# Etudes for the inverse spectral problem

N. Makarov  California Institute of Technology
Department of Mathematics
Pasadena, CA 91125, USA<EMAIL_ADDRESS>and  A. Poltoratski
University of Wisconsin
Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, WI 53706
USA<EMAIL_ADDRESS>
###### Abstract.

In this note we study inverse spectral problems for canonical Hamiltonian
systems, which encompass a broad class of second order differential equations
on a half-line. Our goal is to extend the classical results developed in the
work of Marchenko, Gelfand-Levitan, and Krein to broader classes of canonical
systems and to illustrate the solution algorithms and formulas with a variety
of examples. One of the main ingredients of our approach is the use of
truncated Toeplitz operators, which complement the standard toolbox of the
Krein-de Branges theory of canonical systems.

The first author is supported by N.S.F. Grant DMS-1500821

The second author is supported by NSF Grant DMS-1954085.

The work on Section 7 in the fall of 2021 was supported by the Ministry of
Science and Higher Education of the Russian Federation, agreement No
075-15-2021-602.

##  1\. Introduction

The main object of this note is a canonical Hamiltonian system of differential
equations on a half-line. Such systems were studied around the middle of the
last century by M. G. Krein, who was first to find multiple connections
between spectral problems for such systems and structural problems in certain
spaces of entire functions. The rich complex analytic content of Krein’s
theory was further developed in the work of L. de Branges. At present, the
Krein-de Branges (KdB) theory became a standard tool in the area of spectral
problems. The theory experiences a new peak of popularity in the last 20 years
due to its connections to other areas such as the theory of orthogonal
polynomials, number theory, random matrices and the non-linear Fourier
transform ([8, 6, 16, 17, 29, 34]). In addition to the original book by de
Branges [7] and a chapter in the book by Dym and McKean [10], more recent
monographs by C. Remling [31] and R. Romanov [32] contain the basics of KdB-
theory and further references.

A regular half-line canonical (Hamiltonian) system (CS) is the equation

$\Omega\dot{X}=z{\mathcal{H}}X\qquad{\rm on}\quad[0,\infty).$ (1.1)

Here the Hamiltonian ${\mathcal{H}}={\mathcal{H}}(t)$ is a given $2\times 2$
matrix-function satisfying

${\mathcal{H}}\in L^{1}_{\rm loc}[0,\infty),\quad{\mathcal{H}}\neq 0\quad{\rm
a.e.},\quad{\mathcal{H}}\geq 0\quad{\rm a.e.}$

The first relation means that the entries of ${\mathcal{H}}$ are integrable on
each finite interval. Systems satisfying this condition are called regular.
The matrix $\Omega$ in (1.1) is the symplectic matrix

$\Omega=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},$

and $z\in{\mathbb{C}}$ is the ’spectral parameter’. The unknown function
$X=X(t,z)$ is a two-dimensional vector-function on $[0,\infty)$.

Several important classes of second order differential equations, including
Schrödinger and Sturm-Liouville equations, Dirac and Zaharov-Shabat systems,
string equations and orthogonal polynomials can be rewritten as CS, see for
instance [9, 30, 31] or [32], which makes a study of such systems even more
general.

Spectral problems for differential operators ask to find spectral data for a
given operator (direct problem) or recover an operator from its spectral data
(inverse problem). Such problems stem from theoretical physics and lead to
deep questions in analysis and adjacent areas of mathematics. Among the
classical results in the area of inverse problems are existence and uniqueness
theorems for Schrödinger equations by Borg and Marchenko [4, 5, 23, 24] and
further extensions to broader classes of Dirac systems by Gelfand, Levitan,
Gazymov, Krein and their collaborators [11, 18].

Our goal in this paper is to extend classical theory to broader classes of CS
and to provide explicit algorithms for the inverse spectral problem in those
classes. In our approach spectral measures become symbols of truncated
Toeplitz operators, bringing inverse spectral problems into the general
framework of the Toeplitz approach to problems of the uncertainty principle in
harmonic analysis recently developed in [20, 21, 22, 27].

Truncated Toeplitz operators, defined in Section 4.1, are bounded and
invertible in Paley-Wiener (PW) spaces of entire functions if and only if
their symbols are sampling measures in PW-spaces. We show that the class of CS
whose spectral measures are PW-sampling is broader than those classes
considered in classical theories and use the inverse Toeplitz operators to
solve the inverse spectral problem.

One of the key parts of the Gelfand-Levitan theory is the statement that the
Fourier transform $\hat{\mu}$ of a spectral measure $\mu$ of a regular Dirac
system has the form $\delta_{0}+\phi$, where $\delta_{0}$ is the Dirac point-
mass at 0 and $\phi$ is a locally summable function on ${\mathbb{R}}$.
Recalling that the Fourier transform of the Lebesgue measure $m$ on
${\mathbb{R}}$ is $\delta_{0}$ (after a proper normalization), the theory
considers systems which are close, in some sense, to the free system whose
spectral measure is $m$. The Gelfand-Levitan approach to the inverse spectral
problem then uses the invertibility of the operator

$T:L^{2}([0,L])\to L^{2}([0,L]),\ Tf=f+\phi\ast f,$

to solve the problem. The invertibility of $T$ is established via the Hilbert-
Fredholm Lemma and uses the property that $T$ is of the form $I+K$, where $K$
is compact. The compactness of the convolution operator $Kf=\phi\ast f$
follows from the local summability of $\phi$, which is an important condition
in the classical theory.

Our approach replaces the Fredholm invertibility of $T$ with invertibility of
the truncated Toeplitz operator $L_{\mu}$, see Section 4.1. We show that
spectral measures satisfying the condition

$\hat{\mu}=\delta_{0}+\phi$ (1.2)

form a subclass of sampling measures in PW-spaces, see Lemma 3.1. (We will
call (1.2) Gelfand-Levitan condition since it appears in their work [11, 18].)
The algorithm then proceeds with the use of the inverse operator
$L^{-1}_{\mu}$ which exists when $\mu$ is PW-sampling.

This approach connects inverse spectral problems with the study of sampling
measures in PW-spaces, see [25] for results and further references. While
describing sampling measures in a fixed space $\text{\rm PW}_{a}$ is a deep
and difficult problem, measures which are sampling for all PW-spaces admit an
elementary description, see Theorem 3.1.

Our algorithm takes an especially compact form in the case when the spectral
measure is periodic, see Sections 6.1-6.5. In this case the inverse spectral
problem connects with the theory of orthogonal polynomials on the unit circle,
see Theorem 6.3.

Given a CS with a PW-sampling spectral measure corresponding to a fixed
initial condition at the origin, it is natural to ask if the spectral measures
corresponding to other initial conditions are also PW-sampling. One of our
examples (see Section 5.3) shows that generally this is not true. Theorem 5.3
gives necessary and sufficient conditions for other spectral measures
(measures from the same Aleksandrov-Clark family) to be PW-sampling.

Our approach allows us to extend the classical results developed in the work
of Marchenko, Gelfand-Levitan, and Krein to broader classes of canonical
systems and to illustrate the solution algorithms and formulas with a variety
of examples in the last section of the paper.

The paper is organized as follows:

  * •

In Section 2.1 we review the basics of KdB-theory

  * •

Section 2.2 discusses the definition and main properties of spectral measures
of CS

  * •

In Section 2.3 we define representing measures for de Branges (dB) spaces
generalizing Parseval’s identity

  * •

In Section 2.4 we discuss the Weyl transform which generalizes the Fourier
transform in the settings of KdB-theory

  * •

Section 2.5 contains the definition and basic formulas related to Aleksandrov-
Clark (AC) measures

  * •

In Section 2.6 we discuss PW-spaces as a particular case of dB-spaces
corresponding to a free system

  * •

In Section 3.1 we introduce PW-sampling measures. Theorem 3.1 gives an
elementary description of PW-class

  * •

Section 3.2 provides basic examples of PW-sampling measures

  * •

In Section 3.3 we discuss the class of measures appearing the classical
Gelfand-Levitan theory. Lemma 3.1 shows that such measures form a subclass of
PW-class.

  * •

Section 3.4 introduces the main class of CS considered in this paper, the
systems whose spectral measures are PW-sampling. We formulate Theorems 3.2 and
3.4 showing that PW-sampling measures correspond to Hamiltonians with a.e.
non-vanishing determinants.

  * •

In Section 3.5 we show how symmetries of the Hamiltonian relate to symmetries
of the spectral measure

  * •

The special case of Krein’s string equation, which corresponds to CS with
diagonal Hamiltonians, is discussed in Section 3.6

  * •

In Section 4.1 we define truncated Toeplitz operators, one of the main tools
in our algorithms

  * •

Using Toeplitz operators we are able to supply the proofs of Lemma 3.1 and
Theorem 3.4 in Section 4.2

  * •

Section 4.3 shows how to obtain reproducing kernels of dB spaces from sinc
functions using inverse Toeplitz operators.

  * •

In Section 4.4 we provide an algorithm for the recovery of the upper left
element of a Hamiltonian ${\mathcal{H}}$ from the spectral measure.

  * •

To recover off-diagonal terms of ${\mathcal{H}}$ we define conjugate
reprokernels in Section 4.5

  * •

Section 4.6 contains Theorem 4.5, which gives the formula for the generalized
Hilbert transform, the unitary operator sending a chain of dB-spaces to the
dual chain corresponding to a different boundary condition

  * •

In Section 4.7 we provide an algorithm for the recovery of the off-diagonal
terms of ${\mathcal{H}}$

  * •

In Section 4.8 we give formulas for the Fourier transform of the reproducing
kernel at 0, which is used in our algorithms

  * •

Section 5 discusses algorithms of recovery of the lower right element of
${\mathcal{H}}$

  * •

Section 5.2 discusses the relation between AC-dual measures and a change of
initial condition in a given system

  * •

In Section 5.3 we give an example of a PW-sampling measure whose AC-dual is
not PW-sampling. Theorem 5.3 gives a necessary and sufficient condition for a
PW-sampling measure to have only PW-sampling measures as duals

  * •

Section 6.1 starts the discussion of the periodic case

  * •

Section 6.2 introduces a matrix of trigonometric moments for a periodic
measure

  * •

Section 6.3 contains a simplified algorithm for the recovery of the upper left
element of ${\mathcal{H}}$ in the periodic case

  * •

Section 6.4 gives a similar algorithm for the recovery of off-diagonal terms

  * •

In Section 6.5 we discuss connections of the periodic case with the theory of
orthogonal polynomials

  * •

In Sections 7.1-7.10 we provide a variety of examples illustrating our methods
and formulas

##  2\. Canonical systems and spectral measures

###  2.1. Transfer matrices, Hermite-Biehler functions and de Branges spaces

Instead of a two-dimensional vector function $X$ one may look for a $2\times
2$ matrix-valued solution $M=M(t,z)$ solving (1.1). Such a matrix valued
function satisfying the initial condition $M(0,z)=I$ is called the transfer
matrix or the matrizant of the system. The columns of the transfer matrix $M$
are the solutions for the system (1.1) satisfying the initial conditions
$\begin{pmatrix}1\\\ 0\end{pmatrix}$ (Neumann) and $\begin{pmatrix}0\\\
1\end{pmatrix}$ (Dirichlet) at 0. As a general rule we denote

$M=\begin{pmatrix}A&B\\\ C&D\end{pmatrix}.$ (2.1)

An entire function $F(z)$ belongs to the Hermite-Biehler (HB) class if

$|F(z)|>|F(\bar{z})|\textrm{ for all }z\in{\mathbb{C}}_{+}.$

We say that an entire function is real if it is real on ${\mathbb{R}}$.

For each fixed $t$, the entries of the transfer matrix $M$ of the system
(1.1), $A(z)=A(t,z)\equiv A_{t}(z)$, $B(z),C(z)$ and $D(z)$ are real entire
functions. The functions

$E:=A-iC,\qquad\tilde{E}:=B-iD$

belong to the Hermite-Biehler class, see for instance [31].

For an entire function $G$ we denote by $G^{\\#}$ its Schwartz reflection with
respect to ${\mathbb{R}}$, $G^{\\#}(z)=\bar{G}(\bar{z})$. We denote by $H^{2}$
the standard Hardy space in the upper half-plane.

For every Hermite-Biehler function $F$ one can consider the de Branges (dB)
space ${\mathcal{B}}(F)$, a Hilbert space of entire functions defined as

${\mathcal{B}}(F)=\left\\{G\ |G\text{ is entire, }\frac{G}{F},\
\frac{G^{\\#}}{F}\in H^{2}\right\\}.$

The Hilbert space structure in ${\mathcal{B}}(F)$ is inherited from $H^{2}$:

$<G,H>_{{\mathcal{B}}(F)}=\left<\frac{G}{F},\frac{H}{F}\right>_{H^{2}}=\int_{-\infty}^{\infty}G(t)\bar{H}(t)\frac{dt}{|F(t)|^{2}}.$

It follows that

${\mathcal{B}}(F)\stackrel{{\scriptstyle\rm
iso}}{{\subset}}L^{2}(1/|F|^{2},{\mathbb{R}}).$

One of the important properties of dB-spaces is that they admit an equivalent
axiomatic definition.

######  Theorem 2.1.

[7] Suppose that $H$ is a Hilbert space of entire functions which satisfies

A1) $F\in H$, $F(\lambda)=0$ $\Rightarrow$
$F(z)\frac{z-\overline{\lambda}}{z-\lambda}\in H$ with the same norm,

A2) $\forall\lambda\notin\mathbb{R}$, the point evaluation $F\mapsto
F(\lambda)$ is a bounded linear functional on $H$,

A3) $F\mapsto F^{\sharp}$ is an isometry in $H$.

Then $H={\mathcal{B}}(E)$ for some $E\in HB$.

It is not difficult to show that in every dB-space ${\mathcal{B}}(E)$, (A2)
actually holds for all $\lambda\in{\mathbb{C}}$. Thus, as follows from the
Riesz representation theorem, for each $\lambda\in{\mathbb{C}}$ there exists
$K(\lambda,\cdot)\in{\mathcal{B}}(H)$ such that for any
$F\in{\mathcal{B}}(E)$,

$F(\lambda)=<F,K(\lambda,\cdot)>_{{\mathcal{B}}(E)}.$

The function $K(\lambda,z)$ is called the reproducing kernel (reprokernel) for
the point $\lambda$. In the case of the dB-space ${\mathcal{B}}(E)$,
$K(\lambda,z)$ has the formula

$K(\lambda,z)=\frac{1}{2\pi
i}\frac{E(z)E^{\\#}(\bar{\lambda})-E^{\\#}(z)E(\bar{\lambda})}{\bar{\lambda}-z}=\frac{1}{\pi}\frac{A(z)C(\bar{\lambda})-C(z)A(\bar{\lambda}))}{\bar{\lambda}-z},$

where $A=(E+E^{\\#})/2$ and $C=(E^{\\#}-E)/2i$ are real entire functions such
that $E=A-iC$.

The functions $E,\ \tilde{E}$ corresponding to a canonical system (1.1) give
rise to the family of dB-spaces

${\mathcal{B}}_{t}={\mathcal{B}}(E(t,\cdot)),\qquad\tilde{\mathcal{B}}_{t}={\mathcal{B}}(\tilde{E}(t,\cdot)).$

A value $t$ is ${\mathcal{H}}$-regular if it does not belong to an open
interval on which ${\mathcal{H}}$ is a constant matrix of rank one. The spaces
${\mathcal{B}}_{t},\tilde{\mathcal{B}}_{t}$ form chains, i.e.,
${\mathcal{B}}_{s}\subsetneq{\mathcal{B}}_{t}$ for $s<t$ and the inclusion is
isometric for regular $t$ and $s$.

We denote by $K_{t}(z,w)$ and $\tilde{K}_{t}(z,w)$ their reprokernels;
sometimes, when $t$ is fixed, we write $K_{w}(z)$ and $\tilde{K}_{w}(z)$.

We denote by $\Pi$ the Poisson measure on ${\mathbb{R}}$,
$d\Pi(x)=\frac{dx}{1+x^{2}}$. We will denote by $m$ the Lebesgue measure on
${\mathbb{R}}$.

An entire function $F$ belongs to the Cartwright class ${\mathcal{C}}_{a}$ if
it has exponential type at most $a$ and $\log|F|\in L^{1}(\Pi)$.

A dB-space ${\mathcal{B}}(E)$ is called regular if for any
$F\in{\mathcal{B}}(E)$ and any $w\in{\mathbb{C}}$,
$(F(z)-F(w))/(z-w)\in{\mathcal{B}}(E)$. Any regular space ${\mathcal{B}}(E)$
is a subspace of ${\mathcal{C}}_{a}$ for some $a>0$, see [10, 31].

Regular systems (1.1) give rise to chains of regular dB-spaces
${\mathcal{B}}_{t},\ \tilde{\mathcal{B}}_{t}$.

######  Remark 2.1.

A more general class of $2\times 2$ systems are self-adjoint systems (SAS)
defined as

$\Omega\dot{X}=z{\mathcal{H}}X+QX\qquad{\rm on}\quad[0,\infty).$

Here $\Omega$ and ${\mathcal{H}}$ are as before and $Q$ is another given
$2\times 2$ locally summable symmetric real matrix function. An SAS is a
general form of a second order symmetric real system. It is well known that
every regular system of this form is equivalent to a regular CS in the sense
that they have the same dB spaces ${\mathcal{B}}_{t}$ and therefore same
spectral measures, as defined in the next section. An SAS can be reduced to a
regular CS via a substitution, see [31, 32].

One of the important classes of SAS is the class of Dirac systems for which
${\mathcal{H}}=I$. Such systems play an important role in applications and
constitute one of the main objects in the study of spectral problems, see for
instance [18]. The class of canonical systems considered in this note includes
the systems corresponding to regular Dirac systems.

$\triangle$

###  2.2. Spectral measures

There are several ways to introduce spectral measures of canonical systems.
We’ll make a simplifying assumption that the system has no ”jump intervals”,
i.e., intervals on which the Hamiltonian is rank one and constant. In this
case all $t\in[0,\infty)$ are ${\mathcal{H}}$-regular and all inclusions
${\mathcal{B}}_{s}\subset{\mathcal{B}}_{t},\tilde{\mathcal{B}}_{s}\subset\tilde{\mathcal{B}}_{t}$
are isometric. We can make this assumption because we will be mostly concerned
with the case $\det{\mathcal{H}}\neq 0$ a.e.

We informally identify absolutely continuous measures with functions writing
$\mu=f$ for the measure satisfying $d\mu(x)=f(x)dx$. In particular, instead of
writing $\mu=cm$ for constant multiples of the Lebesgue measure we write
$\mu=c$.

A measure $\mu$ on ${\mathbb{R}}$ is called Poisson-finite ($\Pi$-finite) if

$\int\frac{d|\mu|(x)}{1+x^{2}}<\infty.$

A measure on $\hat{R}={\mathbb{R}}\cup\\{\infty\\}$ is $\Pi$-finite if it is a
sum of a $\Pi$-finite measure on ${\mathbb{R}}$ and a finite point mass at
infinity.

By definition, a positive measure $\mu$ on ${\mathbb{R}}$ is a spectral
measure of the CS (1.1) with the initial condition $\begin{pmatrix}1\\\
0\end{pmatrix}$ at $t=0$ if

$\forall t,\qquad{\mathcal{B}}_{t}\stackrel{{\scriptstyle\rm
iso}}{{\subset}}L^{2}(\mu).$

(The definition is slightly more complicated in presence of jump intervals for
the Hamiltonian, which we do not allow in this paper.) It is well-known that
spectral measures of regular CS are $\Pi$-finite, see for instance [31]. In a
similar way, using $\tilde{B}_{t}$, one can define a spectral measure
$\tilde{\mu}$ for the initial condition $\begin{pmatrix}0\\\ 1\end{pmatrix}$.

Conversely, one of the main results of the Krein-de Branges theory says that
every positive $\Pi$-finite measure is a spectral measure of a regular CS. In
general, the corresponding system may not satisfy $\det{\mathcal{H}}\neq 0$
a.e., the restriction we are assuming in this article. Also, HB functions
corresponding to the systems considered in this paper have no zeros on the
real line. We will assume this restriction in our general discussions of dB-
spaces. (If $E$ vanishes at some point of ${\mathbb{R}}$, then all functions
in ${\mathcal{B}}(E)$ must vanish at the same point, as follows from the
definition. PW-type spaces discussed in this note clearly do not have such a
property.)

Every regular canonical system has a spectral measure; in fact for $\mu$ we
can take any limit point of the family of measures $|E_{t}|^{-2}$ as
$t\to\infty$, [7]. The spectral measure may or may not be unique. It is unique
iff

$\int{\rm trace}\ {\mathcal{H}}(t)dt=\infty.$ (2.2)

The case when the spectral measure is unique is called the limit point case
and the case when it is not, the limit circle case.

