Bigg{(}1-\frac{1}{2\text{i}z}K(x)-\frac{1}{2\text{i}z}&(q(0)+q^{*}(x))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&-q(0)q^{*}(x)+q_{x}^{*}(x)-q^{*}(x)K(x)-q(0)K(x)-q_{x}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\\\
+\frac{1}{2}\mathrm{e}^{\text{i}zx}\Bigg{(}1+\frac{1}{2\text{i}z}K(x)+\frac{1}{2\text{i}z}&(q(x)+q^{*}(0))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda^{*}(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&-q^{*}(0)q(x)+q_{x}(x)-q(x)K(x)-q^{*}(0)K(x)-q_{x}^{*}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\,,\end{aligned}$
(A.23a) $\displaystyle\begin{aligned}
\tilde{y}_{12}(x,z)=\frac{\text{i}}{2}\mathrm{e}^{-\text{i}zx}\Bigg{(}1-\frac{1}{2\text{i}z}K(x)+\frac{1}{2\text{i}z}&(q(0)-q^{*}(x))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&+q(0)q^{*}(x)+q_{x}^{*}(x)-q^{*}(x)K(x)+q(0)K(x)+q_{x}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\\\
-\frac{\text{i}}{2}\mathrm{e}^{\text{i}zx}\Bigg{(}1+\frac{1}{2\text{i}z}K(x)+\frac{1}{2\text{i}z}&(q(x)-q^{*}(0))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda^{*}(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&+q^{*}(0)q(x)+q_{x}(x)-q(x)K(x)+q^{*}(0)K(x)+q_{x}^{*}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\,,\end{aligned}$
(A.23b) $\displaystyle\begin{aligned}
\tilde{y}_{21}(x,z)=-\frac{\text{i}}{2}\mathrm{e}^{-\text{i}zx}\Bigg{(}1-\frac{1}{2\text{i}z}K(x)+\frac{1}{2\text{i}z}&(q^{*}(x)-q(0))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&+q(0)q^{*}(x)-q_{x}^{*}(x)+q^{*}(x)K(x)-q(0)K(x)-q_{x}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\\\
+\frac{\text{i}}{2}\mathrm{e}^{\text{i}zx}\Bigg{(}1+\frac{1}{2\text{i}z}K(x)+\frac{1}{2\text{i}z}&(q^{*}(0)-q(x))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda^{*}(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&+q^{*}(0)q(x)-q_{x}(x)+q(x)K(x)+q^{*}(0)K(x)-q_{x}^{*}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\,,\end{aligned}$
(A.23c) $\displaystyle\begin{aligned}
\tilde{y}_{22}(x,z)=\frac{1}{2}\mathrm{e}^{-\text{i}zx}\Bigg{(}1-\frac{1}{2\text{i}z}K(x)+\frac{1}{2\text{i}z}&(q(0)+q^{*}(x))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&-q(0)q^{*}(x)-q_{x}^{*}(x)+q^{*}(x)K(x)+q(0)K(x)+q_{x}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\\\
+\frac{1}{2}\mathrm{e}^{\text{i}zx}\Bigg{(}1+\frac{1}{2\text{i}z}K(x)-\frac{1}{2\text{i}z}&(q(x)+q^{*}(0))+\frac{1}{4z^{2}}\bigg{(}|q(0)|^{2}+\Lambda^{*}(x)-{\textstyle\frac{1}{2}}K^{2}(x)\\\
&-q^{*}(0)q(x)-q_{x}(x)+q(x)K(x)+q^{*}(0)K(x)+q_{x}^{*}(0)\bigg{)}+O\left(\frac{1}{z^{3}}\right)\Bigg{)}\,,\end{aligned}$
(A.23d)

where

$K(x)=\int_{0}^{x}\
|q(x^{\prime})|^{2}\mathrm{d}x^{\prime}\,,\qquad\Lambda(x)=\int_{0}^{x}q(x^{\prime})q_{x}^{*}(x^{\prime})\mathrm{d}x^{\prime}\,.$
(A.24)

Importantly, similarly to (3.1), these expressions also hold off the real
$z$-axis, for the same reasons as in the proof of Lemma 3.1. The above
expressions are used in the proof of Lemma 3.9.