Finally, let us mention that spectral measures are invariant with respect to
’time’ parametrizations, i.e., a change of variable $t$ in the initial system
(1.1) via an increasing homeomorphism $t\mapsto s(t)$ does not change the
spectral measure.

###  2.3. De Branges measures

If $E$ is an HB-function non-vanishing on ${\mathbb{R}}$, then one can define
a continuous branch of the argument of $E$, $\phi_{E}(x)=-\arg E(x)$. The
function $\phi_{E}$ is called the phase function for the space
${\mathcal{B}}(E)$. It is not difficult to show that $\phi_{E}$ is a growing
real analytic function, $\phi^{\prime}>0$ on ${\mathbb{R}}$.

We call a sequence of real points discrete if it has no finite accumulation
points. A measure on $\hat{\mathbb{R}}={\mathbb{R}}\cap\\{\infty\\}$ is
discrete if its support on ${\mathbb{R}}$ is a discrete sequence.

For a dB-space ${\mathcal{B}}(E)$ there exists a standard one-parameter family
of discrete measures on ${\mathbb{R}}$ such that ${\mathcal{B}}(E)\overset{\rm
iso}{=}L^{2}(\mu)$. Let $\alpha\in{\mathbb{C}},\ |\alpha|=1$ and denote by
$t_{n}$ the sequence of points on ${\mathbb{R}}$ such that

$\frac{E^{\\#}(t_{n})}{E(t_{n})}=\alpha.$

In terms of the phase function, $\phi(t_{n})=\arg\alpha\ (\rm{mod}\ \pi)$.
Consider the measure

$\mu_{\alpha}=\sum_{n}\frac{\pi}{\phi^{\prime}(t_{n})|E(t_{n})|^{2}}\delta_{t_{n}}.$
(2.3)

Then ${\mathcal{B}}(E)\overset{\rm iso}{=}L^{2}(\mu_{\alpha})$, for all values
of $\alpha$ except possibly one, see for instance [7]. It is well known that
in the case considered in this paper, when $E=E_{t}$ is an HB function
corresponding to a regular det-normalized CS, such an exceptional value of
$\alpha$ does not exist.

If $E_{t}$ is the family of HB-functions corresponding to a system (1.1) with
a fixed initial condition at 0, and $E=E_{T}$ for some fixed $T>0$, then the
family of measures $\mu_{\alpha}$ constructed as above is the family of
spectral measures of the system restricted to the finite interval $(0,T)$ with
a corresponding self-adjoint boundary condition at $0$ and various self-
adjoint boundary conditions at $T$.

The Cauchy integrals of the measures $\mu_{\alpha}$ can be expressed through
the elements of the transfer matrix $M$. The following formula will be useful
to us in Section 5.2.

######  Lemma 2.1.

Let $M$ be a transfer matrix (2.1) of a CS (1.1). Let $E=A-iC$ be the HB
function corresponding to the first column of $M$ and let the representing
measures $\mu_{\alpha}$ for ${\mathcal{B}}(E)$ be defined as in (2.3). Suppose
that ${\mathcal{B}}(E)\overset{\rm iso}{=}L^{2}(\mu_{-1})$. Then

${\mathcal{K}}\mu_{-1}(z):=\frac{1}{\pi}\int\left[\frac{1}{\zeta-z}-\frac{\zeta}{1+\zeta^{2}}\right]~{}d\mu_{-1}(\zeta)=-\frac{B}{A}+\text{\rm
const}.$

Note that, in particular, the measure $\mu_{-1}$ is supported at the zeros of
$A(z)$, which can also be seen straight from its definition.

###### Proof.

The inequality $K_{z}(z)=||K_{z}||^{2}>0$ for the reproducing kernel $K_{z}$
of the space ${\mathcal{B}}(A+iB)$ implies that $-\frac{B}{A}$ has positive
imaginary part in ${\mathbb{C}}_{+}$. By the Herglotz representation theorem,

$-\frac{B}{A}={\mathcal{K}}\nu+\text{\rm const}$

for some positive measure $\nu$ on $\hat{R}$. Since $-B/A$ is analytic outside
of the zeros of $A$, $\nu$ is a discrete measure whose support on
${\mathbb{R}}$ is at $\\{A=0\\}=\\{t_{n}\\}$, the same as $\mu_{-1}$. Since
${\mathcal{B}}(E)\overset{\rm iso}{=}L^{2}(\mu_{-1})$, the reprokernels
$K_{t_{n}}$ form an orthogonal basis in ${\mathcal{B}}(E)$, which implies that
$\nu$ cannot have a pointmass at infinity, see for instance the proof of
Theorem 22 in [7]. Examining the behavior of $-B/A$ near $t_{n}$ we find that
$\nu$ has pointmasses of the size $-\pi B(t_{n})/A^{\prime}(t_{n})$ at
$t_{n}$. Since

$\phi=-\arg E=-\arctan\left(-\frac{C}{A}\right)\mod\pi,$

we have

$\phi^{\prime}=\frac{1}{1+(C/A)^{2}}\frac{C^{\prime}A-A^{\prime}C}{A^{2}}=\frac{1}{|E|^{2}}(C^{\prime}A-A^{\prime}C).$

Since $A(t_{n})=0$,

$\phi^{\prime}(t_{n})=-\frac{A^{\prime}C}{|E|^{2}}(t_{n})$

and

$\frac{\pi}{\phi^{\prime}(t_{n})|E(t_{n})|^{2}}=-\frac{\pi}{A^{\prime}(t_{n})C(t_{n})}.$

Since $\det M\equiv 1$, $C(t_{n})=1/B(t_{n})$ and

$\nu(t_{n})=-\pi\frac{B(t_{n})}{A^{\prime}(t_{n})}=-\frac{\pi}{A^{\prime}(t_{n})C(t_{n})}=\frac{\pi}{\phi^{\prime}(t_{n})|E(t_{n})|^{2}}=\mu_{-1}(t_{n}).$

∎

###  2.4. The Weyl transform

On every finite interval $[0,t]$ we define the Hilbert space
$L^{2}({\mathcal{H}},[0,t])$ associated with the Hamiltonian
${\mathcal{H}}\geq 0$. The space consists of of 2-dim vector functions on
$[0,t]$ with the inner product

$<f,g>_{L^{2}_{\mathcal{H}}[0,t]}=\int_{0}^{t}<Hf,g>dt.$

To give an alternative definition of a spectral measure for a CS we can
consider the Weyl transform

${\mathcal{W}}:~{}\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}\mapsto
F(z)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\left<{\mathcal{H}}(t)\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix},\begin{pmatrix}A_{t}(\bar{z})\\\
C_{t}(\bar{z})\end{pmatrix}\right>~{}dt,$

first defined on test functions, and call $\mu$ a spectral measure if
${\mathcal{W}}$ extends to an isometry $L^{2}({\mathcal{H}},[0,t])\to
L^{2}(\mu)$ for every $t\geq 0$. Note that then

${\mathcal{B}}_{t}={\mathcal{W}}L^{2}({\mathcal{H}},[0,t])$ (2.4)

and the reprokernels $K_{\lambda}(z)$ of ${\mathcal{B}}_{t}$ are obtained as
Weyl transforms of the solutions,

$K_{\lambda}={\mathcal{W}}X_{\lambda}.$ (2.5)

The spectral measure $\tilde{\mu}$ can be similarly defined using the
functions $B$ and $D$ in place of $A$ and $C$ in the above formula. Spectral
measures corresponding to other self-adjoint boundary conditions can also be
defined using proper modifications of either one of our two definitions.

###  2.5. Aleksandrov-Clark and dual measures

Let $\varphi$ be a function from the unit ball of
$H^{\infty}({\mathbb{C}}_{+})$ (such functions are often called Schur
functions). Associated with every such $\varphi$ is the family of $\Pi$-finite
positive measures $\\{\sigma^{\varphi}_{\alpha}\\}_{\alpha\in{\mathbb{T}}}$ on
$\hat{\mathbb{R}}$ defined via the formula

$\Re\frac{\alpha+\varphi(z)}{\alpha-\varphi(z)}=py+\frac{1}{\pi}\int{\frac{yd\sigma^{\varphi}_{\alpha}(t)}{(x-t)^{2}+y^{2}}},\hskip
28.45274ptz=x+iy,$ (2.6)

where the number $\pi p$ can be interpreted as the pointmass of
$\sigma^{\varphi}_{\alpha}$ at $\infty$. Such measures $\sigma^{\varphi}_{a}$
are called Aleksandrov-Clark (AC-) measures for $\varphi$. Families of AC-
measures appear in complex function theory and applications to perturbation
theory and spectral problems, see for instance [28].

At most one of the measures
$\\{\sigma^{\varphi}_{\alpha}\\}_{\alpha\in{\mathbb{T}}}$ has a pointmass at
infinity: $\sigma_{\alpha}(\infty)\neq 0$ iff $\varphi(\infty)=\alpha$, in the
sense of non-tangential limit, and $\varphi-\alpha\in L^{2}({\mathbb{R}})$.

It follows from the definition that all of the measures
$\sigma^{\varphi}_{\alpha}$ are singular iff one of them is singular iff
$\varphi$ is inner. All measures are discrete iff one of them is discrete iff
$\varphi$ is a meromorphic inner function (MIF), an inner function which can
be extended meromorphically to the whole plane.

If $\varphi=\theta$ is inner, then

$\sigma_{\alpha}\cdot
1_{\mathbb{R}}=2\pi\sum_{\theta(\xi)=\alpha}\frac{\delta_{\xi}}{|\theta^{\prime}(\xi)|},$
(2.7)

as can be deduced from the definition.

Every HB-function $E(z)$ gives rise to a MIF, $\theta_{E}=E^{\\#}/E$. In this
case the last formula is closely related to (2.3) and the measures involved in
the two formulas satisfy

$\sigma_{a}=|E|^{2}\mu_{\alpha}.$

We will call two $\Pi$-finite positive measures $\mu$ and $\tilde{\mu}$ on
${\mathbb{R}}$ (AC-) dual if there exists a Schur function $\varphi$ such that
$\mu=\sigma^{\varphi}_{-1}$ and $\tilde{\mu}=\sigma^{\varphi}_{1}$. We discuss
dual measures in Section 5.2 in more detail.

Connection between duality of measures, as defined above, to spectral problems
is provided by the property that if $\mu$ is a spectral measure of a CS with
the Neumann initial condition at $0$, then the spectral measure $\tilde{\mu}$
of the same system with the Dirichlet initial condition is dual to $\mu$.
Similarly, measures satisfying (2.6) with other $\alpha$ correspond to other
initial conditions for the same system. Note also that any two measures
$\sigma_{\alpha}$ and $\sigma_{\beta}$, $\alpha\neq\beta$, from the same AC-
family are dual up to constant multiples, i. e., there exist $C_{1},C_{2}>0$
such that $C_{1}\sigma_{\alpha}$ and $C_{2}\sigma_{\beta}$ are dual.

###  2.6. Paley-Wiener spaces

Let us discuss the ’free’ case ${\mathcal{H}}\equiv I$.

This case illustrates the connection between the Weyl transform and the
Fourier transform. We will use the following definition for the Fourier
transform in $L^{2}({\mathbb{R}})$:

$({\mathcal{F}}f)(\xi)\equiv\hat{f}(\xi)=\frac{1}{\sqrt{2\pi}}\int e^{-i\xi
x}f(x)dx,$

first defined on test functions and then extended to a unitary operator
${\mathcal{F}}L^{2}({\mathbb{R}})\to L^{2}({\mathbb{R}})$ via Parseval’s
theorem. The Paley-Wiener space $\text{\rm PW}_{t}$ of entire functions is
defined as the image

${\rm PW}_{t}={\mathcal{F}}L^{2}[-t,t].$

By the Paley-Wiener theorem, $\text{\rm PW}_{t}$ can be equivalently defined
as the space of entire functions of exponential type at most $t$ which belong
to $L^{2}({\mathbb{R}})$. The Hilbert structure in $\text{\rm PW}_{t}$ is
inherited from $L^{2}({\mathbb{R}})$.

Verifying the axioms in Theorem 2.1, one can show that any $\text{\rm PW}_{a}$
is a dB-space. Alternatively, one can arrive at the same conclusion using the
first definition of a dB-space with $E(z)=e^{-iaz}\in$HB.

For the transfer matrix of the free system ${\mathcal{H}}=I$ we have

$\begin{pmatrix}A&B\\\ C&D\end{pmatrix}=\begin{pmatrix}\cos tz&-\sin tz\\\
\sin tz&\cos tz\end{pmatrix}.$

Using the standard notation $S(z)=e^{iz}$,

$E_{t}(z)=S^{-t}(z)\equiv e^{-itz},$

and the spectral measure of ${\mathcal{H}}\equiv I$ is $\mu=m$, the Lebesgue
measure on ${\mathbb{R}}$. The spaces ${\mathcal{B}}_{t}$ are PW-spaces,

${\rm PW}_{t}={\mathcal{B}}(S^{-t})$

The reproducing kernels of ${\rm PW}_{t}$ are (constant multiples of) sinc
functions

$K_{t}(z,w)=\frac{1}{\pi}{\text{\rm
sinc}}_{t}(z-\bar{w})=\frac{1}{\pi}\frac{\sin t(z-\bar{w})}{z-\bar{w}}.$

We will denote the kernel at $w=0$ by

$\mathring{k_{t}}(z):=\frac{1}{\pi}\frac{\sin tz}{z}.$

The Weyl transform in the free case ${\mathcal{H}}\equiv I$ becomes the
classical Fourier transform:

${\mathcal{W}}:~{}\begin{pmatrix}f_{1}\\\ f_{2}\end{pmatrix}\mapsto
F(z)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\left<{\mathcal{H}}(t)\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix},\begin{pmatrix}A_{t}(\bar{z})\\\
C_{t}(\bar{z})\end{pmatrix}\right>~{}dt=$
$\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\left<\begin{pmatrix}f_{1}\\\
f_{2}\end{pmatrix},\begin{pmatrix}\cos t\bar{z}\\\ \sin
t\bar{z}\end{pmatrix}\right>~{}dt=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}(f_{1}\cos
tz+f_{2}\sin tz)dt=$
$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)e^{-itz}dt=\hat{f}(z),$

where the function $f$ is defined on ${\mathbb{R}}$ as

$f(t)=\begin{cases}\frac{1}{\sqrt{2}}(f_{1}(t)+if_{2}(t))&\textrm{ for }t>0\\\
\frac{1}{\sqrt{2}}(f_{1}(t)-if_{2}(t))&\textrm{ for }t<0\end{cases}.$

It follows that the Lebesgue measure $m$ on ${\mathbb{R}}$ is the spectral
measure for the free system. The measure is unique because $trace\
{\mathcal{H}}\equiv 2$ and (2.2) holds.

##  3\. PW-sampling measures and Hamiltonians of PW-type

###  3.1. PW-sampling measures

By definition, a positive measure $\mu$ on ${\mathbb{R}}$ is PW-sampling
($\mu\in\text{\rm PW}$) if it is sampling for all Paley-Wiener spaces PWt:

$\forall t\quad\exists C>0,\qquad\forall f\in{\rm PW}_{t},\qquad
C^{-1}\|f\|\leq\|f\|_{L^{2}(\mu)}\leq C\|f\|.$

Note that any PW-sampling measure must satisfy

$\sup_{x\in{\mathbb{R}}}\mu((x,x+1))<\infty,$ (3.1)

because otherwise

$\sup_{s\in{\mathbb{R}}}||f(\cdot-s)||_{L^{2}(\mu)}=\infty$

for every non-zero $f\in\text{\rm PW}_{a}$.

Indeed, first notice that for any such $f$ there exists an interval
$(x,x+\delta),\ \delta>0,$ on which $|f|>\varepsilon>0$. If (3.1) is not
satisfied then there exists a sequence of intervals
$I_{n}=(x_{n},x_{n}+\delta)$ such that $\mu(I_{n})\to\infty$. But then

$||f(\cdot-(x_{n}+\delta/2))||^{2}_{L^{2}(\mu)}\geq\varepsilon^{2}\mu(I_{n})\to\infty,$

which shows that the norms are not equivalent because

$||f(\cdot-(x_{n}+\delta/2))||_{PW_{a}}=\text{\rm const}.$

It follows from (3.1) that a PW-sampling measure is $\Pi$-finite. If
$\mu=h\cdot m$ with $h^{\pm 1}\in L^{\infty}$, then $\mu$ is obviously a PW-
sampling measure. More examples will be given at the end of this section.

While the description of sampling measures for a fixed $PW_{a}$-space is a
difficult problem (see for instance [25]), PW-sampling measures, as defined
above, admit a complete metric characterization.

Given $\mu$ and $\delta>0$ we say that an interval $l\subset{\mathbb{R}}$ is
$\delta$-massive with respect to $\mu$ if

$\mu(l)\geq\delta\textrm{ and }|l|\geq\delta.$

The $\delta$-capacity of an interval $I\subset{\mathbb{R}}$ with respect to
$\mu$, denoted by $C_{\delta}(I)$, is the maximal number of disjoint
$\delta$-massive intervals intersecting $I$.

######  Theorem 3.1.

$\mu$ is PW-sampling if and only if

(i) For any $x\in{\mathbb{R}}$, $\mu(x,x+1)\leq\text{\rm const}$;

(ii) For any $t>0$ there exist $L$ and $\delta$ such that for all $I,\ |I|\geq
L$,

$C_{\delta}(I)\geq t|I|.$

To prove Theorem 3.1 we need to recall the following well-known results.

A sequence $\Lambda=\\{\lambda_{n}\\}\subset{\mathbb{C}}$ is called separated
if $|\lambda_{m}-\lambda_{n}|>c>0$ for a fixed $c$ and all $m\neq n$. A
separated sequence is sampling for $PW_{a}$ if

$||f(\lambda_{n})||_{l^{2}}\asymp||f||_{PW_{a}}$

for $f\in PW_{a}$. Sampling measures on ${\mathbb{R}}$ and real sampling
sequences are closely related.

######  Proposition 3.1.

[25] A positive measure $\mu$ on ${\mathbb{R}}$ is sampling for $\text{\rm
PW}_{a}$ iff

1) $\mu((x,x+1))<C<\infty$ for some $C$ and all $x\in{\mathbb{R}}$ and

2) There exists a sampling sequence $\\{\lambda_{n}\\}$ for $\text{\rm
PW}_{a}$ and $\delta>0$ such that the points $\lambda_{n}$ have disjoint
neighborhoods $U(\lambda_{n})$ satisfying $\mu(U_{n})>\delta$.

For $\Lambda\subset{\mathbb{R}}$ define the lower Beurling uniform density of
$\Lambda$ as

$D_{-}(\Lambda)=\lim_{L\to\infty}\min_{x\in{\mathbb{R}}}\frac{\\#(\Lambda\cap(x,x+L))}{L}.$

######  Proposition 3.2.

A real separated sequence $\Lambda$ is sampling for $PW_{a}$ if $\pi
D_{-}(\Lambda)>a$ and not sampling if $\pi D_{-}(\Lambda)<a$.

This statement is essentially contained in the work of Beurling [3]; see also
[13] for a different version.

###### Proof of Theorem 3.1.

Let $\mu$ satisfy the conditions of the theorem and let $t$ be a positive
number. Then for $\delta=\delta(t)$ from (ii) one can choose a sequence of
disjoint $\delta$-massive intervals whose centers $\lambda_{n}$ form a
sequence $\Lambda$ satisfying $D_{-}(\Lambda)>t/2$. Hence it is a sampling
sequence for $PW_{t/2}$ by Proposition 3.2. Together with (i), Proposition 3.1
implies that $\mu$ is a sampling measure for $PW_{t/2}$.

In the opposite direction, if (i) does not hold, then $\mu$ is not a sampling
measure for any $PW_{a}$ as was discussed above. Suppose that (i) holds but
$\delta$ from (ii) does not exist for some $t>0$. If $\mu$ is sampling for
$\text{\rm PW}_{t+\varepsilon}$, then by Proposition 3.1 one can choose
$U(\lambda_{n})$ such that $\Lambda=\\{\lambda_{n}\\}$ is sampling for
$\text{\rm PW}_{t+\varepsilon}$. Then $D(\Lambda)>t$ by Proposition 3.2. Since
the sequence $\Lambda$ is separated, one can choose $U(\lambda_{n})$ to be
$\delta$-massive and obtain a contradiction. Therefore, $\mu$ is not sampling
for $PW_{t+\varepsilon}$. ∎

For a PW-sampling measure $\mu$ we denote by PW${}_{a}(\mu)$ the Hilbert
spaces of entire functions which have the same elements as the spaces PWa but
the $L^{2}(\mu)$ scalar product. From the axiomatic characterization of dB-
spaces, it follows that PW${}_{a}(\mu)$ are dB-spaces. By construction, all
spaces PW${}_{a}(\mu)$ are embedded isometrically in $L^{2}(\mu)$.

###  3.2. Basic examples of PW-sampling measures

  1. (1)

As was mentioned before, any measure $\mu$ of the form

$d\mu=w(x)dx,\ w(x)\asymp 1$

is PW-sampling. The upper bound $w<C$ can be replaced with (3.1). This can be
seen directly from the estimate

$||f^{\prime}||_{\infty}\leq a\sqrt{2a}||f||_{PW_{a}}$

which holds for any $f\in\text{\rm PW}_{a}$.

  2. (2)

Any measure that decays to 0 at infinity, i.e., any measure satisfying

$\mu((x,x+1))\to 0\ {\rm as}\ x\to\infty,$

is not PW-sampling. To see this, note that shifts of the sinc function have
the same PW-norm but

$\left|\left|\frac{\sin(z-N)}{z-N}\right|\right|^{2}_{L^{2}(\mu)}\lesssim\sum_{n\in{\mathbb{Z}}}\frac{\mu((n,n+1))}{(N-n)^{2}+1}\to
0\ \text{ as }\ N\to\infty.$

In particular, any finite measure is not PW-sampling.

  3. (3)

Let $\mu=m+\sum_{0}^{\infty}\alpha_{n}\delta_{x_{n}},$ where $0<\alpha_{n}<C$
and $\\{x_{n}\\}\subset{\mathbb{R}}$ is a separated sequence. Then $\mu$ is
PW-sampling.

Also, if $\mu\in\text{\rm PW}$ and $\nu\geq 0$ decays at infinity, in
particular if $\nu$ has compact support, then $\mu+\nu\in\text{\rm PW}$. This
and further examples of that kind are obtained from the following argument.

If $\mu$ satisfies (ii) from Theorem 3.1 and $\nu$ is any non-negative
measure, then $\mu+\nu$ satisfies (ii). If $\mu$ and $\nu$ both satisfy (i),
then so does $\mu+\nu$. Hence, if $\mu\in\text{\rm PW}$ and $\nu\geq 0$
satisfies (i), then $\mu+\nu\in\text{\rm PW}$.

In particular, addition of finite positive measures with compact support or
locally finite positive measures decaying at infinity does not change the PW-
sampling property.

  4. (4)

It follows from Theorem 3.1 that a positive periodic measure is PW-sampling if
and only if it has locally infinite support. In particular, the measures
$\mu(x)=1+a\cos x,\ a\in[-1,1]$, considered in Section 7, are PW-sampling, but
a measure which has only finitely many point masses on each period is not.

  5. (5)

Any discrete measure $\mu$ satisfying

$D_{-}(\text{\rm supp}\,\mu)<\infty$

is not PW-sampling, as follows from Propositions 3.1 and 3.2. In particular,
any measure supported on a separated sequence is not PW-sampling.

  6. (6)

PW-sampling measures of the form $\mu=1_{S}\cdot m,\ S\subset{\mathbb{R}}$ are
described by the Paneah-Logvinenko-Sereda theorem [26, 19]. A set
$S\subset{\mathbb{R}}$ is relatively dense if there exists $r,\delta>0$ such
that

$|(x-r,x+r)\cap S|>\delta$

for all $x$. An equivalent reformulation of the theorem in one dimension (the
case studied by Paneah) is that $\mu$ is PW-sampling iff $S$ is relatively
dense. This statement also follows from Theorem 3.1.

###  3.3. Relations to Gelfand-Levitan theory

We will say that a positive $\Pi$-finite non-discrete measure $\mu$ is a GL-
measure ($\mu\in$GL) if it satisfies the Gelfand-Levitan-type condition

${\mathcal{F}}\mu=C\delta_{0}+f,$ (3.2)

for some function $f\in L^{1}_{\rm loc}({\mathbb{R}})$ and $C>0$. Here the
Fourier transform ${\mathcal{F}}\mu$ is understood in the sense of
distributions. The condition means that $\mu$ is, in some sense, close to a
constant multiple of the Lebesgue measure, $\frac{C}{\sqrt{2\pi}}m$, whose
Fourier transform is $C\delta_{0}$. As was mentioned in the introduction,
measures of this class are considered in the classical Gelfand-Levitan theory
of inverse spectral problems. The relation between the class of PW-sampling
measures and the class GL defined above is established by the following
statement.

######  Lemma 3.1.

$\textrm{GL}\subset PW.$

The inverse inclusion clearly does not hold. The measure $1+\cos x$ considered
in Section 7 is obviously PW-sampling but not GL because its Fourier transform
consists of three pointmasses. Furthermore, any periodic measure with locally
infinite support is a PW-sampling measure, but not a GL-measure, unless it is
a constant multiple of $m$. For a non-periodic example, If $\nu$ is any even
finite measure with a non-trivial singular part, then $f=\hat{\nu}$ is a real
bounded function and $\mu=(C+f)m\in\text{\rm PW}$ for large enough $C$. But
$\mu\not\in$GL because ${\mathcal{F}}(\mu)=D\delta_{0}-\nu$ and $\nu\not\in
L^{1}_{\rm loc}$.

The proof of Lemma 3.1 is postponed until Section 4.1.

######  Remark 3.1.

The measures considered in Gelfand-Levitan theory (see for instance [18],
Chapter XII) are the spectral measures of Dirac systems with regular
potentials on the half-line. Such measures have non-trivial absolutely
continuous parts, satisfy (3.2) and some additional conditions.

The class GL as defined above is the class of measures for which some of the
methods of the classical theory may still work, although it is substantially
broader than the set of measures actually considered in [18] or [30].

As will be seen in the proof of Lemma 3.1, the restriction that a GL-measure
should not be discrete can be further weakened to the condition that it should
not be supported on a zero set of a PW-function (a sequence of finite
Beurling-Malliavin exterior density, see for instance [15]).

$\triangle$

###  3.4. Canonical systems of PW-type

We say that two Hilbert spaces $H_{1}$ and $H_{2}$ are equal as sets, and
write $H_{1}\doteq H_{2}$, if the spaces have the same elements, but not
necessarily the same scalar product. We say that a dB-space ${\mathcal{B}}$
has exponential type $T$ if $T$ is the smallest constant with the property
that all functions in ${\mathcal{B}}$ have exponential type at most $T$.

A formula by Krein, see for instance [31, 32], allows one to calculate the
type $T$ of a dB-space ${\mathcal{B}}_{t}$ corresponding to a Hamiltonian
${\mathcal{H}}(t)$:

$T=\int_{0}^{t}\sqrt{\det{\mathcal{H}}(t)}dt.$ (3.3)

Two SAS are equivalent if the corresponding chains of dB-spaces
${\mathcal{B}}^{1}_{t}$ and ${\mathcal{B}}^{2}_{t}$ satisfy

${\mathcal{B}}^{1}_{t}={\mathcal{B}}^{2}_{s(t)}$

(norms and all) for some increasing homeomorphism
$s:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$. The systems are strongly equivalent
if $s(t)=t$. Basic results of KdB theory imply that two CS are equivalent iff
they have the same spectral measures.

The condition ${\mathcal{H}}\neq 0$ a.e., imposed on every CS (1.1), together
with (2.4), implies that for any $t_{2}>t_{1}\geq 0$,
${\mathcal{B}}_{t_{1}}\subsetneq{\mathcal{B}}(t_{2})$.

Let ${\mathcal{H}}$ be a Hamiltonian of a canonical system (1.1). We say that
${\mathcal{H}}$ is of PW-type (${\mathcal{H}}\in\text{\rm PW}$) if for any
$t>0$ there exists $s=s(t)>0$ such that $s(t)\to\infty$ as $t\to\infty$ and

${\mathcal{B}}_{t}({\mathcal{H}})\doteq{\rm PW}_{s}.$ (3.4)

We call the corresponding system a PW-type system.

######  Theorem 3.2.

If ${\mathcal{H}}$ is of PW-type, then

$\det{\mathcal{H}}\neq 0\quad{\rm
a.e.},\qquad\int_{0}^{\infty}\sqrt{\det{\mathcal{H}}(t)}dt=\infty.$ (3.5)

Also, we have

${\mathcal{B}}_{t}\doteq{\rm PW}_{s(t)},\qquad
s(t):=\int_{0}^{t}\sqrt{\det({\mathcal{H}})}.$

###### Proof.

We can assume that ${\mathcal{H}}$ is trace-normalized. Let $\mu$ be a
spectral measure corresponding to ${\mathcal{H}}$. Then $\mu\in\text{\rm PW}$
and $\det{\mathcal{H}}\neq 0$ a.e., see the proof of Theorem 3.4 in Section
4.2. The equation in (3.5) follows from $s(t)\to\infty$ and Krein’s type
formula, together with the remaining equations in the statement. ∎

The change of time $t\mapsto s=s(t)$ in the above theorem allows us to
transform any PW-type system, or more generally any canonical system
satisfying (3.5), into a canonical system with

$\det{\mathcal{H}}=1\quad{\rm a.e.}$

We will call such systems det-normalized. A regular (locally summable)
Hamiltonian will remain regular under det-normalization.

######  Theorem 3.3.

Consider a regular CS with the Hamiltonian ${\mathcal{H}},\
\det{\mathcal{H}}\neq 0$ a.e. Then there exists an equivalent regular system
with the Hamiltonian $\tilde{\mathcal{H}}$ such that
$\det\tilde{\mathcal{H}}=1$ a.e.

###### Proof.

The function

$s(t)=\int_{0}^{t}\sqrt{\det{\mathcal{H}}}$

is a $W^{1,1}_{\rm loc}$-diffeomorphism because $\sqrt{\det{\mathcal{H}}}$ is
in $L^{1}_{\rm loc}$ and $\det{\mathcal{H}}>0$ a.e. Consider the inverse
diffeomorphism $t(s)$ and set

$\tilde{H}(s)=({\mathcal{H}}\circ t)(s)\dot{t}(s).$

Then $\tilde{\mathcal{H}}\geq 0$, is locally integrable, and
$\det\tilde{\mathcal{H}}=1$ a.e. Clearly, the systems corresponding to
$\tilde{\mathcal{H}}$ and ${\mathcal{H}}$ are equivalent. ∎

As was mentioned before, a change of time variable, in particular det-
normalization, does not affect the spectral measure of the system.

The converse of the last theorem is not true: our next example shows that
there are det-normalized canonical systems which are not of PW-type, see also
[2].

######  Example 3.1.

Consider the Hamiltonian

${\mathcal{H}}=\begin{pmatrix}t^{m}&0\\\ 0&t^{-m}\end{pmatrix},\qquad
m\in(-1,1),$

and denote

$m=2\nu-1.$

Let $J_{\nu}$ denote the Besssel function of the first kind and let

$J_{\nu}(\lambda)=\lambda^{\nu}F_{\nu}(\lambda).$

Then $F_{\nu}$ is an entire function. For the solution
$\begin{pmatrix}A//C\end{pmatrix}$ of the CS with the Hamiltonian
${\mathcal{H}}$ we have:

######  Lemma 3.2.

$A=F_{\nu-1}(zt),\qquad C=t^{2\nu}zF_{\nu}(zt).$

The lemma can be established by direct calculations.

We will use the following asymptotic of the Bessel functions on the positive
axis:

$J_{\nu}(\lambda)=\frac{\cos\left[\lambda-(2\nu+1)\frac{\pi}{4}\right]}{\sqrt{\lambda}}+O\left(\lambda^{-3/2}\right),\qquad\lambda\to\infty.$

The amplitudes of $|A|^{2}$ and $|C|^{2}$ are $t^{m}x^{-m}$ and $t^{-m}x^{-m}$
respectively. Hence for each fixed $t$,

$\frac{1}{|E(x)|^{2}}\approx x^{m},\qquad x\to\infty.$

It follows that the measure $|E_{t}(x)|^{-2}dx$ is not sampling for $\text{\rm
PW}_{t}$ for any $t$, except in the case $m=0$. Therefore the corresponding
spaces ${\mathcal{B}}_{t}$ are not equal to $\text{\rm PW}_{t}$ as sets and
the system is not of PW-type.

######  Remark 3.2.

(a) All regular Dirac systems are of PW-type as follows from Lemma 3.1 and the
results of [18], see also [2].

(b) Characterization of PW-Hamiltonians is a difficult problem even in the
diagonal case. So far we know that

$\det{\mathcal{H}}\neq 0\qquad\not\Rightarrow\qquad PW$

but

$\det{\mathcal{H}}\neq 0,\quad{\mathcal{H}}\in W^{1,1}_{\rm
loc}\qquad\Rightarrow\qquad PW,$

as follows from the property that regular Dirac systems correspond to CS with
$W^{1,1}_{\rm loc}$-Hamiltonians.

(c) We will also see that

$\begin{pmatrix}h_{11}&h_{12}\\\ h_{12}&h_{22}\end{pmatrix}\in\text{\rm PW}\
\not\Rightarrow\ \begin{pmatrix}h_{22}&-h_{12}\\\
-h_{12}&h_{11}\end{pmatrix}\in\text{\rm PW},$

Example 5.1. The second Hamiltonian corresponds to the switch between the
Dirichlet and Neumann initial conditions, see Section 3.5.

$\triangle$

The next statement relates PW-type systems and PW-sampling measures.

######  Proposition 3.3.

Suppose $\det({\mathcal{H}})=1$ a.e., and let $\mu$ be the (unique) spectral
measure of the corresponding CS. Then

$\mu\in{\rm(PW)}\qquad\Leftrightarrow\qquad{\mathcal{H}}\in{\rm(PW)}.$

Moreover, if either holds, then

$\forall t,\quad{\mathcal{B}}_{t}({\mathcal{H}})={\rm PW}_{t}(\mu).$

###### Proof.

A $\Pi$-finite measure $\mu$ admits a unique, up to parametrization, chain of
regular de Branges spaces isometrically embedded in $L^{2}(\mu)$, as follows
from the results of [7]. Since both ${\mathcal{B}}_{t}({\mathcal{H}})$ and
$\text{\rm PW}_{s}(\mu)$ form such chains, we must have

${\mathcal{B}}_{t}({\mathcal{H}})=\text{\rm PW}_{s}(\mu)\doteq\text{\rm
PW}_{s}.$

By Krein’s formula, the space ${\mathcal{B}}_{t}$ consists of functions of
exponential type at most

$\int_{0}^{t}\sqrt{\det{\mathcal{H}}(x)}dx,$

which is equal to $t$ due to the det-normalization. Since $\text{\rm
PW}_{s}(\mu)$ consists of functions of type $s$, we must have
${\mathcal{B}}_{t}({\mathcal{H}})=\text{\rm PW}_{t}(\mu)$. ∎

By one of the main theorems of Krein-de Branges theory, every positive
$\Pi$-finite measure on ${\mathbb{R}}$ is a spectral measure for a canonical
system with a locally summable (trace-normalized) Hamiltonian [7]. We obtain
the following corollary.

######  Theorem 3.4.

Let $\mu$ be a PW-sampling measure. Then there exists a det-normalized
${\mathcal{H}}$ such that $\mu$ is its spectral measure. Two det-normalized
Hamiltonians have the same spectral measure iff there is $k\in{\mathbb{R}}$
such that

$\check{\mathcal{H}}={\mathcal{H}}_{k}:=T_{k}^{*}{\mathcal{H}}T_{k},\qquad
T_{k}=\left(\begin{array}[]{cc}1&k\\\ 0&1\\\ \end{array}\right).$

We postpone the proof until Section 4.2.

######  Remark 3.3.

In the above formula

${\mathcal{H}}_{k}=\begin{pmatrix}h_{11}&h_{12}+kh_{11}\\\
h_{12}+kh_{11}&h_{22}+2kh_{12}+k^{2}h_{11}\end{pmatrix}$

and

$M_{k}:=T^{-1}_{k}MT_{k}=\begin{pmatrix}A-kC&B+kA-kD-k^{2}C\\\
C&kC+D\end{pmatrix}$

is the matrizant for ${\mathcal{H}}_{k}$.

$\triangle$

###  3.5. Symmetries

For a Hamiltonian

${\mathcal{H}}=\begin{pmatrix}h_{11}&h_{12}\\\ h_{12}&h_{22}\end{pmatrix}$

we denote

$\breve{\mathcal{H}}=\begin{pmatrix}h_{11}&-h_{12}\\\
-h_{12}&h_{22}\end{pmatrix}{\rm and}\
\tilde{\mathcal{H}}=\begin{pmatrix}h_{22}&-h_{12}\\\
-h_{12}&h_{11}\end{pmatrix}.$

For a measure $\mu$, $\breve{\mu}$ is the measure satisfying
$\breve{\mu}(e)=\mu(-e),\ e\subset{\mathbb{R}},$ and $\tilde{\mu}$ is the AC-
dual measure as defined in Section 2.5.

We denote by $\mu_{\mathcal{H}}$ the spectral measure corresponding to the
system with the Hamiltonian ${\mathcal{H}}$.

######  Lemma 3.3.

(i) The involution ${\mathcal{H}}\mapsto\breve{{\mathcal{H}}}$ descends to the
involution $\mu\mapsto\breve{\mu}$ for the corresponding spectral measures:
$\mu=\mu_{\mathcal{H}},\ \breve{\mu}=\mu_{\breve{\mathcal{H}}}$.

(ii) The involution $H\mapsto\tilde{H}$ descends to the involution
$\mu\mapsto\tilde{\mu}$. In other words,

$\mu_{\tilde{H}}=\tilde{\mu}_{H}$

###### Proof.

(i) Let $\tau=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$. Then $\tau^{2}=$id and
$\Omega\tau=-\tau\Omega$. Note that

$\tau H\tau=\breve{H}$

Computations show that $N(z)=M^{(\tau)}(-z),{\rm where}\ M^{(\tau)}:=\tau
M\tau,$ is the matrizant of $\breve{H}$: from $\tau\Omega\dot{M}\tau=z\tau
HM\tau$ we get

$-\Omega\dot{M}^{(\tau)}=z\tau H\tau\tau M\tau=z\breve{H}M^{(\tau)}.$

Furthermore,

$N(x)=\begin{pmatrix}A(-z)&-B(-z)\\\ -C(-z)&D(-z)\end{pmatrix}$

and

$|E_{N}(x)|^{2}=|A(-x)+iC(-x)|^{2}=|E(-x)|^{2}$

It follows that $\breve{\mu}$ is the spectral measure of $\breve{H}$.

(ii) We use the same argument but with $\tau=\Omega$. We have

$\Omega(\Omega\dot{M}\Omega)=z\Omega HM\Omega=z(-\Omega H\Omega)(\Omega
M\Omega)$

It follows that

$\Omega M\Omega=\begin{pmatrix}-D&C\\\ B&-A\end{pmatrix}$

is the matrizant of $\tilde{H}=-\Omega H\Omega$. ∎

###  3.6. Even measures and diagonal Hamiltonians

We call a measure $\mu$ on ${\mathbb{R}}$ even if $\mu=\breve{\mu}$, i.e.,
$\mu(x)=\mu(-x)$.

######  Theorem 3.5.

If the Hamiltonian ${\mathcal{H}}$ is diagonal, then the spectral measure
$\mu_{\mathcal{H}}$ is even. Conversely, if $\mu$ is an even positive
$\Pi$-finite measure, then there exists a diagonal Hamiltonian ${\mathcal{H}}$
such that $\mu=\mu_{\mathcal{H}}$.

###### Proof.

The first part follows from the last lemma. Alternatively, we can argue as
follows.

Let ${\mathcal{H}}={\rm diag}(h_{1},h_{2})$, then the system is

$\dot{C}=zh_{1}A,\qquad\dot{A}=-zh_{2}C.$

Denote

$\tilde{A}(z,t)=A(-z,t),\qquad\tilde{C}(z,t)=-C(-z,t).$

We have $\tilde{A}=A$ and $\tilde{C}=C$ because the functions satisfy the same
system and the same ICs. We derive the reality condition

$E(-z)=E^{\\#}(z),$

and therefore the symmetry of $\mu$.

As follows from Theorem 3.4 and Lemma 3.3, if the measure is symmetric, then
there is a number $k$ such that

$\breve{H}-H=\begin{pmatrix}0&kh_{11}\\\ kh_{11}&\ast\end{pmatrix}$

which implies

$-2h_{12}=kh_{11},$

so the ratio of the functions $h_{12}$ and $h_{11}$ is constant. Applying
Theorem 3.4 again we see that the Hamiltonian can be made diagonal. ∎

######  Remark 3.4.

CS with diagonal Hamiltonians are equivalent to so-called Krein’s strings, see
the book by Dym and McKean [10]. In particular, the converse statement in
Theorem 3.4 also follows from Krein’s theorem which says that if $\mu$ is a
$\Pi$-finite even measure, then it is a spectral measure of a string.