###  A.4 Asymptotics of the modified Bloch-Floquet solutions as $z\to\infty$

Here we derive the expressions in Proposition 3.5 for $\psi^{\pm}(x,z)$, as
well as their asymptotic behavior as $z\to\infty$ given in Propositions 3.6
and 3.9.

###### Proof of Proposition 3.5.

Since $\tilde{y}_{1}$ and $\tilde{y}_{2}$ are independent solutions, we can
expand each Bloch eigenfunction $\psi=(\psi_{1},\psi_{2})^{T}$ in terms of
them:

$\psi(x,z)=c_{1}\tilde{y}_{1}(x,z)+c_{2}\tilde{y}_{2}(x,z)$ (A.25)

We can also express
$\tilde{y}_{1}(x+L,z)=(\tilde{y}_{11}(x+L,z),\tilde{y}_{21}(x+L,z))^{T}$ in a
similar way:

$\displaystyle\tilde{y}_{1}(x+L,z)=\tilde{y}_{11}(L,z)\tilde{y}_{1}(x,z)+\tilde{y}_{21}(L,z)\tilde{y}_{2}(x,z)\,,$
(A.26a)
$\displaystyle\tilde{y}_{2}(x+L,z)=\tilde{y}_{11}(L,z)\tilde{y}_{1}(x,z)+\tilde{y}_{22}(L,z)\tilde{y}_{2}(x,z)\,,$
(A.26b)

where the coefficients of the expansion are easily obtained by evaluating the
above expressions at $x=0$. Evaluating (A.25) at $x+L$ and using (A.26), we
then have

$\psi(x+L,z)=c_{1}(\tilde{y}_{11}(L,z)\tilde{y}_{1}(x,z)+\tilde{y}_{21}(L,z)\tilde{y}_{2}(x,z))+c_{2}(\tilde{y}_{12}(L,z)\tilde{y}_{1}(x,z)+\tilde{y}_{22}(L,z)\tilde{y}_{2}(x,z))\,.$
(A.27)

Hence,

$c_{1}(\tilde{y}_{11}(L)-\rho)+c_{2}\tilde{y}_{12}(L)=0,\qquad
c_{1}\tilde{y}_{21}(L)+c_{2}(\tilde{y}_{22}(L)-\rho)=0.$ (A.28)

Recalling that the Floquet multipliers $\rho^{\pm 1}$ are precisely the values
for which the determinant of the above system is zero, solving for $c_{1}$ and
$c_{2}$, and imposing the normalization $\psi_{1}(0)=1$, one then obtains
(3.5). ∎

###### Proof of Proposition 3.6.

From (2.11) and (3.1), as $z\to\infty$, we have

$\rho^{-1}(z)=\begin{cases}\mathrm{e}^{-\text{i}zL}(1-\frac{1}{2\text{i}z}K_{o}+O(1/z^{2})),&z\in\overline{\mathbb{C}^{+}}\,,\\\\[4.30554pt]
\mathrm{e}^{\text{i}zL}(1+\frac{1}{2\text{i}z}K_{o}+O(1/z^{2})),&z\in\overline{\mathbb{C}^{-}}\,,\end{cases}$
(A.29)

with $K_{o}=K(L)$ and $K(x)$ is as in (A.24), implying

$\displaystyle\log(\rho(z))=\begin{cases}\text{i}zL+\frac{1}{2\text{i}z}K_{o}+O(1/z^{2}),&z\in\overline{\mathbb{C}^{+}}\,,\\\\[4.30554pt]
-\text{i}zL-\frac{1}{2\text{i}z}K_{o}+O(1/z^{2}),&z\in\overline{\mathbb{C}^{-}}\,.\end{cases}$
(A.30)

We thus have the following asymptotic behavior for $\rho^{-x/L}(z)$:

$\rho^{-x/L}(z)=\begin{cases}\mathrm{e}^{-\text{i}zx}\big{(}1-\frac{K_{o}x}{2\text{i}Lz}+O(1/z^{2})\big{)},&z\in\overline{\mathbb{C}^{+}}\,,\\\\[8.61108pt]
\mathrm{e}^{\text{i}zx}\big{(}1+\frac{K_{o}x}{2\text{i}Lz}+O(1/z^{2})\big{)},&z\in\overline{\mathbb{C}^{-}}\,.\end{cases}$
(A.31)