$\triangle$

Combining the last statement with Theorem 3.4, in the PW-situation we arrive
to the following conclusion.

######  Theorem 3.6.

There is a 1-to-1 correspondence between even PW-sampling measures and
diagonal det-normalized Hamiltonians of PW-type.

##  4\. Solution formulae for the inverse problem

###  4.1. Truncated Toeplitz operators

Let $t>0$ and let $\mu$ be a positive measure such that PW${}_{t}\subset
L^{2}(\mu)$. Then we can define a bounded operator

$L\equiv L_{\mu,t}:~{}{\rm PW}_{t}\to{\rm PW}_{t}$

by the (symbolic) formula

$f\mapsto P_{t}(f\mu),$

where $P_{t}$ is the ”projection” to PWt. The formula makes perfect sense if
$\mu\in L^{\infty}$ but we need to interpret it in terms of quadratic forms in
the general case:

$\forall f,g\in{\rm PW}_{t},\qquad(Lf,g)=\int f\bar{g}d\mu.$ (4.1)

Clearly, $L$ is a well defined bounded non-negative operator.

The following statement can be deduced directly from the definition.

######  Lemma 4.1.

$L_{\mu,t}$ is an invertible (positive) operator iff $\mu$ is sampling for
${\rm PW}_{t}$.

In regard to the family of PW-sampling measures we obtain the following

######  Corollary 4.1.

If $\mu\in\text{\rm PW}$, then the operator $L_{\mu}$ defined by (4.1) is
bounded and invertible in every $\text{\rm PW}_{t},\ t>0$.

Consider now the dB spaces

${\mathcal{B}}_{t}={\rm PW}_{t}(\mu)$

and the operators

$j=j_{t}:~{}{\rm PW_{t}}\to{\mathcal{B}}_{t},\qquad f\mapsto f.$

######  Theorem 4.1.

If $\mu\in$PW, then for any $t>0$,

$j_{t}^{*}j_{t}=L_{\mu}.$

###### Proof.

If $f,g\in{\rm PW}_{t}$, then

$(j^{*}jf,g)_{\rm PW}=(jf,jg)_{\mathcal{B}}=(jf,jg)_{\mu}=\int
f\bar{g}~{}d\mu.$

∎

###  4.2. Postponed proofs

Using the operators $L_{\mu}$ we can now supply the proofs of the lemma on GL-
measures and the theorem on Hamiltonians with PW spectral measures.

###### Proof of Lemma 3.1.

Let $\mu\in GL$, $\hat{\mu}=\delta_{0}+\phi,\ \phi\in L^{1}_{loc}$.

Let us first show that $L_{\mu}$ has trivial kernel in every $\text{\rm
PW}_{a}$. If $f\in\text{\rm PW}_{a},\ f\not\equiv 0$ is in the kernel, then
the measure $f\mu$ annihilates $\text{\rm PW}_{a}$. In particular

$\int f^{\\#}fd\mu=\int|f|^{2}d\mu=0,$

which implies that $\mu$ is a discrete measure supported on the zero set of
$f$. Since GL-measures are non-discrete, we obtain that $L\mu$ is injective on
every $\text{\rm PW}_{a}$.

Let $\mu_{a}$ be the measure such that $\hat{\mu}_{a}=1_{[-2a,2a]}\hat{\mu}$.
Then $\mu_{a}=(1+\psi)m$ where $\psi={\mathcal{F}}^{-1}(\phi_{a}),\
\phi_{a}=1_{[-2a,2a]}\phi,$ is a bounded function. Note that the restriction
of $L_{\mu}$ onto $\text{\rm PW}_{a}$ is equal to $L_{\mu_{a}}$. Therefore
$L_{\mu_{a}}$ has trivial kernel in $\text{\rm PW}_{a}$.

Note that

${\mathcal{F}}(L_{\mu_{a}}f)=1_{[-a,a]}(\hat{f}+\hat{f}\ast\phi_{a})=\hat{f}+1_{[-a,a]}\hat{f}\ast\phi_{a}.$

Hence the operator

$\hat{L}_{\mu_{a}}:L^{2}([-a,a])\to L^{2}([-a,a]),\
\hat{L}_{\mu_{a}}\hat{f}={\mathcal{F}}(L_{\mu_{a}}f),$

is of the Fredholm form

$\hat{L}_{\mu_{a}}=I+K,\ \ Kf=P_{a}(f\ast\phi_{a}),$

where $P_{a}$ is the orthogonal projection $L^{2}([-3a,3a])\to L^{2}([-a,a])$.
Since $\phi_{a}$ is summable, and $P_{a}$ is bounded, $K$ is compact. Since
the kernel of $L_{\mu_{a}}$ is trivial, so is the kernel of
$\hat{L}_{\mu_{a}}$. By the Fredholm alternative, $\hat{L}_{\mu_{a}}$ is
invertible.

Since $\phi_{a}$ is summable, $\hat{L}_{\mu_{a}}$ is bounded.

Hence $L_{\mu}$ is bounded and invertible in any $\text{\rm PW}_{a}$. By Lemma
4.1, $\mu\in\text{\rm PW}$. ∎

Let us now prove Theorem 3.4.

We denote by $\text{\rm PW}_{[a,b]}(\mu)$ the orthogonal difference $\text{\rm
PW}_{b}(\mu)\ominus\text{\rm PW}_{a}(\mu)$. We will need the following

######  Lemma 4.2.

Let $\mu\in\text{\rm PW}$. Then for any $0\leq a<b\leq c$ there exists a
constant $d=d(c,\mu)>0$ such that for any $f\in\text{\rm PW}_{[a,b]}(\mu)$

$\int_{a}^{b}|\hat{f}(x)|^{2}dx>d||f||^{2}_{2}.$

###### Proof.

Let $f\in\text{\rm PW}_{[a,b]}(\mu)$. Note that then $f\in\text{\rm PW}_{b}$.
Denote by $f_{a}$ the orthogonal projection of $f$ onto $\text{\rm PW}_{a}$ in
$\text{\rm PW}_{b}$ and let $f_{b}=f-f_{a}$. Then the integral in the
statement is $||f_{b}||_{2}^{2}$.

Note that $f\in\text{\rm PW}_{[a,b]}(\mu)$ is equivalent to
$L_{\mu,b}f\perp\text{\rm PW}_{a}$, which implies that $\text{\rm
supp}\,{\mathcal{F}}(L_{\mu,b}f)\cap(-a,a)=\emptyset$. Since $L_{\mu,a}$ is
invertible,

$||L_{\mu,b}f_{b}||_{2}^{2}=\int_{-b}^{b}|{\mathcal{F}}(L_{\mu,b}f_{b})(x)|^{2}dx\geq\int_{-a}^{a}|{\mathcal{F}}(L_{\mu,b}f_{b})(x)|^{2}dx=$
$\int_{-a}^{a}|{\mathcal{F}}(L_{\mu,b}(f-f_{a}))(x)|^{2}dx=\int_{-a}^{a}|{\mathcal{F}}(L_{\mu,b}f_{a})(x)|^{2}dx=||L_{\mu,a}f_{a}||_{2}^{2}>d_{1}||f_{a}||_{2}^{2},$

for some $d_{1}=d_{1}(c)$. Since $L_{\mu,b}$ is bounded, it follows that

$||f_{b}||_{2}^{2}>d_{2}||f_{a}||_{2}^{2},$

for some $d_{2}=d_{2}(c)$. ∎

###### Proof of Theorem 3.4.

Since PW-sampling measures are $\Pi$-finite, $\mu$ is the spectral measure of
some regular trace-normalized canonical system with Hamiltonian
${\mathcal{H}}$. We need to show that the set $\\{t|\det{\mathcal{H}}(t)=0\\}$
has Lebesgue measure zero. If that is the case, we can renormalize the system
with the change of variable

$s(t)=\int_{0}^{t}\sqrt{\det{\mathcal{H}}}.$

To show that $|\\{t|\det{\mathcal{H}}(t)=0\\}|=0$, recall that any
$\Pi$-finite measure admits a unique chain of regular de Branges spaces. Since
both $\text{\rm PW}_{s}(\mu)$ and the spaces ${\mathcal{B}}_{t}$ corresponding
to ${\mathcal{H}}$ constitute such chains, $\text{\rm
PW}_{s}={\mathcal{B}}_{t}$ for some $s=s(t)$. According to Krein’s formula for
the exponential type (3.3), $s(t)$ is defined as above.

Assume now that $|\\{t|\det{\mathcal{H}}(t)=0\\}\cap(0,T)|>0$ for some $T>0$.
Let $f\in L^{2}({\mathcal{H}})$ be a non-zero function such that

$\text{\rm supp}\,f\subset\\{t|\det{\mathcal{H}}(t)=0\\}\cap(0,T).$

It follows from the type formula that for any $\varepsilon>0$ one can choose
disjoint intervals

$[a_{k},b_{k}]\subset(0,T)\setminus\text{\rm supp}\,f,\ 0\leq
a_{1}<b_{1}<a_{2}<...<b_{n}\leq T$

such that

$\sum_{k=1}^{n}s(b_{k})-s(a_{k})=s(T)-\varepsilon.$

Put $F={\mathcal{W}}f$. Then $F\in\text{\rm PW}_{s(T)}$ and $\text{\rm
supp}\,\hat{F}\subset[0,s(T)]$. According to the last lemma,

$\int_{0}^{s(a_{1})}|\hat{F}(x)|^{2}dx+\int_{s(b_{1})}^{s(a_{2})}|\hat{F}(x)|^{2}dx+...+\int_{s(b_{n})}^{s(T)}|\hat{F}(x)|^{2}dx>d||F||_{2}^{2}.$

Recall that the Weyl transform ${\mathcal{W}}$ is unitary and therefore
$||F||_{2}^{2}>0$. Since

$\varepsilon=\left|\ [0,s(T)]\setminus\cup[s(a_{k}),s(b_{k})]\ \right|$

can be chosen arbitrarily small, we obtain a contradiction.

For the second part of the statement, notice that if ${\mathcal{H}}$ and
$\check{\mathcal{H}}$ have the same PW-sampling spectral measure, then the
det-normalization for both Hamiltonians implies
${\mathcal{B}}_{t}=\check{\mathcal{B}}_{t}=\text{\rm PW}_{t}(\mu)$ for all
$t$. Then for the HB functions $E_{t}=A_{t}-iC_{t}$ and
$\check{A}_{t}-i\check{C}_{t}$ we have

$\begin{pmatrix}\check{A}_{t}\\\
\check{C}_{t}\end{pmatrix}=R^{-1}\begin{pmatrix}A_{t}\\\ C_{t}\end{pmatrix}$
(4.2)

fore some constant matrix $R\in{\rm SL}(2,{\mathbb{R}})$, see for instance
[31], Theorem 2.2.

Note that if $\begin{pmatrix}A_{t}\\\ C_{t}\end{pmatrix}$ is a solution to a
CS with a Hamiltonian ${\mathcal{H}}$ then $R^{-1}\begin{pmatrix}A_{t}\\\
C_{t}\end{pmatrix}$ is a solution to a system with the Hamiltonian
$R^{*}{\mathcal{H}}R$. Here we use the property that any $SL(2,{\mathbb{R}})$
matrix is symplectic, i.e., $R^{*}\Omega R=\Omega$.

From the uniqueness of systems on $(0,t)$ with the same HB-function
corresponding to $t$ (see for instance Theorem 13 in [32]) it follows that all
$\begin{pmatrix}\check{A}_{s}\\\ \check{C}_{s}\end{pmatrix},\ s<t$ satisfy
(4.2) with $t=s$ and the same constant matrix $R$. Choosing larger $t$ we
conclude that (4.2) must hold with the same $R$ for all $t$. Finally, the
property that

$\begin{pmatrix}A_{t}\\\ C_{t}\end{pmatrix}(0)=\begin{pmatrix}\check{A}_{t}\\\
\check{C}_{t}\end{pmatrix}(0)=\begin{pmatrix}1\\\ 0\end{pmatrix}$

together with $R\in{\rm SL}(2,{\mathbb{R}})$ imply that $R$ is upper
triangular with $1$ on the diagonal. ∎

###  4.3. Reprokernels

Let $\mu$ be a sampling measure for $\text{\rm PW}_{t}$ for some fixed $t>0$.
Consider the de Branges space ${\mathcal{B}}=\text{\rm PW}_{t}(\mu)$,
${\mathcal{B}}\doteq{\rm PW}_{t}$. Then the truncated Toeplitz operator
$L=L_{\mu,t}$, $L:{\rm PW}_{t}\to{\rm PW}_{t}$ defined as in the last section
is bounded and invertible. For $w\in{\mathbb{C}}$ we denote by $K_{w}$ and
$\mathring{K}_{w}$ the reprokernels in ${\mathcal{B}}$ and PWt respectively.

######  Theorem 4.2.

For all $w\in{\mathbb{C}}$,

$j\left[L^{-1}\mathring{K}_{w}\right]=K_{w}.$

###### Proof.

For any $f$ and $g$ in PWt we have

$(jf,jg)_{\mathcal{B}}=(f,Lg)_{\rm PW}.$

For $g=j^{-1}K_{w}$ we obtain

$(jf,K_{w})_{\mathcal{B}}=(f,Lj^{-1}K_{w})_{\rm PW}.$

The left-hand side is equal to

$f(w)=(f,\mathring{K}_{w})_{\rm PW},$

and since $f$ is arbitrary, we get

$Lj^{-1}K_{w}=\mathring{K}_{w}.$

∎

In the formulas for the inverse spectral problem the last statement will only
be used in the case $w=0$. We will use the notations $k_{t},\
\mathring{k}_{t}$ for the reprokernels $K_{0},\ \mathring{K}_{0}$ in
${\mathcal{B}}_{t}$ and PWt respectively. Since

$\mathring{k}_{t}(z)=\frac{1}{\pi}~{}\frac{\sin tz}{z},$

Theorem 4.2 implies

$\qquad k_{t}=j_{t}\left[L^{-1}_{\mu,t}\mathring{k}_{t}\right].$ (4.3)

###  4.4. Recovering the term $h_{11}(t)$

In this section we obtain a formula which recovers the element $h_{11}$ of the
Hamiltonian

${\mathcal{H}}=\begin{pmatrix}h_{11}&h_{12}\\\ h_{12}&h_{22}\end{pmatrix}$

in CS (1.1) from the spectral measure.

Note that for det-normalized CS with even spectral measures this amounts to a
full solution for the inverse spectral problem as $h_{12}\equiv 0$ by Theorem
3.5 and $h_{11}h_{22}=1$. As was mentioned before, systems with even measures
represent Krein’s string operators.

As before, we assume that $\mu$ is a PW-sampling measure. Recall that by
Theorem 3.4, $h_{11}$ is determined by $\mu$ in a unique way (unlike $h_{22}$
or $h_{12}$).

######  Theorem 4.3.

Let $\mu\in\text{\rm PW}$ be the spectral measure of a system (1.1) with the
Hamiltonian ${\mathcal{H}}$. Then $t\mapsto k_{t}(0)$ is an absolutely
continuous function and

$h_{11}(t)=h^{\mu}(t):=\pi\frac{d}{dt}k_{t}(0),$

where the kernel $k_{t}$ is given by (4.3).

###### Proof.

From (2.5) we obtain the ”Christoffel-Darboux” formula

$\int_{0}^{t}\langle{\mathcal{H}}(s)X_{z}(s),X_{w}(s)\rangle~{}ds=\pi
K_{w}(z,t).$

It follows that

$\langle{\mathcal{H}}(t)X_{z}(t),X_{w}(t)\rangle=\pi\dot{K}_{w}(z,t).$

If $z=w=0$, then $X_{0}(t)=(1,0)^{\scriptscriptstyle T}$, and

$\langle{\mathcal{H}}(t)X_{0}(t),X_{0}(t)\rangle=h_{11}(t).$

∎

######  Remark 4.1.

The nesting property ${\mathcal{B}}_{t_{1}}\overset{\rm
iso}{\subset}{\mathcal{B}}_{t_{2}}$ for $t_{t}<t_{2}$, together with

$k_{t}(0)=\sup\\{f(0)|\ f\in{\mathcal{B}}_{t},||f||_{{\mathcal{B}}_{t}}=1\\},$

implies that the function $t\mapsto k_{t}(0)$ is increasing, so the derivative
in the definition of $h^{\mu}$ exists a.e. The same can be seen from our
proof. The det-normalization implies $h^{\mu}>0$ a.e.

$\triangle$

######  Corollary 4.2.

If $\mu\in$PW is even, then

${\mathcal{H}}=\begin{pmatrix}h^{\mu}&0\\\ 0&h_{\mu}^{-1}\end{pmatrix},$

where $h_{\mu}$ is from the last theorem, is the unique det-normalized
diagonal Hamiltonian such that $\mu$ is the spectral measure of (1.1).

###  4.5. Conjugate kernels

Let ${\mathcal{H}}$ be a det-normalized PW Hamiltonian and let
${\mathcal{B}}_{t}$ and $\tilde{\mathcal{B}}_{t}$ be the corresponding chains
of dB-spaces generated by HB functions $E_{t}=A_{t}-iC_{t}$ and
$\tilde{E}_{t}=B_{t}-iD_{t}$. We drop the subindex $t$ in our formulas when
$t$ is fixed.

As before, we denote by $K_{t}(z,w)$ and $\tilde{K}_{t}(z,w)$ the
corresponding reprokernels, sometimes using abbreviated notations $K_{w}(z),\
\tilde{K}_{w}(z)$ when $t$ is fixed. Due to a special role played by
reprokernels at 0, we also use the notation

$k_{t}(z):=K_{t}(z,0)=\frac{1}{\pi}\frac{C_{t}(z)}{z}.$

We now introduce ”conjugate” kernels
$\tilde{L}_{w}(z)\equiv\tilde{L}_{t}(z,w)\in\tilde{\mathcal{B}}_{t}$ defined
as

$\tilde{L}_{w}(z)=\frac{1}{\pi}~{}\frac{[D(z)A(\bar{w})-B(z)C(\bar{w})]-1}{z-\bar{w}}.$

In particular

$\tilde{l}_{t}(z):=\tilde{L}_{t}(z,0)=\frac{1}{\pi}~{}\frac{D(z)-1}{z}.$

As before, we denote by $X_{z}(t)$ and $Y_{z}(t)$ the solutions to CS (1.1)
with Neumann and Dirichlet initial conditions correspondingly,

$X_{z}(t):=\begin{pmatrix}A_{t}(z)\\\
C_{t}(z)\end{pmatrix},Y_{z}(t):=\begin{pmatrix}B_{t}(z)\\\
D_{t}(z)\end{pmatrix}.$

The following relation presents a special case of Lagrange identities.

######  Lemma 4.3.

$\pi\tilde{L}_{t}(z,w)=(Y_{z},X_{w})_{L^{2}({\mathcal{H}},[0,t])}$

To prove the lemma we first establish the following relation for the transfer
matrix.

######  Lemma 4.4.

$\frac{\partial}{\partial t}[M_{t}^{*}(w)\Omega
M_{t}(z)]=(z-\bar{w})M^{*}_{t}(w)\Omega M_{t}(z).$

###### Proof.

$\frac{\partial}{\partial t}[M_{t}^{*}(w)\Omega
M_{t}(z)]=+M_{t}^{*}(w)\Omega\frac{\partial}{\partial
t}M_{t}(z)]+\frac{\partial}{\partial t}[M_{t}^{*}(w)\Omega]M_{t}(z)=$
$zM^{*}_{t}(w){\mathcal{H}}M_{t}(z)-\bar{w}M_{t}^{*}(w){\mathcal{H}}^{*}M_{t}(z)=(z-\bar{w})M^{*}_{t}(w)\Omega
M_{t}(z).$

∎

###### Proof of Lemma 4.3.

$(Y_{z},X_{w})_{L^{2}({\mathcal{H}},[0,t])}=\int_{0}^{t}<{\mathcal{H}}Y_{z}(s),X_{w}(s)>ds=$
$=\int_{0}^{t}\left<{\mathcal{H}}M_{s}(z)\begin{pmatrix}0\\\
1\end{pmatrix},M_{s}(w)\begin{pmatrix}1\\\ 0\end{pmatrix}\right>ds=$
$=\int_{0}^{t}\left<M^{*}_{s}(w){\mathcal{H}}M_{s}(z)\begin{pmatrix}0\\\
1\end{pmatrix},\begin{pmatrix}1\\\ 0\end{pmatrix}\right>ds=$
$\int_{0}^{t}\left<\frac{\frac{\partial}{\partial s}[M_{s}^{*}(w)\Omega
M_{s}(z)]}{z-\bar{w}}\begin{pmatrix}0\\\ 1\end{pmatrix},\begin{pmatrix}1\\\
0\end{pmatrix}\right>ds=$ $=\frac{\frac{\partial}{\partial
t}[M_{t}^{*}(w)\Omega
M_{t}(z)]-1}{z-\bar{w}}=\frac{[D(z)A(\bar{w})-B(z)C(\bar{w})]-1}{z-\bar{w}}.$

∎

######  Corollary 4.3.

$h_{12}(t)=\pi\frac{d}{dt}~{}\tilde{l}_{t}(0).$

###### Proof.

If $z=w=0$, then

$(Y_{z},X_{w})_{L^{2}({\mathcal{H}},[0,t])}=\int_{0}^{t}\left\langle{\mathcal{H}}\begin{pmatrix}0\\\
1\end{pmatrix},\begin{pmatrix}1\\\
0\end{pmatrix}\right\rangle=\int_{0}^{t}h_{12}.$

∎

Our next goal is to express the conjugate kernel $\tilde{l}_{t}$ in terms of
the spectral measure which will allow us to recover the element $h_{12}$ of
the Hamiltonian via the last corollary, see Theorem 4.6.

######  Theorem 4.4.

For each $t$ there exists a linear unitary operator

$T_{t}:~{}{\mathcal{B}}_{t}\to\tilde{\mathcal{B}}_{t}$

such that

$\forall w\in{\mathbb{C}},\quad K_{w}\mapsto\tilde{L}_{w}.$

If $t_{1}<t_{2}$, then $T_{t_{1}}$ coincides with the restriction of
$T_{t_{2}}$ to ${\mathcal{B}}_{t_{1}}$.

###### Proof.

The operator $T_{t}$ can be defined via the formula

$\tilde{W}_{t}=T_{t}\circ{\mathcal{W}}_{t},$

where ${\mathcal{W}}_{t}:L^{2}({\mathcal{H}},[0,t])\to{\mathcal{B}}_{t}$ and
$\tilde{\mathcal{W}}_{t}:L^{2}({\mathcal{H}},[0,t])\to\tilde{\mathcal{B}}_{t}$
are the restrictions of the corresponding Weyl transforms.∎

######  Remark 4.2.

We emphasize the operators $T_{t}$ (and the spaces $\tilde{\mathcal{B}}_{t}$)
are not determined by the spectral measure; they also depend on the choice of
${\mathcal{H}}$. If we replace ${\mathcal{H}}$ by
$\check{\mathcal{H}}={\mathcal{H}}_{k}$ (see Remark 3.3), then we get

$\check{T}_{t}=T_{t}+kI.$

$\triangle$

###  4.6. Hilbert transform

For a $\Pi$-finite measure $\mu$ and $f\in L^{2}(|\mu|)$ we will use the
notation

$K(f\mu)(z)=\frac{1}{\pi}\int\frac{f(s)~{}d\mu(s)}{s-z},$

and

${\mathcal{K}}\mu(z)=\frac{1}{\pi}\int\left[\frac{1}{s-z}-\frac{s}{1+s^{2}}\right]~{}d\mu(s),$

where $z\in{\mathbb{C}}\setminus{\mathbb{R}}$. If $f\in L^{2}(\mu)$ is an
entire function, then we define

$H^{\mu}f=K(f\mu)-f{\mathcal{K}}\mu.$

It is clear that $H^{\mu}f$ extends to an entire function:

$(H^{\mu}f)(z)=\frac{1}{\pi}\int\left[\frac{f(s)-f(z)}{s-z}+\frac{sf(z)}{1+s^{2}}\right]~{}d\mu(s).$

######  Example 4.1.

If $\mu=m$, then $H^{\mu}f$ is the analytic continuation of the Hilbert
transform

$Hf(z)=\frac{1}{\pi}{\rm v.p.}\int_{\mathbb{R}}\frac{f(s)ds}{s-z}$

and

$H^{\mu}:~{}{\rm PW}_{t}\to{\rm PW}_{t}$

is a unitary operator.

######  Remark 4.3.

If $\mu$ is a periodic locally finite measure then

$\int\frac{s}{1+s^{2}}~{}d\mu(s)=c$

exists and $H^{\mu}$ takes on the form

$H^{\mu}f=Kf\mu-fK\mu+cf=\frac{1}{\pi}\int\frac{f(s)-f(z)}{s-z}d\mu(s)+cf.$

$\triangle$

######  Theorem 4.5.

Let ${\mathcal{H}}$ be a det-normalized PW Hamiltonian, and let $\mu$ be the
spectral measure. Let ${\mathcal{B}}_{t},\ \tilde{\mathcal{B}}_{t}$ be the
corresponding dB-spaces and let $T_{t}$ be the unitary operators from Theorem
4.4. Then there exists a real constant $c=c({\mathcal{H}})$ such that

$\forall t,\qquad T_{t}=(H^{\mu}-cI)\Big{|}_{{\mathcal{B}}_{t}}.$

######  Remark 4.4.

Note that $c({\mathcal{H}}_{k})=c({\mathcal{H}})-k$, where ${\mathcal{H}}_{k}$
is like in Remark 3.3.

$\triangle$

###### Proof.

1) Let us fix $t>0$. Let $\nu=\mu_{-1}$ be the representing measure for the
space ${\mathcal{B}}_{t_{0}}$ for some $t_{0}>t$ and $\alpha=-1$, as defined
in Section 2.3. Let us first show that

$T_{t}=(H^{\nu}-cI)\Big{|}_{{\mathcal{B}}_{t}}.$

Indeed, for $F\in{\mathcal{B}}_{t}$,

$T_{t}F(z)=<T_{t}F,\tilde{K}_{z}>_{\tilde{\mathcal{B}}_{t}}=<T_{t}F,T_{t}L_{z}>_{\tilde{\mathcal{B}}_{t}}==<F,L_{z}>_{{\mathcal{B}}_{t}}=$
$\int F(s)\frac{[A(s)D(z)-C(s)B(z)]-1}{s-z}d\nu(s)=$
$=-K(F\nu)(z)-B(z)\int\frac{F(s)C(s)}{s-z}d\nu(s),$

because by construction $\nu=\mu_{-1}$ is supported at the zeros of $A$. On
the other hand,

$F(z)=<F,K_{z}>_{{\mathcal{B}}_{t}}=-\int
F(s)\frac{A(s)C(z)-C(s)A(z)}{s-z}d\nu(s)=$ $A(z)\int\frac{F(s)C(s)}{s-z}d\nu$

Therefore

$T_{t}F=-K(F\nu)-\frac{B}{A}F.$

It is left to recall that by Lemma 2.1

${\mathcal{K}}\nu={\mathcal{K}}\mu_{-1}=-\frac{B}{A}+\text{\rm const}.$

2) We will now show that $H^{\nu}=H^{\mu}$ on ${\mathcal{B}}_{t}$.

Fix any function $G\in{\mathcal{B}}_{t}$ such that $G(i),G(-i)\neq 0$.
Consider the expression

$\int\left[\frac{F(s)-F(z)}{s-z}-\frac{1}{2}\left(\frac{F(z)}{G(i)}\frac{G(s)-G(i)}{s-i}+\frac{F(z)}{G(-i)}\frac{G(s)-G(-i)}{s+i}\right)\right]d\nu(s).$
(4.4)

Let us denote the function under the integral by $S(z,s)$. Note that since
${\mathcal{B}}_{t}$ is regular, for each fixed $z$ the functions $S(z,w)$ and
$wS(z,w)$ belong to ${\mathcal{B}}_{t}$ as functions of $w$. Hence, the
integral must remain the same if $\nu$ is replaced with any measure $\eta$
such that ${\mathcal{B}}_{t}\overset{\rm iso}{\subset}L^{2}(\eta)$. Indeed,
let $Q\in{\mathcal{B}}_{t}$ be any function such that $Q(0)=1$. Once again,
since ${\mathcal{B}}_{t}$ is regular, $R(z)=(Q(z)-1)/z\in{\mathcal{B}}_{t}$.
Therefore

$\int S(z,s)d\eta(s)=\int S(z,s)(\bar{Q}(s)-s\bar{R}(s))d\eta(s)=$
$<S(z,s),Q(s)>_{\eta}-<sS(z,s),R(s)>_{\eta}=$
$<S(z,s),Q(s)>_{\nu}-<sS(z,s),R(s)>_{\nu}.$

Since $\mu$ is the spectral measure, it can replace $\nu$ in (4.4). It is left
to notice that when $\nu=\mu$ the expression in (4.4) is equal to

$KF\mu(z)-F(z){\mathcal{K}}\mu(z)+F(z)\int\left[\frac{1}{G(i)}\frac{G(s)}{s-i}+\frac{1}{G(-i)}\frac{G(s)}{s+i}\right]d\mu.$

Recall that $G$ was a fixed function, independent of $F$. It remains to put

$c_{\mu}=d_{\mu}-d_{\nu},$

where $d_{\mu}$ is equal to the last integral and $d_{\nu}$ is equal to the
last integral with $\mu$ replaced by $\nu$. ∎

###  4.7. Recovering the term $h_{12}(t)$

######  Theorem 4.6.

Let $\mu\in\text{\rm PW}$, and let the functions $k_{t}\in\text{\rm
PW}_{t}(\mu)$ be obtained from $\mu$ by via the formula (4.3). Define
$\tilde{l}_{t}=H^{\mu}k_{t}$. Then $\mu$ is the spectral measure of the
Hamiltonian

${\mathcal{H}}=\begin{pmatrix}h^{\mu}&g^{\mu}\\\
g^{\mu}&\frac{1+g^{2}_{\mu}}{h^{\mu}}\end{pmatrix},$

where

$g^{\mu}(t):=\pi\frac{d}{dt}\tilde{l}_{t}(0).$

###### Proof.

Let $\check{\mathcal{H}}$ be the Hamiltonian provided by Theorem 3.4.
Uniqueness of the regular chain of dB-spaces in $L^{2}(\mu)$ implies that
$\check{\mathcal{B}}_{t}=\text{\rm PW}_{t}(\mu)$. Let $\check{T}_{t}$ be the
operators from Theorem 4.4. Denote

$\check{l}_{t}=\check{T}_{t}k_{t}.$

By Theorem 4.5 there is a real number $c$ such that $\check{T}_{t}=H^{\mu}-cI$
on ${\mathcal{B}}_{t}$, so

$\check{l}_{t}=\tilde{l}_{t}-ck_{t}.$

Differentiating, we get

$\check{h}_{12}=g^{\mu}-ch^{\mu}.$

It follows that

${\mathcal{H}}=T_{c}^{*}\check{\mathcal{H}}T_{c}.$

∎

###  4.8. Equations for the Fourier transform of $k_{t}$

How to compute the functions $h^{\mu}$ and $g^{\mu}$ from Theorem 4.5?
Sometimes it is helpful to work with the functions

$\psi_{t}:=\hat{k}_{t},$

so that

$\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\psi_{t}=k_{t}(0).$

If $f\in{\rm PW}_{t}$, then

$f(0)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}\hat{f}(\xi)~{}d\xi$

while also

$f(0)=(f,\mathring{k}_{t})_{PW_{t}}=\left(\hat{f},{\mathcal{F}}(\mathring{k}_{t})\right)_{L^{2}(-t,t)}.$

This leads us to the well known formula for the Fourier transform of the sinc
function:

${\mathcal{F}}\mathring{k}_{t}={\mathcal{F}}\left(\frac{\sin tz}{\pi
z}\right)=\frac{1}{\sqrt{2\pi}}~{}1_{(-t,t)}.$

The Fourier transform $\psi_{t}$ of the reprokernel $k_{t}$ can be found using
the following statement.

######  Theorem 4.7.

$\psi=\psi_{t}$ satisfies

$\psi\ast\hat{\mu}=1\quad{\rm on}\quad(-t,t)$

and

$\psi=0\quad{\rm on}\quad{\mathbb{R}}\setminus[-t,t].$

###### Proof.

We denote by $P_{t}$ the orthogonal projection
$P_{t}:L^{2}({\mathbb{R}})\to\text{\rm PW}_{t}$. Somewhat formally we
interpret the equation $L_{\mu}k_{t}=\mathring{k}_{t}$ as

$P_{t}(k_{t}\mu)~{}=~{}\mathring{k}_{t},$

and taking Fourier transforms we obtain

$1_{(-t,t)}~{}{\mathcal{F}}(k_{t}\mu)~{}=~{}\frac{1}{\sqrt{2\pi}}1_{(-t,t)}.$

Note that

${\mathcal{F}}(k_{t}\mu)~{}=~{}\frac{1}{\sqrt{2\pi}}\hat{\mu}\ast\psi_{t}.$

∎

######  Remark 4.5.

For $\mu\in\text{\rm PW}$ and $f\in\text{\rm PW}_{a}$ the map

$g\mapsto\int g\bar{f}d\mu$

defines a bounded linear functional on any $\text{\rm PW}_{t},\ t>0$. It
follows that the restriction of $\hat{f}\ast\hat{\mu}$ to any interval
$(-t,t)$ belongs to $L^{2}([-t,t])$. In particular, so does
$\psi_{t}\ast\hat{\mu}$ because $k_{t}\in\text{\rm PW}_{t}(\mu)\doteq\text{\rm
PW}_{t}$.

$\triangle$

##  5\. Different initial conditions

###  5.1. The term $h_{22}(t)$

Let ${\mathcal{H}}$ be a Hamiltonian such that $\det({\mathcal{H}})=1$ a.e.
and let $\tilde{\mu}$ be the spectral measure of ${\mathcal{H}}$ for the
initial condition $(0,1)^{\scriptscriptstyle T}$. Assume that
$\tilde{\mu}\in$(PW). Then we can recover the term $h_{22}$ from $\tilde{\mu}$
exactly as we did it for $h_{11}$ and the spectral measure $\mu$:

$h_{22}(t)=\pi\frac{d}{dt}\tilde{k}_{t}(0),\qquad\tilde{k}_{t}:=j_{t}\left[L^{-1}_{\tilde{\mu},t}\mathring{k}_{t}\right]$

or simply,

$h_{22}=h^{\tilde{\mu}}.$

######  Theorem 5.1.

Let $\mu$ and $\tilde{\mu}$ be the spectral measures of the det-normalized
Hamiltonian $H$ corresponding to the Neumann and Dirichlet boundary conditions
at 0 correspondingly. Assume that both $\mu$ and $\tilde{\mu}$ are PW-sampling
measures. Then there exists a real number $c$ such that

${\mathcal{H}}=\begin{pmatrix}h^{\mu}&g^{\mu}-ch^{\mu}\\\
g^{\mu}-ch^{\mu}&h^{\tilde{\mu}}\end{pmatrix},$

######  Corollary 5.1.

If ${\mathcal{H}}$ is diagonal and both $\mu$ and $\tilde{\mu}$ are PW-
sampling measures, then

$h^{\mu}h^{\tilde{\mu}}=1\qquad{\rm a.e.~{}on}\quad\mathbb{R}_{+}.$

###  5.2. AC duality

In view of the last theorem, we wold like to describe all possible measures
$\tilde{\mu}$ for a given $\mu$.

As was defined in Section 2.5, $\tilde{\mu}$ is AC-dual to $\mu$ if there
exists a function $\phi\in H^{\infty}({\mathbb{C}}_{+})$,
$\|\phi\|_{\infty}\leq 1,$ such that

$\mu=\sigma_{-1}^{\phi},\qquad\tilde{\mu}=\sigma_{1}^{\phi},$

i.e.,

${\mathcal{P}}\mu=\Re\frac{1-\phi}{1+\phi},\qquad{\mathcal{P}}\tilde{\mu}=\Re\frac{1+\phi}{1-\phi}.$
(5.1)

For simplicity we’ll assume $\mu(\infty)=\tilde{\mu}(\infty)=0$; as we
mentioned earlier our spectral measures will satisfy this condition. Given
$\mu$ there is a 1-parameter family of dual measures
$\tilde{\mu}=\sigma_{1}^{\phi_{b}}$ where

$\phi_{b}=\frac{i-{\mathcal{K}}\mu-b}{i+{\mathcal{K}}\mu+b}.\qquad(b\in{\mathbb{R}}).$
(5.2)

We will sometimes use the notation $\tilde{\mu}_{b}$ if
$\mu=\sigma^{\phi_{b}}_{-1}$, $\tilde{\mu}_{b}=\sigma^{\phi_{b}}_{1}$ for
$\phi_{b}$ as in (5.2). In particular,

${\mathcal{P}}\tilde{\mu}_{b}=\Re\frac{i}{{\mathcal{K}}\mu+b}.$ (5.3)

######  Theorem 5.2.

Let $\mu\in\text{\rm PW}$ be given. Then $\tilde{\mu}$ is an AC-dual measure
of $\mu$ iff there exists a Hamiltonian ${\mathcal{H}}$ such that $\mu$ and
$\tilde{\mu}$ are the spectral measures of ${\mathcal{H}}$ with the Neumann
and Dirichlet initial conditions respectively.

###### Proof.

(i) Let $\mu$ and $\tilde{\mu}$ be the spectral measures of some
${\mathcal{H}}$, and let $\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ be the
transfer matrix of ${\mathcal{H}}$. For each $t>0$ we define the measure
$\mu_{t}\equiv\mu^{E_{t}}$ as the representing measure for the space
${\mathcal{B}}(A_{t}-iC_{t})$ supported on $\\{A_{t}=0\\}$ and
$\tilde{\mu}_{t}\equiv\mu^{\tilde{E}_{t}}$ as the representing measure for the
space ${\mathcal{B}}(B_{t}-iD_{t})$ supported on $\\{B_{t}=0\\}$. If we denote

$\phi_{t}=\frac{A_{t}-iB_{t}}{A_{t}+iB_{t}},$

then

$\mu_{t}=\sigma_{-1}^{\phi_{t}},\qquad\tilde{\mu}_{t}=\sigma_{1}^{\phi_{t}}$

(note that ${\mathcal{K}}\mu_{t}=-B/A$ like in Lemma 2.1). Since $\mu_{t}$ is
a representing measure for ${\mathcal{B}}_{t}={\mathcal{B}}(E_{t})$, we have

${\mathcal{B}}_{s}\overset{\rm iso}{\subset}L^{2}(\mu_{t})\ {\rm for\ all}\
s\leq t.$

Since $\mu$ is the unique measure for which the above inclusion holds for all
$s>0$ and since $\cup_{t}{\mathcal{B}}_{t}$ is dense in $L^{2}(\mu)$, we
obtain that $\mu_{t}\to\mu$, $*$-weakly as $t\to\infty$. Similarly,
$\tilde{\mu}_{t}\to\tilde{\mu}$. Since

$\phi_{t}=\frac{1+i{\mathcal{K}}\mu_{t}}{1-i{\mathcal{K}}\mu_{t}},$

the functions $\phi_{t}$ tend to $\phi$,

$\phi=\frac{1+i{\mathcal{K}}\mu}{1-i{\mathcal{K}}\mu}$

pointwise in ${\mathbb{C}}_{+}$. Then $\mu$, $\tilde{\mu}$ and $\phi$ satisfy
(5.1) and we see that $\mu$ and $\tilde{\mu}$ are dual.

(ii) Consider $\check{\mathcal{H}}={\mathcal{H}}_{k}$ defined as in Remark
3.3. Then $\check{A}=A$ and $\check{B}=B+kA$, so

$\check{\phi}_{t}=\frac{A_{t}-ikA_{t}-iB_{t}}{A_{t}+ikA_{t}+iB_{t}}.$

Recalling that, according to Lemma 2.1, ${\mathcal{K}}\mu=-B/A+\text{\rm
const}$ and using (5.2), we obtain all AC-dual measures as $k$ ranges over
${\mathbb{R}}$. ∎

###  5.3. PW-sampling in AC-families

The statement of the Theorem 5.1 includes an assumption that both measures
$\mu$ and $\tilde{\mu}$ are PW-sampling. This brings up a natural question of
whether this assumption is redundant, i.e, if $\mu\in\text{\rm PW}$ implies
$\tilde{\mu}\in\text{\rm PW}$.

Our next goal is to study this question in more detail. First we give an
example showing that the answer, in general, is negative.

We denote by ${\rm sign}\ x$ the sign-function defined as $\pm 1$ on
${\mathbb{R}}_{\pm}$ and as 0 at 0.

######  Example 5.1.

Consider $\mu(x)=2+{\rm sign}\ x$. Clearly, $\mu\in\text{\rm PW}$. Let
$\phi\in H^{\infty},\ ||\phi||_{\infty}\leq 1$ be the function such that
$\phi(i)\in{\mathbb{R}}$ and (2.6) is satisfied for $\mu$ and $\tilde{\mu}$
with $\alpha=\mp 1$ respectively. Then

${\mathcal{K}}\mu=i\frac{1-\phi}{1+\phi}=-\frac{1}{{\mathcal{K}}\tilde{\mu}}.$

Since ${\mathcal{K}}\mu(z)\to\infty$ as $z\to\infty$ the boundary values of
${\mathcal{K}}\tilde{\mu}(z)$ on ${\mathbb{R}}$ tend to 0 at $\pm\infty$. It
follows that

$\tilde{\mu}((x,x+1))\leq\int_{x}^{x+1}|{\mathcal{K}}\tilde{\mu}(t)|dt\to 0\
{\rm as}\ x\to\pm\infty.$

As was discussed in Section 3.1, this implies $\tilde{\mu}\not\in\text{\rm
PW}$.

######  Remark 5.1.

This example has the following interpretation: even though $\tilde{\mu}$ is
not a PW-measure, the corresponding inverse spectral problem can be solved by
switching to its dual $\mu\in\text{\rm PW}$.

If, on the other hand, all (or two) of the measures in an AC-family are PW-
sampling then a combination of Theorems 4.6 and 5.1 provides a continuum of
often rather non-trivial identities between solutions to the inverse spectral
problems corresponding to different measures from the family.

$\triangle$

We now formulate a necessary and sufficient condition for all measures dual to
a given $\mu\in\text{\rm PW}$ to be PW-sampling.

######  Theorem 5.3.

Let $\mu\in\text{\rm PW}$. All measures dual to $\mu$ are PW-sampling if and
only if there exists $d\neq 0$ such that ${\mathcal{K}}\mu(z)$ is bounded on
the line $\\{\Re z=d\\}$.

Note that ${\mathcal{K}}\mu$ is bounded on some line $\\{\Re z=d\\},\ d\neq
0$, iff ${\mathcal{K}}\mu$ is bounded on every such line.

First we establish the following

######  Lemma 5.1.

Let $\nu$ be a positive measure on ${\mathbb{R}}$,

$\frac{1}{\pi}\int\frac{d\nu(x)}{1+x^{2}}\leq C,$ (5.4)

and let $I_{1},I_{2},I_{3}$ be three adjacent intervals, enumerated left to
right, $|I_{j}|=L>0$. Let $I=I_{1}\cup I_{2}\cup I_{3}$ be an open interval.
Suppose that

1) $|{\mathcal{K}}\nu|<C$ on ${\mathbb{R}}+i$

2) $\nu(I_{1}),\nu(I_{3})>\varepsilon$.

Let $\tilde{\nu}=\tilde{\nu}_{b},\ b\in{\mathbb{R}}$ be a dual measure. Then
$\tilde{\nu}(I)>d$, where $d>0$ depends only on $C,L,\varepsilon$ and $b$, but
not on the position of the intervals on ${\mathbb{R}}$.

###### Proof.

First, let $I$ be an interval centered at zero (i.e., we do not aim to prove
that $d$ does not depend on the position of $I$ at this stage). Denote by
$d_{0}$ the infimum of $\tilde{\nu}(I)$ taken over all $\nu$ satisfying the
conditions. Suppose that $d$ is zero and that $\nu^{n}$ is a sequence of
measures such that $\tilde{\nu}^{n}_{b}(I)\to 0$. Choosing a subsequence of
$\nu^{n}$ if necessary, we assume that $\nu^{n}$ converges $*$-weakly in the
space of $\Pi$-finite measures on $\hat{\mathbb{R}}$ to a $\Pi$-finite measure
$\eta$. Note that $\eta$ must still satisfy 2), and therefore is non-trivial.
The relation between Cauchy integrals for dual measures implies
$\tilde{\nu}^{n}_{b}\to\tilde{\eta}_{b}$. Since $I$ is open,
$\tilde{\eta}(I)=\lim\tilde{\nu}^{n}(I)=0$. Since $\tilde{\eta}$ is a positive
measure vanishing on $I$, ${\mathcal{K}}\tilde{\eta}$ increases monotonically
on $I$. It follows that $\eta$ may have only one point-mass on $I$, which
contradicts 2). Therefore $d>0$.

Now let $I$ be an arbitrary interval and let us show that because of 1), $d$
does not depend on its position on ${\mathbb{R}}$.

Suppose that there exists a sequence $I^{n}$ of open intervals and a sequence
$\mu^{n}$ of measures such that each of them satisfies the conditions of the
lemma in place of $\mu$ and $I$, but $\tilde{\mu}^{n}_{b}(I^{n})\to 0$.
Suppose that $I^{n}$ is centered at $x^{n}$. Consider the sequence of measures
$\nu^{n}=\mu^{n}(x-x^{n})$. Since

${\mathcal{K}}\nu^{n}(z)={\mathcal{K}}\mu^{n}(z-x^{n})+b^{n},$

where

$b^{n}=\int\frac{t}{1+t^{2}}d\nu(t)-\int\frac{t-x^{n}}{1+(t-x^{n})^{2}}d\nu(t)=\Im{\mathcal{K}}\nu(i)-\Im{\mathcal{K}}\nu(i+x^{n}),$

we obtain $\tilde{\nu}^{n}_{b-b^{n}}=\tilde{\nu}_{b}(z-x^{n})$. Note that
because of 1), $|b^{n}|<2C$.

Choosing a subsequence if necessary, we can assume that $b^{n}\to b_{0}$. It
follows from (5.2) and (5.1) that
${\mathcal{P}}\nu^{n}_{b-b_{0}}-{\mathcal{P}}\nu^{n}_{b-b^{n}}\to 0$ pointwise
in ${\mathbb{C}}_{+}$. Hence $\nu^{n}_{b-b_{0}}-\nu^{n}_{b-b_{n}}\to 0$,
$*$-weakly and $\nu^{n}_{b-b_{0}}(I)-\nu^{n}_{b-b_{n}}(I)\to 0$. By our choice
of the measures $\nu^{n}=\mu^{n}(x-x^{n})$, $\tilde{\nu}^{n}_{b-b^{n}}(I)\to
0,$ which implies $\nu^{n}_{b-b_{0}}(I)\to 0$. Since $\nu^{n}$ satisfy the
conditions of the lemma on the interval $I$ centered at 0, this contradicts
the first part of the proof. ∎

###### Proof of Theorem 5.3.

Let $\mu\in\text{\rm PW}$ satisfy the conditions of the theorem. Then
$\tilde{\mu}((x,x+1))$ is uniformly bounded. Indeed, if it is not, then
$P\tilde{\mu}$ is unbounded on ${\mathbb{R}}+id$. Let $x_{n}\in{\mathbb{R}}$
be a sequence of points such that $P\tilde{\mu}(x_{n}+id)\to\infty$. It
follows that $\phi(x_{n}+id)\to 1$ and $P\mu(x_{n}+id)\to 0$. Then one can
choose intervals $J_{n}$ centered at $x_{n}$ such that $|J_{n}|\to\infty$ and
$\mu(J_{n})\to 0$. Existence of such intervals contradicts $\mu\in\text{\rm
PW}$ because it implies

$||f(x-x_{n})||_{L^{2}(\mu)}\to 0$

for any $f\in\text{\rm PW}_{a}$.

Let $I_{1},I_{2},I_{3}$ be consequent $\delta$-massive intervals for $\mu$,
from the statement of Theorem 3.1. Then by the last lemma $I=I_{1}\cup
I_{2}\cup I_{3}$ is a $\delta^{\prime}$-massive interval for $\tilde{\mu}$ for
some $\delta^{\prime}>0$. Since $\mu$ is sampling for every $\text{\rm
PW}_{3a}$, $\tilde{\mu}$ is sampling for every $\text{\rm PW}_{a}$.

In the opposite direction, suppose that $K\mu$ is unbounded on some
${\mathbb{R}}+i\varepsilon$. Similar to above, $P\tilde{\mu}$ is then not
bounded from below by a positive constant, which implies the existence of
intervals $J_{n}$ such that $|J_{n}|\to\infty$ but $\tilde{\mu}(J_{n})\to 0$.
Hence, $\tilde{\mu}\not\in\text{\rm PW}$. ∎

##  6\. Periodic spectral measures

###  6.1. Measures from $M^{+}({\mathbb{T}})$

Let $M^{+}({\mathbb{T}})$ be the set of all finite positive measures on the
unit circle ${\mathbb{T}}\sim[0,2\pi]$ with infinite support. We identify
measures on the circle with $2\pi$-periodic measures on ${\mathbb{R}}$,
$M^{+}({\mathbb{T}})\subset M^{+}({\mathbb{R}})$, and define $M^{+}_{\rm
even}({\mathbb{T}})\subset M^{+}({\mathbb{R}})$ as the subset of even periodic
measures.

Notice that all measures in $M^{+}({\mathbb{T}})$ are $\Pi$-finite.
Furthermore, as was mentioned before, it follows from Theorem 3.1 that
$M^{+}({\mathbb{T}})\subset\text{\rm PW}$. Let us list two more important
properties of $M^{+}({\mathbb{T}})$ pertaining to our problems.

######  Proposition 6.1.

If $\mu\in M^{+}({\mathbb{T}})$, then all dual measures are in
$M^{+}({\mathbb{T}})$

###### Proof.

Periodicity of $\tilde{\mu}_{b}$ follows from (5.2) and (5.1). A well-known
property of dual measures is that supports of $\mu$ and $\tilde{\mu}$
alternate, i.e., between any two points of one support there is a point of
another. It follows that, since the support $\mu$ is infinite, so is the
support of $\tilde{\mu}$. ∎

######  Proposition 6.2.

If $\mu\in M^{+}_{\rm even}({\mathbb{T}})$, then there is a unique
$\tilde{\mu}\in M^{+}_{\rm even}({\mathbb{T}})$ such that $\mu$ and
$\tilde{\mu}$ are dual: $\tilde{\mu}=\tilde{\mu}_{b},\
b=-\Re{\mathcal{K}}\mu(0)$.

###### Proof.

From (5.1) we see that in order for $\tilde{\mu}_{b}$ to be even,
$\Re\frac{1+\phi_{b}}{1-\phi_{b}}$ must be even. From (5.3) we obtain that

${\mathcal{P}}\tilde{\mu}_{b}=\Re\frac{i}{{\mathcal{K}}\mu+b}=\frac{{\mathcal{P}}\mu}{|{\mathcal{K}}\mu+b|^{2}}.$

In the last expression, ${\mathcal{P}}\mu$ is even because $\mu$ is even. In
order for the whole fraction to be even, $|{\mathcal{K}}\mu+b|$ must be even.
If $b=-\Im{\mathcal{K}}\mu(0)$, then

$\Im({\mathcal{K}}\mu+b)=\Im{\mathcal{K}}\mu={\mathcal{P}}\mu$

is even, $\Re({\mathcal{K}}\mu+b)$ is odd and $|{\mathcal{K}}\mu+b|$ is even.
For any other value of $b$, $\Re({\mathcal{K}}\mu(0)+b)\neq 0$,
$\Re({\mathcal{K}}\mu+b)$ is not odd and $[\Re({\mathcal{K}}\mu+b)]^{2}$ is
not even. Hence, $|{\mathcal{K}}\mu+b|$ is not even. ∎

Note that for the unique even dual measure $\tilde{\mu}$ we have

${\mathcal{K}}\tilde{\mu}=-\frac{1}{{\mathcal{K}}\mu},\
{\mathcal{P}}\tilde{\mu}=-\Re\frac{1}{{\mathcal{K}}\mu}.$ (6.1)

###  6.2. Moments

If $\mu\in M^{+}({\mathbb{T}})$, we can consider the Fourier series

$\mu\sim\sum_{-\infty}^{\infty}\gamma_{k}e^{ikx},$

where

$\gamma_{k}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ikx}~{}d\mu(x)=\alpha_{k}+i\beta_{k}$

are the ”trigonometric moments” of $\mu$. Clearly,

$\gamma_{-k}=\overline{\gamma_{k}},$

so

$\mu\sim\gamma_{0}+2\sum_{1}^{\infty}(\alpha_{k}\cos kx-\beta_{k}\sin kx),$

and $\mu$ is even iff all moments are real, i.e., all $\beta_{k}=0$.

The moments can be arranged into the Toeplitz matrix

$\Gamma(\mu)=\begin{pmatrix}\gamma_{0}&\gamma_{1}&\gamma_{2}&\dots\\\
\gamma_{-1}&\gamma_{0}&\gamma_{1}&\dots\\\
\gamma_{-2}&\gamma_{-1}&\gamma_{0}&\dots\\\
\dots&\dots&\dots&\dots\end{pmatrix}$

We will denote by $\Gamma_{n}(\mu)$ the $n\times n$ matrix in the upper left
corner of $\Gamma$. It is well known that a sequence $\\{\gamma_{k}\\}$ is the
sequence of moments of some $\mu\in M^{+}({\mathbb{T}})$ iff

$\forall n,\qquad\det\Gamma_{n}>0.$

Finally let us recall the relation between the moments of a periodic measure
and its Fourier transform: if $\mu\sim\sum\gamma_{k}e^{ikx}$, then

$\hat{\mu}=\sqrt{2\pi}\sum_{-\infty}^{\infty}\gamma_{k}\delta_{k}.$

###  6.3. Computation of $h^{\mu}(t)$

Recall that if $\mu\in\text{\rm PW}$, then

$h_{\mu}(t):=\pi\frac{d}{dt}\left[L_{\mu,t}^{-1}\stackrel{{\scriptstyle\circ}}{{k_{t}}}(0)\right]$

If $A=(a_{jk})$ is a matrix we use the notation $\SS[A]$ for the sum of the
elements of $A$:

$\SS[A]=\sum_{j,k}a_{jk}.$

######  Theorem 6.1.

If $\mu\in M^{+}({\mathbb{T}})$, then the function $h^{\mu}(t)$ is locally
constant on ${\mathbb{R}}_{+}\setminus\frac{1}{2}\mathbb{N}$, i.e.,

$h_{\mu}(t)=h_{0},\;h_{1},\;h_{2},\;\dots\quad{\rm
on}\quad\left(0,\frac{1}{2}\right),\;\left(\frac{1}{2},1\right),\;\left(1,\frac{3}{2}\right),\;\dots,$

and

$h_{n}=\SS\left[\Gamma_{n+1}(\mu)^{-1}\right]-\SS\left[\Gamma_{n}(\mu)^{-1}\right],\quad
h_{0}=\SS[\Gamma_{1}^{-1}]=\gamma_{0}^{-1}.$

###### Proof.

Since

$\hat{\mu}=\sqrt{2\pi}\sum\gamma_{k}\delta_{k},$

the Fourier transform $\phi_{t}$ of $k_{t}$ satisfies the relations in Theorem
4.7. Let

$f=f_{t}~{}=~{}\sqrt{2\pi}~{}\psi_{t}.$

Then $f$ satisfies

$\sum\gamma_{k}f(\xi-k)=1\quad{\rm on}\quad(-t,t)\quad{\rm and}\quad
f=0\quad{\rm on}\quad{\mathbb{R}}\setminus[-t,t].$ (6.2)

Recall

$h^{\mu}(t)=\pi\frac{d}{dt}\frac{1}{\sqrt{2\pi}}\int_{{\mathbb{R}}}\psi_{t}=\frac{1}{2}\frac{d}{dt}\int_{{\mathbb{R}}}f_{t}.$

Let us now find $f_{t}$ from (6.2).

Case $0<t<\frac{1}{2}$. If $\xi\not\in(-t,t)$, then $f(\xi)=0$. If
$\xi\in(-t,t)$, then $\xi\pm 1,\xi\pm 2,\dots$ are not in $(-t,t)$, so

$\sum\gamma_{k}f(\xi-k)=\gamma_{0}f(\xi)\quad{\rm must~{}be}\;=1.$

Thus

$f_{t}=\gamma_{0}^{-1}1_{(-t,t)},\qquad\int f_{t}=2t\gamma_{0}^{-1},\qquad
h^{\mu}(t)=\gamma_{0}^{-1}.$

Case $\frac{1}{2}<t<1$. Consider the partition of the interval $(-t,t)$ by the
points from $\pm t+{\mathbb{Z}}$:

$-t<t-1<-t+1<t.$

We claim that $f=f_{t}$ is constant on each interval of the partition. For
instance, let $\xi\in(-t,t-1)$. Then $f(\xi+k)=0$ unless $k=0$ or $k=1$, so
equating the value of $f\ast\hat{\mu}$ to 1 on the first and third intervals
of the partition we get

$\gamma_{0}f(\xi)+\gamma_{-1}f(\xi+1)=1$

and

$\gamma_{0}f(\xi+1)+\gamma_{1}f(\xi)=1.$

The linear system for $f(\xi)$ and $f(\xi+1)$ have the matrix $\Gamma_{2}$,
which implies

$\begin{pmatrix}f(\xi)\\\
f(\xi+1)\end{pmatrix}=\Gamma_{2}^{-1}\begin{pmatrix}1\\\ 1\end{pmatrix}.$

In particular it follows that the values $f(\xi)$ and $f(\xi+1$ are constant
and

$f(\xi)+f(\xi+1)=\SS[\Gamma_{2}^{-1}].$

(Here we use the formula $\SS[M]=M\vec{1}\cdot\vec{1}$, where
$\vec{1}=(1,1,...,1)^{\scriptscriptstyle T}$.)

On the other hand, if $\xi$ is in the middle interval $(t-1,-t+1)$, then

$f(\xi)=\gamma_{0}^{-1}=\SS[\Gamma_{1}^{-1}]$

by the same argument as in the previous case. Note that the middle interval is
shrinking with velocity 2, while the two other intervals are expanding with
velocity 2. Integrating $f$ from $-t$ to $t$ and then differentiating the
result with respect to $t$ we get

$h^{\mu}(t)=\SS[\Gamma_{2}^{-1}]-\SS[\Gamma_{1}^{-1}].$

Let us now consider the general case $\frac{n}{2}<t<\frac{n+1}{2}$.

There is a simple way to visualize the partition of the interval $(-t,t)$, see
the picture below. In the $(\xi,t)$-plane consider the interval (blue) on the
line $t=\text{\rm const}$ lying in the sector $t>|\xi|$. The points of the
partition are given by intersections of the interval with the lines
$t=\pm\xi+n,\ n\in{\mathbb{Z}}$, which form a square lattice in the sector.

$\xi$$t$

If we enumerate the $2n+1$ intervals of the partition from left to right, then
odd numbered intervals

$I_{1}=(-t,t-n),\ I_{3}=(-t+1,t-(n-1)),...,\ I_{2n+1}=(-t+n,t)$

are expanding while the even numbered intervals

$I_{2}=(t-n,-t+1),\ I_{4}=(t-(n-1),-t+2),...,\ I_{2n}=(t-1,-t+n)$

are shrinking. If $\xi\in I_{1}$ then $\xi+k\in I_{2k+1},\ k=1,2,..,n$.
Equating the values of $f\ast\hat{\mu}$ on each odd interval to 1 we obtain
the system of $n+1$ linear equations

$\begin{pmatrix}f(\xi)\\\ f(\xi+1)\\\ \vdots\\\
f(\xi+n)\end{pmatrix}=\Gamma_{n+1}^{-1}\begin{pmatrix}1\\\ 1\\\ \vdots\\\
1\end{pmatrix},$

from which we deduce

$f(\xi)+...+f(\xi+n)=\SS[\Gamma_{n+1}^{-1}]$ (6.3)

for the values on the expanding intervals. Similarly for $\xi\in I_{2}$,

$\begin{pmatrix}f(\xi)\\\ f(\xi+1)\\\ \vdots\\\
f(\xi+(n-1))\end{pmatrix}=\Gamma_{n}^{-1}\begin{pmatrix}1\\\ 1\\\ \vdots\\\
1\end{pmatrix},$

which yields

$f(\xi)+...+f(\xi+(n-1))=\SS[\Gamma_{n}^{-1}]$ (6.4)

for the values on the shrinking intervals. Integrating $f$ from $-t$ to $t$
and differentiating the result with respect to $t$ we obtain

$h^{\mu}(t)=\SS[\Gamma_{n+1}^{-1}]-\SS[\Gamma_{n}^{-1}]$

as claimed. ∎

###  6.4. Computation of $g^{\mu}(t)$

Let $\mu\in M^{+}({\mathbb{T}})$ and let

$\Delta(\mu)=\begin{pmatrix}0&\gamma_{1}&\gamma_{2}&\dots\\\
-\gamma_{-1}&0&\gamma_{1}&\dots\\\ -\gamma_{-2}&-\gamma_{-1}&0&\dots\\\
\dots&\dots&\dots&\dots\end{pmatrix}$

where, as before, $\gamma_{k}$ are trigonometric moments of $\mu$. We denote
by $\Delta_{n}$ the $n\times n$ matrix in the upper left corner of $\Delta$.

######  Theorem 6.2.

If $\mu\in M^{+}({\mathbb{T}})$, then the function $g^{\mu}(t)$ is locally
constant on ${\mathbb{R}}_{+}\setminus\frac{1}{2}\mathbb{N}$, i.e.

$g^{\mu}(t)=g_{0},\;g_{1},\;g_{2},\;\dots\quad{\rm
on}\quad\left(0,\frac{1}{2}\right),\;\left(\frac{1}{2},1\right),\;\left(1,\frac{3}{2}\right),\;\dots$

and

$g_{n}=\SS\left[\Delta_{n+1}\Gamma_{n+1}^{-1}\right]-\SS\left[\Delta_{n}\Gamma_{n}^{-1}\right].$

###### Proof.

To use Theorem 4.6, we need to calculate $\tilde{l}_{t}(0)$. Once again we
will switch to the Fourier transform and use the identity
$\tilde{l}_{t}(0)=\int_{-t}^{t}{\mathcal{F}}(\tilde{l}_{t})$. Recall that
$\tilde{l}_{t}=H^{\mu}k_{t}$. For $\mu\in M^{+}({\mathbb{T}})$,

$H^{\mu}k_{t}=Kk_{t}\mu-k_{t}K\mu+ck_{t},$

see Remark 4.3. The last term can be disregarded, see Theorem 3.4 and Remark
3.3.

The first term $Kk_{t}\mu$ is equal to the Hilbert transform $Hf\mu$ on
${\mathbb{R}}$. Recall that ${\mathcal{F}}(k_{t}\mu)=1$ on $(-t,t)$ and that
${\mathcal{F}}(Hg)(s)={\rm sign}\ s\cdot\hat{g}$. Hence,

$\int_{-t}^{t}{\mathcal{F}}(Kk_{t}\mu)(s)ds=\int_{-t}^{t}{\rm sign}\
s\cdot{\mathcal{F}}(k_{t}\mu)ds=\int_{-t}^{t}{\rm sign}\ s\ ds=0.$

Therefore

$\tilde{l}_{t}(0)=\int_{-t}^{t}{\mathcal{F}}(k_{t}K\mu)(s)ds=\int_{-t}^{t}f_{t}\ast({\rm
sign}\ s\cdot\hat{\mu}(s))ds,$

where $f_{t}=\hat{k}_{t}$ like in the last proof.

Suppose that $t\in(\frac{n}{2},\frac{n+1}{2})$. Let $I_{k},\ k=1,2,...,2n+1$
be the intervals of the partition of $(-t,t)$ like in the last proof. As was
established there, $f_{t}$ is constant on the intervals $I_{k}$ and its values
on the odd and even intervals are given by (6.3) and (6.4) respectively. It
follows that the values of $q=f_{t}\ast({\rm sign}\ s\cdot\hat{\mu}(s))$ on
odd $I_{k}$ are given by

$\begin{pmatrix}q(\xi)\\\ q(\xi+1)\\\ \vdots\\\
q(\xi+n)\end{pmatrix}=\Delta_{n+1}\Gamma_{n+1}^{-1}\begin{pmatrix}1\\\ 1\\\
\vdots\\\ 1\end{pmatrix}$

for $\xi\in I_{1}$ and for even $I_{k}$ by

$\begin{pmatrix}q(\xi)\\\ q(\xi+1)\\\ \vdots\\\
q(\xi+(n-1))\end{pmatrix}=\Delta_{n}\Gamma_{n}^{-1}\begin{pmatrix}1\\\ 1\\\
\vdots\\\ 1\end{pmatrix},$

for $\xi\in I_{2}$. Finishing like in the last proof we obtain the formula in
the statement. ∎

###  6.5. Canonical systems and orthogonal polynomials on the unit circle

In this section we study the connection between the inverse spectral problems
for CS with periodic spectral measures and orthogonal polynomials on the unit
circle. We show that the formula for $h_{n}$ from Theorem 6.1 can be
interpreted as point evaluations of such polynomials.

For $\mu\in M_{+}({\mathbb{T}})$ we denote by $\text{\rm ONP}(\mu)$ the family
of polynomials of $z$ on the unit circle orthonormal in $L^{2}(\mu)$. By
$\phi_{n}\in\text{\rm ONP}(\mu)$ we denote the polynomial of degree n.

######  Theorem 6.3.

Let $\mu\in M^{+}({\mathbb{T}})$ and let $\\{\phi_{n}\\}$ be ONP($\mu$). We
have

$h^{\mu}_{n}=|\phi_{n}(1)|^{2}.$

In the proof we will use the following notation. We denote

${\mathcal{P}}_{n}(\mu)={\rm span}\\{1,\dots,z^{n}\\}\subset L^{2}(\mu).$

Then

$K^{w}_{n}\equiv K_{n}(z,w)=\sum_{j=0}^{n}\phi_{j}(z)\overline{\phi_{k}(w)}$

is the reprokernel of ${\mathcal{P}}_{n}(\mu)$.

In this section $m=m_{\mathbb{T}}$ stands for the normalized Lebesgue measure
on ${\mathbb{T}}$, $m({\mathbb{T}})=1$. For $\text{\rm ONP}(m)$ we have
$\phi_{n}(z)=z^{n}$ and

$\mathring{K}(z,w)=\sum_{j=0}^{n}z^{j}\bar{w}^{j}.$

If $w=1$, then we’ll write $k_{n}(z)$ for $K_{n}(z,1)$, in particular

$\mathring{k}_{n}(z)=\sum_{j=0}^{n}z^{j}.$ (6.5)

We consider truncated Toeplitz operators on
${\mathcal{P}}_{n}={\mathcal{P}}_{n}(m)$:

$T_{\mu}=T_{\mu}^{n}:~{}{\mathcal{P}}_{n}\to{\mathcal{P}}_{n}$

defined by

$(T_{\mu}p,q)=\int p\bar{q}~{}d\mu\quad\forall\ p,q\in{\mathcal{P}}_{n}.$

For absolutely continuous measures $\mu$ with $L^{2}(m)$-densities
$T_{\mu}:p\mapsto P_{n}(p\mu)$ where $P_{n}$ is the orthogonal projection onto
${\mathcal{P}}_{n}$ in $L^{2}(m)$. Note that the matrix of $T_{n}$ in the
basis $\\{z^{j}\\}$ is

$T_{jk}=(Tz^{k},z^{j})=\int z^{k}\bar{z}^{j}~{}d\mu=\gamma_{j-k},$

which is the matrix $\Gamma^{\scriptscriptstyle T}_{n+1}(\mu)$, the transpose
of $\Gamma_{n+1}(\mu)$ from Section 6.2.

######  Lemma 6.1.

If $j_{n}:{\mathcal{P}}_{n}\to{\mathcal{P}}_{n}(\mu)$ denote the ’identity’
embedding operators, then

$j_{n}^{*}j_{n}=T_{\mu}\ {\rm on}\ {\mathcal{P}}_{n}.$

###### Proof.

$(j_{n}^{*}j_{n}p,q)=(j_{n}p,j_{n}q)_{\mu}=\int p\bar{q}~{}d\mu=(T_{\mu}p,q).$

∎

######  Lemma 6.2.

$\forall w\in{\mathbb{C}},\quad
j_{n}\left[T^{-1}_{\mu}\mathring{K}_{w}\right]=K_{w}$

In particular,

$j_{n}\left[T^{-1}_{\mu}\mathring{k}_{n}\right]=k_{n}.$

###### Proof.

Recall that

$\forall p,q\in{\mathcal{P}}_{n},\quad(jp,jq)_{\mu}=(T_{\mu}p,q).$

Let $p=j^{-1}K_{w}$. Then we have

$(\mathring{K}_{w},q)=\bar{q}(w)=(K_{w},jq)_{\mu}=(Tj^{-1}K_{w},q).$

∎

###### Proof of Theorem 6.3.

We will first show that

$\Sigma\left(\Gamma^{-1}_{n+1}(\mu)\right)=k_{n}(1).$

Indeed, for an $n\times n$ matrix $A$ and the standard basis
$e_{0},\dots,e_{n}$ in ${\mathbb{R}}^{n}$,

$\Sigma(A)=\sum_{j}\sum_{k}(Ae_{j},e_{k})=(A{\bf 1},{\bf 1}),\ {\rm where}\
{\bf 1}=\sum_{j=0}^{n}e_{j}.$

Applying this to the matrix $\Gamma^{T}_{n+1}(\mu)$ of $T_{\mu}$ with respect
to the basis $1,...,z^{n}$ and taking into account that ${\bf
1}=\stackrel{{\scriptstyle\circ}}{{k}}_{n}$, see (6.5), we get

$\Sigma\left(\Gamma^{-1}_{n+1}(\mu)\right)=\Sigma\left(\left[\Gamma^{t}_{n+1}(\mu)\right]^{-1}\right)=\left(T_{\mu}^{-1}\stackrel{{\scriptstyle\circ}}{{k}}_{n},\stackrel{{\scriptstyle\circ}}{{k}}_{n}\right)=\left(k_{n},\stackrel{{\scriptstyle\circ}}{{k}}_{n}\right)=k_{n}(1).$

To finish the proof of the theorem we observe that

$k_{n}(1)=K_{n}(1,1)=\sum_{j=0}^{n}\left|\phi_{j}(1)\right|^{2},$

and

$h_{n}=\Sigma\left[\Gamma_{n+1}^{-1}\right]-\Sigma\left[\Gamma_{n}^{-1}\right]=k_{n}(1)-k_{n-1}(1)=|\phi_{n}(1)|^{2}.$

∎

Theorem 6.3 allows us to describe the change of $h_{n}$ corresponding to a
shift of a periodic spectral measure. For $\eta\in\mathbb{T}$ we define
$\mu^{(\eta)}(\zeta)=\mu(\eta\zeta)$.

######  Corollary 6.1.

$h_{n}^{(\eta)}=|\phi_{n}(\eta)|^{2}.$

###### Proof.

ONP of $\mu^{(\eta)}$ are the polynomials $\phi_{n}(\eta\zeta)$.∎

##  7\. Examples of spectral problems

The goal of this section is to illustrate our formulas with explicit examples
and calculations.

###  7.1. Constant Hamiltonians

Let us first consider spectral problems, inverse and direct, for systems with
constant Hamiltonians.

######  Lemma 7.1.

Let the constants $h_{1}>0,h_{2}>0,g\in{\mathbb{R}}$ satisfy
$h_{1}h_{2}=1+g^{2}$, and

${\mathcal{H}}=\begin{pmatrix}h_{1}&g\\\ g&h_{2}\end{pmatrix}\in{\rm
SL}(2,\mathbb{R}).$

Then

$M(t,z)=\begin{pmatrix}C-gS&-h_{2}S\\\ h_{1}S&C+gS\end{pmatrix},$

where $C:=\cos zt$ and $S:=\sin zt$

###### Proof.

Just verify $M(0)=I$ and $\Omega\dot{M}=z{\mathcal{H}}M$ using
$\det({\mathcal{H}})=1$. To guess the solution use undetermined coefficients.∎

######  Corollary 7.1.

$\mu=h_{1}^{-1}$.

###### Proof.

Notice that the solution can be rewritten as

$\begin{pmatrix}u_{z}(t)\\\ v_{z}(t)\end{pmatrix}\begin{pmatrix}C-gS\\\
h_{1}S\end{pmatrix}=\begin{pmatrix}1&-g\\\
0&h_{1}\end{pmatrix}\begin{pmatrix}C\\\
S\end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{h_{1}}}&-\frac{g}{\sqrt{h_{1}}}\\\
0&\sqrt{h_{1}}\end{pmatrix}\begin{pmatrix}\sqrt{h_{1}}C\\\
\sqrt{h_{1}}S\end{pmatrix}$

Since the determinant of the last $2\times 2$ matrix is 1, by Theorem 2.2 from
[31] $F_{t}(z)=\sqrt{h_{1}}C-i\sqrt{h_{1}}S$ generates the same chain of dB
spaces as $E_{t}(z)=u_{z}(t)-iv_{z}(t)$. But

${\mathcal{B}}(F_{t})={\mathcal{B}}\left(\sqrt{h_{1}}e^{-izt}\right),$

and $h_{1}^{-1}=1/|F_{t}|^{2}$ is the spectral measure. ∎

Alternatively, to prove the last corollary we can apply the general algorithm
described in Section 6.3 to the even measure $\mu=a,\ a=h_{1}^{-1}$. We have
$\Gamma(\mu)=aI$, so $\Gamma_{n}^{-1}=\frac{1}{a}I_{n}$ and
$S\left(\Gamma_{n}^{-1}\right)=\frac{n}{a}$. Then
$h_{n}=\frac{n+1}{a}-\frac{n}{a}=\frac{1}{a}$.

To practice calculating $h_{2}$ using the dual measure, notice that for
$\mu=a$, $i{\mathcal{K}}\mu+ib=a+ib$ and
${\mathcal{P}}\tilde{\mu}_{b}=\tilde{a}:=\frac{a}{a^{2}+b^{2}}$. We have

$h=h^{\mu}=\frac{1}{a},\quad
g^{\mu}=0,\quad\tilde{h}=h^{\tilde{\mu}}=\frac{1}{\tilde{a}},\quad
g^{\tilde{\mu}}=0.$

Hamiltonians corresponding to the pair of measures $\mu$, $\tilde{\mu}_{b}$
are given by the constant matrix

$H=\begin{pmatrix}h&kh\\\ kh&\tilde{h}\end{pmatrix},\ k\in{\mathbb{R}}.$

###  7.2. Spectral measure $\mu=1+\cos x$

Let us solve the inverse spectral problem for this particular measure. Recall
that by Theorem 6.1, for a $2\pi$-periodic even measure $\mu=1+\cos x$ the
Hamiltonian is diagonal and constant on

$\left(0,\frac{1}{2}\right),\;\left(\frac{1}{2},1\right),\;\left(1,\frac{3}{2}\right),\;\dots$

To find the values of the Hamiltonian on these intervals using Theorem 6.1
notice that

$\Gamma(\mu)=\begin{pmatrix}1&\frac{1}{2}&0&\dots\\\
\frac{1}{2}&1&\frac{1}{2}&\dots\\\ 0&\frac{1}{2}&1&\dots\\\
\vdots&\vdots&\vdots&\ddots\end{pmatrix},$ $\Gamma_{1}^{-1}=(1),\ \
\Gamma_{2}^{-1}=\frac{4}{3}\begin{pmatrix}1&-\frac{1}{2}\\\
-\frac{1}{2}&1\end{pmatrix},~{}\dots$

and

$\SS_{1}=1,\ \SS_{2}=\frac{4}{3},~{}\dots$

for $\SS_{n}=\SS[\Gamma_{n}^{-1}]$.

We conclude that the values of $h_{0},h_{1},h_{2}...$ are

$1,\ \frac{1}{3},\ \frac{2}{3},\ \frac{2}{5},\ \frac{3}{5},\ \frac{3}{7},\
\frac{4}{7},\ \frac{4}{9},...$
$...,h_{2n}=\frac{n+1}{2n+1},h_{2n+1}=\frac{n+1}{2n+3},...$

Since the Hamiltonian ${\mathcal{H}}$ is diagonal and $\det{\mathcal{H}}=1$,
the the values for the second diagonal term can be calculated as $1/h_{n}$.

From (6.1) we can find the even dual measure $\tilde{\mu}$:

${\mathcal{P}}\mu=\Re(1+e^{iz}),\quad{\mathcal{P}}\tilde{\mu}=\Re\frac{1}{1+e^{iz}}=\frac{1}{2}+\frac{1}{2}\Re\frac{1-S}{1+S},$

so

$\tilde{\mu}=\frac{1}{2}+\pi\sum_{-\infty}^{\infty}\delta_{\pi+2\pi
k},\quad\Gamma(\tilde{\mu})=\begin{pmatrix}1&-\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&\dots\\\
-\frac{1}{2}&1&-\frac{1}{2}&\frac{1}{2}&\dots\\\
\frac{1}{2}&-\frac{1}{2}&1&-\frac{1}{2}&\dots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

Computation gives $\tilde{h}_{0}=1$, $\tilde{h}_{1}=3$,
$\tilde{h}_{2}=\frac{3}{2},\tilde{h}_{3}=\frac{5}{2}$, etc., as expected from
the above calculation for $h_{k}$.

Note that one can find $h_{n}$ for $\mu=1+\cos x$ and many other periodic
measures using tables of orthogonal polynomials on the unit circle available
in the literature, together with Theorem 6.3. To find $\tilde{h}_{n}$ one can
apply Theorem 6.3 to the dual measure, see Section 7.4. Further examples based
on that approach will be given below, see Sections 7.5 and 7.6.

###  7.3. Four steps by hand

Suppose $\mu$ is even (then the moments $\gamma_{k}$ are real). As before,
denote $\SS_{n}=\SS[\Gamma_{n}^{-1}]$.

######  Lemma 7.2.

$\SS_{1}=\frac{1}{\gamma_{0}},\quad\SS_{2}=\frac{2}{\gamma_{0}+\gamma_{1}},\quad\SS_{3}=\frac{3\gamma_{0}-4\gamma_{1}+\gamma_{2}}{(\gamma_{0}+\gamma_{2})\gamma_{0}-2\gamma_{1}^{2}},$
$\SS_{4}=\frac{2(2\gamma_{0}-\gamma_{1}-2\gamma_{2}+\gamma_{3})}{(\gamma_{0}+\gamma_{3})(\gamma_{0}+\gamma_{1})-(\gamma_{1}+\gamma_{2})^{2}}.$

###### Proof.

The first formula is obvious. The second follows from

$\Gamma_{2}^{-1}=\frac{1}{\gamma_{0}^{2}-\gamma_{1}^{2}}\begin{pmatrix}\gamma_{0}&-\gamma_{1}\\\
-\gamma_{1}&\gamma_{0}\end{pmatrix}$

Third formula: let $\vec{x}$ be defined by

$\begin{pmatrix}\gamma_{0}&\gamma_{1}&\gamma_{2}\\\
\gamma_{1}&\gamma_{0}&\gamma_{1}\\\
\gamma_{2}&\gamma_{1}&\gamma_{0}\end{pmatrix}\begin{pmatrix}x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix}=\begin{pmatrix}1\\\ 1\\\ 1\end{pmatrix}$

By symmetry we have $x_{1}=x_{3}$, and

$\SS_{3}=(\Gamma_{3}^{-1}\vec{1},\vec{1})=x_{1}+x_{2}+x_{3}=2x_{1}+x_{2}.$

The variables $2x_{1}$ and $x_{2}$ satisfy the system

$\frac{\gamma_{0}+\gamma_{2}}{2}(2x_{1})+\gamma_{1}x_{2}=1,\qquad\gamma_{1}(2x_{1})+\gamma_{0}x_{2}=1$

(the first two equations in the $3\times 3$ system above), and we have

$\SS_{3}=2x_{1}+x_{2}=\SS(M^{-1}),$

where $M$ is the 2$\times$2 matrix of the last system. The derivation of the
forth formula is as simple – we use the symmetry $x_{1}=x_{4}$ and
$x_{2}=x_{3}$, so again we only need to invert a 2$\times$2 matrix. ∎

######  Example 7.1.

If $\gamma_{0}=1,\;\gamma_{1}=\frac{1}{2},\;\gamma_{\geq 2}=0$, then we have

$\SS_{1}=1,\quad\SS_{2}=\frac{4}{3},\quad\SS_{3}=2,\quad\SS_{4}=\frac{12}{5},$

and if $\gamma_{0}=1,\;\gamma_{1}=-\frac{1}{2},\;\gamma_{2}=\frac{1}{2}$, then

$\SS_{1}=1,\quad\SS_{2}=4,\quad\SS_{3}=\frac{11}{2}.$

###  7.4. Finding dual measures

Recall that if $\phi$ is a Schur function, then we define Clark’s measures
$\sigma_{\alpha}^{\phi}$ on $\hat{\mathbb{R}}$ for $\alpha\in{\mathbb{T}}$ via
the equation

${\mathcal{P}}\sigma_{\alpha}=\Re\frac{\alpha+\phi}{\alpha-\phi},$

see Section 2.5.

Let us find all dual measures for $\mu=1+\cos x$ using the formulas from
Section 5.2. It is clear the the corresponding Schur function $\phi$ is a
periodic non-constant function and therefore $\phi-\alpha\not\in
L^{2}({\mathbb{R}})$ for any $\alpha$. Hence the dual measures have no masses
at $\infty$.

For $S(z):=e^{iz}$ we have

${\mathcal{P}}\mu=\Im{\mathcal{K}}\mu=\Re(1+S)\ {\rm and}\
{\mathcal{K}}\mu=i(1+S).$

Hence by (5.3),

${\mathcal{P}}\tilde{\mu}_{b}=\Re\frac{1}{1+S-ib}.$ (7.1)

Case $b=0$ (the only choice such that $\tilde{\mu}$ is even):

${\mathcal{P}}\tilde{\mu}=\frac{1}{2}+\frac{1}{2}\Re\frac{1-S}{1+S}={\mathcal{P}}\left[\frac{1}{2}+\frac{1}{2}\sigma^{S}_{-1}\right],$

and by (2.7),

$\tilde{\mu}=\frac{1}{2}+\pi\sum_{n\in{\mathbb{Z}}}\delta_{(2n+1)\pi}.$

Let us also find the moments of $\tilde{\mu}$:

$\gamma_{k}=\frac{1}{4\pi}\int_{0}^{2\pi}e^{-ikx}dx+\frac{1}{2}\sum_{n=-\infty}^{\infty}\int_{0}^{2\pi}e^{-ikx}\delta_{(2n+1)\pi}(x)=I+II.$

In $II$, only the term with $n=0$ has an atom in $(0,2\pi)$, so

$II=\frac{1}{2}e^{-ik\pi}=\frac{1}{2}(-1)^{k}.$

Of course, $I=\frac{1}{2}$ if $k=0$ and $I=0$ otherwise.

Case $b\neq 0$. From (7.1) one can conclude that
${\mathcal{P}}\tilde{\mu}_{b}$ is bounded near ${\mathbb{R}}$ and therefore
$\tilde{\mu}$ is absolutely continuous. Hence, on ${\mathbb{R}}$,

$\tilde{\mu}=\Re\frac{1}{1+S-ib}=\Re\frac{1}{1+\cos x+i(\sin
x-b)}=\frac{1+\cos x}{2+2\cos x-2b\sin x+b^{2}}.$

(Note that formally setting $b=0$ in the last formula we get the wrong answer
because $\tilde{\mu}_{0}$ is not a.c.) To find the moments, we may use
geometric progression,

$\tilde{\mu}=\Re\left[\frac{1}{1-ib}~{}\frac{1}{1+\frac{e^{ix}}{1-ib}}\right]=\Re\left[\frac{1}{1-ib}-\frac{e^{ix}}{(1-ib)^{2}}+\frac{e^{2ix}}{(1-ib)^{3}}-\dots\right],$

and we get

$\gamma_{0}=\frac{1}{1+b^{2}},\quad\gamma_{1}=-\frac{1}{2}\frac{1}{(1-ib)^{2}},\quad\gamma_{2}=\frac{1}{2}\frac{1}{(1-ib)^{3}},\quad\dots$

(and $\gamma_{-k}=\overline{\gamma_{k}}$). This follows from the
representations like

$\Re\frac{e^{ix}}{(1-ib)^{2}}=\frac{1}{2}\frac{e^{ix}}{(1-ib)^{2}}+\frac{1}{2}\frac{e^{-ix}}{(1+ib)^{2}}.$

Note that the formal limit $b=0$ gives the right moments.

Using the moments and Theorem 6.1 one can recover the Hamiltonians (in the
case $b=0$ it was done in Section 7.2).

Generalizing the example from Section 7.2, let us consider $\mu=1+a\cos$ where
$|a|<1$. Then ${\mathcal{P}}\mu=\Re(1+aS)$, and

${\mathcal{P}}\tilde{\mu}=\Re\frac{1}{1+aS}$

for a symmetric case ($b=0$). Thus

$\tilde{\mu}=\frac{1+a\cos x}{1+2a\cos x+a^{2}},$

and

$\Gamma(\tilde{\mu})=\left\\{1,~{}-\frac{a}{2},~{}\frac{a^{2}}{2},~{}\frac{a^{3}}{2},~{}\dots\right\\}.$

(Here we use the notation $M=\\{a_{1},a_{2},...\\}$ for a symmetric Toeplitz
matrix $M$ with the first row $a_{1},a_{2},...$.)

This follows from the representation $\tilde{\mu}=\Re[1-aS+a^{2}S^{2}-\dots]$.
Note that we get the right moments in the limit $a=1$. Of course,

$\Gamma(\mu)=\left\\{1,~{}\frac{a}{2},~{}0,~{}0,~{}\dots\right\\}.$

###  7.5. The Poisson measure

Fix $a\in{\mathbb{D}}$ and consider the $\mu\in M^{+}({\mathbb{T}}),\
\mu=P_{a}$, where $P_{a}$ is the Poisson kernel on ${\mathbb{T}}$,

$P_{a}(\xi)=\frac{1-|a|^{2}}{|\xi-a|^{2}}.$

Recall that a polynomial is called monic if its main coefficient is equal to
1.

######  Lemma 7.3 (Szego, Bernstein).

The monic orthogonal polynomials on ${\mathbb{T}}$ with respect to $P_{a}$ are

$\Phi_{0}(z)=1,\qquad\Phi_{n}(z)=z^{n}-az^{n-1},\quad(n\geq 1),$

are orthogonal with respect to $\mu$. Their norms are

$\|\Phi_{0}\|_{\mu}=1,\qquad\|\Phi_{n}\|^{2}_{\mu}=1-|a|^{2}.$

###### Proof.

If $n>k\geq 1$, then

$\displaystyle(\Phi_{n},\Phi_{k})_{\mu}$
$\displaystyle=\int_{{\mathbb{T}}}(z^{n}-az^{n-1})(z^{-k}-\bar{a}z^{1-k})P_{a}(z)$
$\displaystyle=a^{n-k}+|a|^{2}a^{n-k}-aa^{n-k-1}-\bar{a}a^{n-k+1}=0.$

Also,

$(\Phi_{n},\Phi_{0})_{\mu}=\int_{\mathbb{T}}(z^{n}-az^{n-1})P_{a}=a^{n}-aa^{n-1}=0.$

Finally,

$\|\Phi_{n}\|^{2}=\int_{\mathbb{T}}(1+|a|^{2}-a\bar{z}-\bar{a}z)P_{a}=1+|a|^{2}-|a|^{2}-|a|^{2}.$

∎

For ONP we have

$\phi_{0}(z)=1,\qquad\phi_{n}(z)=\frac{z^{n-1}(z-a)}{\sqrt{1-|a|^{2}}}\quad(n\geq
1).$

######  Corollary 7.2.

$h_{0}=1,\qquad h_{n}=\frac{|1-a|^{2}}{1-|a|^{2}}\quad(n\geq 1).$

Remarks. (a) The periodic measure
$\mu^{\mathbb{R}}(x)=\mu^{\mathbb{T}}(e^{ix})$ is even if
$\mu^{\mathbb{T}}(z)=\mu^{\mathbb{T}}(\bar{z})$. This is the case when $a$ is
real; then

$h_{n}=\frac{1-a}{1+a}\qquad(n\geq 1).$

(b) Shift of the spectral measure,
$\mu(\xi)\mapsto\mu_{\eta}(\xi)=\mu(\eta\xi),\ \eta\in{\mathbb{T}}$, results
in the Hamiltonian with

$h_{n}^{(\eta)}=\frac{|\eta-a|^{2}}{1-|a|^{2}}\qquad(n\geq 1).$

If $a$ is real, then

$h^{(-1)}_{n}=\frac{1+a}{1-a}=h_{n}^{-1}$

which suggests that the measures $P_{a}$ and $P_{-a}$ are dual. This relation
can also be established directly for all $a\in{\mathbb{T}}$:

######  Lemma 7.4.

For any $a\in{\mathbb{D}}$, the measures $P_{a}^{\mathbb{R}}$ and
$P_{-a}^{\mathbb{R}}$ are dual.

###### Proof.

The harmonic function

$\Re\frac{1+\bar{a}S}{1-\bar{a}S}=\frac{1-|a|^{2}|S|^{2}}{|1-\bar{a}S|^{2}}$

is positive in the upper halfplane and its boundary values are
$P_{a}^{\mathbb{R}}(x)$. One of the dual measures can then be found from

$\Re\frac{1-\bar{a}S}{1+\bar{a}S}=P_{-a}\qquad({\rm on}\;{\mathbb{R}}).$

∎

######  Corollary 7.3.

We have

$h_{n}\tilde{h}_{n}=\frac{|1-a^{2}|^{2}}{(1-|a|^{2})^{2}}\qquad(n\geq 1)$

and therefore

$g_{n}^{2}=h_{n}\tilde{h}_{n}-1=\frac{4(\Im a)^{2}}{(1-|a|^{2})^{2}}.$

To illustrate our formulas further let us use the above calculations and
obtain the measure $P_{a}$ in the case of real $a$ solving the direct spectral
problem. In this case the Hamiltonian is diagonal and, as was shown above,
equal to $I$ on $\left(0,\frac{1}{2}\right)$ and to

$\begin{pmatrix}h&0\\\ 0&h^{-1}\end{pmatrix}$

on the rest of ${\mathbb{R}}_{+}$, where

$h=\frac{1-a}{1+a}.$

Using the formulas for constant Hamiltonians from Section 7.1, we obtain that
the matrizant is equal to

$\begin{pmatrix}\cos(z/2)&-\sin(z/2)\\\ \sin(z/2)&\cos(z/2)\end{pmatrix}$

on $\left(0,\frac{1}{2}\right)$ and to

$\begin{pmatrix}\cos zt&-h^{-1}\sin zt\\\ h\sin zt&\cos
zt\end{pmatrix}\begin{pmatrix}\cos(z/2)&-\sin(z/2)\\\
\sin(z/2)&\cos(z/2)\end{pmatrix}$

at ”time” $\frac{1}{2}+t$.

The corresponding (normalized) HB function is

$E=A-iC$

where

$A=\cos zt\cos(z/2)-h^{-1}\sin zt\sin(z/2),\quad C=h\sin zt\cos(z/2)-\cos
zt\sin(z/2).$

The function $E$ has the same dB space as the function

$E^{*}=\sqrt{h}A-\frac{i}{\sqrt{h}}C.$

A small miracle happens in the following computation:

$|E^{*}|^{2}=h|A|^{2}+\frac{1}{h}|C|^{2}=h\cos^{2}\frac{z}{2}+\frac{1}{h}\sin^{2}\frac{z}{2},$

with the final expression showing no dependence on $t$. It follows that the
spectral measure is

$\mu(x)=\frac{1}{|E^{*}(x)|^{2}}=\frac{1}{h\cos^{2}\frac{x}{2}+\frac{1}{h}\sin^{2}\frac{x}{2}}.$

This is equal to

$P_{a}(x)=\frac{1-a^{2}}{(\cos x-a)^{2}+\sin^{2}x}.$

###  7.6. Delta measure plus a constant

Let now $\gamma\in(0,1)$ be a parameter, and consider $\mu\in
M^{+}({\mathbb{T}})$ defined as

$\mu^{\mathbb{T}}=(1-\gamma)+2\pi\gamma\delta_{1}.$

As before, we identify $\mu$ with a periodic measure on ${\mathbb{R}}$,

$\mu=(1-\gamma)+2\pi\gamma\sum_{n}\delta_{2\pi n}.$

######  Lemma 7.5 ([33]).

The monic orthogonal polynomials on ${\mathbb{T}}$ with respect to $\mu$ are

$\Phi_{0}(z)=1,\qquad\Phi_{n}(z)=z^{n}-\alpha_{n-1}(1+\dots+z^{n-1}),\quad(n\geq
1),$

where

$\alpha_{n}=\frac{\gamma}{1+n\gamma}.$

The norms are

$\|\Phi_{n}\|^{2}=1-n\alpha_{n-1}\gamma,\quad(n\geq 0).$

Since the measure is even, the corresponding Hamiltonian is diagonal with the
diagonal entries on $\left(n,n+\frac{1}{2}\right)$ given by Theorem 6.3:

######  Corollary 7.4.

$h_{0}=1,\qquad
h_{n}=\frac{|\Phi_{n}(1)|^{2}}{\|\Phi_{n}\|^{2}}=\frac{(1-n\alpha_{n-1})^{2}}{1-n\alpha_{n-1}\gamma},$
$\tilde{h}_{0}=1,\qquad\tilde{h}_{n}=\frac{1}{h_{n}}=\frac{1-n\alpha_{n-1}\gamma}{(1-n\alpha_{n-1})^{2}}.$

###  7.7. Spectral measure $\mu=\alpha+\beta\pi\delta_{0},\ \alpha>0,\beta\geq
0$

Let us now consider this example of a non-periodic even spectral measure.
Using the formulas

$\hat{\delta}=\frac{1}{\sqrt{2\pi}}\cdot
1,\qquad\hat{1}=\sqrt{2\pi}\cdot\delta,$

we obtain

$\hat{\mu}=\alpha\sqrt{2\pi}\cdot\delta+\frac{\beta\pi}{\sqrt{2\pi}}\cdot 1.$

Next, we use Theorem 4.7 to find the Fourier transform $\psi_{t}$ of the
reproducing kernel. Since $\psi\ast 1=\int\psi$, Theorem 4.7 gives

$\forall
x\in(-t,t),\qquad\alpha\sqrt{2\pi}\psi_{t}(x)+\frac{\beta\pi}{\sqrt{2\pi}}\int_{-t}^{t}\psi_{t}=1.$

Thus

$\psi_{t}=c(t)1_{(-t,t)},\qquad\int_{-t}^{t}\psi_{t}=2tc(t),$

and

$\alpha\sqrt{2\pi}c(t)+\frac{\beta\pi}{\sqrt{2\pi}}2tc(t)=1,$

so

$c(t)=\frac{1}{\alpha\sqrt{2\pi}+\frac{\beta\pi}{\sqrt{2\pi}}2t}=\frac{\sqrt{2\pi}}{2\pi\alpha+2\pi
t\beta}$

and

$k_{t}(0)=\frac{1}{\sqrt{2\pi}}\int\psi_{t}=\frac{2tc(t)}{\sqrt{2\pi}}=\frac{1}{\pi}\frac{t}{\alpha+t\beta}.$

It follows that

$h^{\mu}(t)=\pi\frac{d}{dt}k_{t}(0)=\frac{d}{dt}\frac{t}{\alpha+t\beta}=\frac{\alpha}{(\alpha+t\beta)^{2}}$

and that

$H(t)=\begin{pmatrix}\frac{\alpha}{(\alpha+t\beta)^{2}}&0\\\
0&\frac{(\alpha+t\beta)^{2}}{\alpha}\end{pmatrix}$

is a unique diagonal Hamiltonian with spectral measure $\mu$.

######  Remarks 7.1.

  *   * •

Note that in the limiting case $\alpha\to 0$ (and $\beta=1$):

$\mu(x)\to\pi\delta(x),\qquad h^{\mu}(t)\to\delta(t).$

Indeed,

$\int_{0}^{\infty}\frac{\alpha}{(\alpha+t)^{2}}~{}dt=-\frac{\alpha}{\alpha+t}\Big{|}_{t=0}^{\infty}=1$

This shows how using our methods developed for PW-measures one can solve the
inverse problem for a non-PW measure $\mu=\pi\delta$

  * •

The dual spectral measure of $H$ is

$\tilde{\mu}(x)=\frac{\alpha x^{2}}{\alpha^{2}x^{2}+\beta^{2}}.$

###### Proof.

${\mathcal{P}}\mu(z)=\alpha+\beta\frac{y}{x^{2}+y^{2}}=\Re\left[\alpha+\frac{i\beta}{z}\right]$

We have

$\frac{1}{\alpha+\frac{i\beta}{z}}=\frac{z}{\alpha z+i\beta};$

on ${\mathbb{R}}$ the real part is $\frac{\alpha
x^{2}}{\alpha^{2}x^{2}+\beta^{2}}$.∎

  * •

It is a good exercise to show directly that if $\tilde{\mu}=x^{2}/(1+x^{2})$,
then $h^{\tilde{\mu}}=(1+t)^{2}$. Note that

${\mathcal{F}}\left(\frac{1}{1+x^{2}}\right)(\xi)=\sqrt{\frac{\pi}{2}}e^{-|\xi|},$

and we need to solve equations like

$\psi+ce^{-|\xi|}\ast\psi=1$

(this convolution is called the Beurling transform of $\psi$ in [12]). Hint:

$\int\psi_{t}(y)e^{-|x-y|}dy=\int_{-t}^{x}+\int_{x}^{t}.$

###  7.8. Adding an eigenvalue at the origin

Here is a generalization of the last example studied with different methods by
H. Winkler in [35].

######  Theorem 7.1.

Let $\mu$ be a PW-sampling measure. For $r\geq 0$, let
$\mu_{r}=\mu+r\pi\delta$. Then

$h^{\mu_{r}}(t)=\frac{h^{\mu}(t)}{\left(1+r\int_{0}^{t}h^{\mu}\right)^{2}}$

######  Example 7.2.

$\mu=\alpha m$, $h^{\mu}=\alpha^{-1}$, and

$h^{\mu_{r}}(t)=\frac{\alpha^{-1}}{\left(1+\alpha^{-1}rt\right)^{2}}=\frac{\alpha}{(\alpha+rt)^{2}}$

###### Proof.

Fix $r>0$. Let $\psi_{t}$ and $f_{t}$ be solutions of the equations from
Theorem 4.7,

$\psi_{t}\ast\hat{\mu}_{r}=1,\qquad f_{t}\ast\hat{\mu}_{r}=1\qquad{\rm
on}\quad(-t,t),$

so that

$h^{\mu}=\sqrt{\frac{\pi}{2}}\frac{d}{dt}\int\psi_{t}$

and

$h^{\mu_{r}}=\sqrt{\frac{\pi}{2}}\frac{d}{dt}\int f_{t}.$ (7.2)

From $f\ast\hat{\mu}_{r}=1$ we find

$f_{t}\ast\hat{\mu}+r\sqrt{\frac{\pi}{2}}\int f_{t}=1,$

or

$f_{t}\ast\hat{\mu}=1-c(t),\qquad c(t)=r\sqrt{\frac{\pi}{2}}\int f_{t}.$

It follows that

$f_{t}=(1-c(t))\psi_{t},$

where $c(t)$ satisfies the equation

$c(t)=r\sqrt{\frac{\pi}{2}}(1-c(t))\int\psi_{t}.$

Hence, $c(t)=a/(1+a)$ where $a(t)=r\sqrt{\pi/2}\int\psi_{t}$ and

$\frac{d}{dt}a(t)=rh^{\mu}(t),\ a(0)=0.$

From (7.2) we have

$h^{\mu_{r}}(t)=\frac{d}{dt}\frac{c(t)}{r},$

which yields the statement.∎

######  Remark 7.1.

Note that $\mu_{x+y}$ can be obtained as $\mu_{x+y}=\mu_{x}+y\delta_{0}$. For
$h_{x}=h^{\mu_{x}}$ the last theorem yields the equation

$h_{x+y}(t)=\frac{h_{x}(t)}{\left(1+y\int_{0}^{t}h_{x}\right)^{2}}$

with the initial condition $h_{0}=h^{\mu}$.

$\triangle$

###  7.9. Spectral measure $\mu=\alpha+\beta\pi\delta_{\lambda}$

Here $\alpha>0$, $\beta>0$, and $\lambda\in{\mathbb{R}}$ are parameters.
Unlike Section 7.7, if $\lambda\neq 0$, then the measure is not even.

(i) We want to find

$h^{\mu}=\sqrt{\frac{\pi}{2}}\frac{d}{dt}\int\psi_{t},$

where

$\hat{\mu}\ast\psi_{t}=1\qquad{\rm on}\quad(-t,t).$

We will use the notation $e_{\lambda}(x)=e^{i\lambda x}$. We have

$\hat{\mu}=\alpha\sqrt{2\pi}\delta_{0}+\frac{\pi\beta}{\sqrt{2\pi}}e_{-\lambda}$

Note that

$(e_{-\lambda}\ast\psi)(x)=e_{-\lambda}(x)\int e_{\lambda}\psi,$

so

$\psi_{t}=\left[\frac{1}{\alpha\sqrt{2\pi}}-\frac{\beta}{2\alpha}c(t)e_{-\lambda}\right]\cdot
1_{(-t,t)},$

where

$c(t)=\int e_{\lambda}\psi_{t}.$

Also,

$\int_{-t}^{t}e_{\lambda}=\frac{2\sin\lambda t}{\lambda}.$

From the last 3 equations we get

$c(t)=\sqrt{\frac{2}{\pi}}\frac{1}{\alpha+\beta t}\frac{\sin\lambda
t}{\lambda},$

and

$\sqrt{\frac{\pi}{2}}\int\psi_{t}=\frac{t}{\alpha}-\frac{\beta}{\alpha}\frac{1}{\alpha+\beta
t}\left(\frac{\sin\lambda t}{\lambda}\right)^{2}.$

Hence,

$h^{\mu}(t)=\frac{d}{dt}\left[\frac{t}{\alpha}-\frac{\beta}{\alpha}