Next, by Floquet’s theorem, we can write

$(\psi^{-}(x,z),\psi^{+}(x,z))=:P(x,z)\,\rho^{-x/L\sigma_{3}}(z)\,,$ (A.32)

where $P(x,z)=(p^{-}(x,z),p^{+}(x,z))$ and where $p^{\pm}(x,z)$ are periodic
in $x$ with period $L$. Inserting (A.32) into (2.16) we have

$P_{x}(x,z)=L^{-1}\log(\rho(z))P(x,z)\sigma_{3}+\text{i}z\sigma_{2}P(x,z)+\tilde{Q}(x)P(x,z)\,,$
(A.33)

where $\tilde{Q}(x)=UQU^{-1}$. We then convert the above first-order system of
ODEs into a scalar second-order ODE in order to compute the asymptotic
behavior of its solutions. Consider $p^{-}(x,z)$ first for simplicity.
Equation (A.33) implies

$p_{x}^{-}=L^{-1}\log(\rho(z))P(x,z)+\text{i}z\sigma_{2}P(x,z)+\tilde{Q}(x)P(x,z)\,,$
(A.34)

which can be converted to integral form as

$p^{-}(x,z)=p^{-}(0,z)-\frac{L}{2\log\rho}p_{x}^{-}(0,z)(1-\rho^{2x/L})\\\
-\frac{L}{2\log\rho}\int_{0}^{x}(1-\rho^{-2(x^{\prime}-x)/L})(-\frac{1}{L^{2}}(\log\rho)^{2}-z^{2}+|q|^{2}+\tilde{Q}_{x})p^{-}(x^{\prime},z)\mathrm{d}x^{\prime}\,.$
(A.35)

Now we consider the Neumann series expansion

$\displaystyle p^{-}(x,z)=\sum_{n\in\mathbb{N}}p^{(n)}(x,z),\qquad
p^{(0)}(x,z)=p^{-}(0,z)-\frac{L}{2\log\rho}p_{x}^{-}(0,z)(1-\rho^{2x/L}).$
(A.36a) $\displaystyle
p^{(n+1)}(x,z)=-\frac{L}{2\log\rho}\int_{0}^{x}(1-\rho^{-2(x^{\prime}-x)/L})\bigg{(}-\frac{1}{L^{2}}(\log\rho)^{2}-z^{2}+|q|^{2}+\tilde{Q}_{x}\bigg{)}\,p^{(n)}(x^{\prime},z)\,\mathrm{d}x^{\prime}\,,\qquad
n\in\mathbb{N}\,.$ (A.36b)

The asymptotic behavior of $\log\rho$ in (A.30) implies ††margin:  [???]

$\displaystyle-\frac{1}{L^{2}}(\log\rho)^{2}-z^{2}=-K_{o}/L+O(1/z),$ (A.37a)
$\displaystyle\frac{L}{2\log\rho}=\begin{cases}\frac{1}{2\text{i}z}+\frac{K_{o}}{4\text{i}z^{3}}+O(z^{-4})\,,&z\in\overline{\mathbb{C}^{+}}\,,\\\\[8.61108pt]
-\frac{1}{2\text{i}z}-\frac{K_{o}}{4\text{i}z^{3}}+O(z^{-4})\,,&z\in\overline{\mathbb{C}^{-}}\,.\end{cases}$
(A.37b)

Equation (A.37a) allows one to prove that the Neumann series (A.36) is
convergent. Next we compute the asymptotic behavior of $p^{(0)}$. From (3.5)
we have

$p^{-}(0,z)=\psi^{-}(0,z)=\left(\begin{array}[]{c}1\\\
\frac{\rho^{-1}-\tilde{y}_{11}(L,z)}{\tilde{y}_{12}(L,z)}\end{array}\right)\,.$
(A.38)

Moreover, from (2.16), we have
$\psi^{\pm}_{x}=(-\text{i}z\sigma_{2}+\tilde{Q}(x))\psi^{\pm}$, which yields

$\psi_{x}^{-}(0,z)=\tilde{y}_{1,x}(0,z)+\frac{\rho^{-1}-\tilde{y}_{11}(L,z)}{\tilde{y}_{12}(L,z)}\tilde{y}_{2,x}(0,z)=\left(\begin{array}[]{c}q_{\mathrm{re}}(0)+(z+q_{\mathrm{im}}(z))\psi_{2}^{-}(0,z)\\\
-z+q_{\mathrm{im}}(0)-q_{\mathrm{re}}(0)\psi^{-}_{2}(0,z)\end{array}\right).$
(A.39)

Thus we have

$p_{x}^{-}(0,z)=\frac{1}{L}\log\rho(z)\,\psi^{-}(0,z)+\psi_{x}^{-}(0,z)\,.$
(A.40)

Next, we compute the asymptotic behavior of the two components of $p^{(0)}$
separately. Equation (A.31) implies $\rho^{2x/L}(z)=1+o(1)$ as $z\to\infty$
for $z\in\overline{\mathbb{C}^{+}}$ and $x\in(0,L)$ and for
$z\in\overline{\mathbb{C}^{-}}$ and $x\in(-L,0)$. Combining (A.36), (A.37b)
and (A.40), the asymptotic behavior of the first component of $p^{(0)}$ can be
expressed as

$\displaystyle
p^{(0)}_{1}(x,z)=1-{\textstyle\frac{1}{2}}-\frac{L}{2\log\rho}q_{\mathrm{re}}(0)-\frac{L}{2\log\rho}(z+q_{\mathrm{im}}(0))\psi^{-}_{2}(0,z)=1-\frac{q^{*}(0)}{2\text{i}z}+O(1/z^{2})$
(A.41)

as $z\in\overline{\mathbb{C}^{+}}$ and $x\in(0,L)$, where the subscripts 1 and
2 denote the components of $p^{(0)}$. Similarly, as
$z\in\overline{\mathbb{C}^{-}}$ and $x\in(-L,0)$, we have

$p^{(0)}_{1}(x,z)=1+\frac{q(0)}{2\text{i}z}+O(1/z^{2})\,.$ (A.42)

For the second component of $p^{(0)}$, as $z\in\overline{\mathbb{C}^{+}}$ and
$x\in(0,L)$ we have

$\displaystyle
p^{(0)}_{2}(x,z)=\psi^{-}_{2}(0,z)-{\textstyle\frac{1}{2}}\psi_{2}^{-}(0,z)-\frac{L}{2\log\rho}(z-q_{\mathrm{im}}(0)+q_{\mathrm{re}}(0)\psi_{2}^{-}(0,z))=-\text{i}-\frac{q^{*}(0)}{2z}+O(1/z^{2})\,.$
(A.43)

Similarly, as $z\in\overline{\mathbb{C}^{-}}$ and $x\in(-L,0)$, we have

$p^{(0)}_{2}(x,z)=\text{i}-\frac{q(0)}{2z}+O(1/z^{2})\,.$ (A.44)

Next we consider the asymptotic behavior of $p^{(1)}$. It is obvious that
$1-\rho^{-2(x^{\prime}-x)/L}=1+O(1/z)$ for $z\in\overline{\mathbb{C}^{+}}$ and
$0<x^{\prime}\leq x<L$ and for $z\in\overline{\mathbb{C}^{-}}$ and $-L<x\leq
x^{\prime}<0$, so that

$p^{(1)}(x,z)=-\frac{L}{2\log\rho}\int_{0}^{x}(-K_{o}/L+|q|^{2}+\tilde{Q}_{x})\,p^{(0)}\mathrm{d}x^{\prime}\,.$
(A.45)

Therefore, we can write the first element of $p^{(1)}$ as

$p^{(1)}_{1}(x,z)=-\frac{L}{2\log\rho}\left[\int_{0}^{x}(-K_{o}/L+|q|^{2}+q_{\mathrm{re},x})p^{(0)}_{1}\mathrm{d}x^{\prime}+\int_{0}^{x}(q_{\mathrm{im},x})p^{(0)}_{2}\mathrm{d}x^{\prime}\right]\,.$
(A.46)

Plugging the asymptotic behaviors of $p^{(0)}_{1}$, $p^{(0)}_{2}$ and
${L}/({2\log\rho})$ into the above equation, we obtain the asymptotic
expression for $p^{(1)}_{1}$:

$p^{(1)}_{1}=\frac{K_{o}x}{2\text{i}zL}-\frac{1}{2\text{i}z}\int_{0}^{x}|q|^{2}\mathrm{d}x^{\prime}-\frac{1}{2\text{i}z}(q^{*}(x)-q^{*}(0))+O(1/z^{2})\,.$
(A.47a) Morover, following the same method, we have the following asymptotic
behavior of $p^{(1)}_{2}$: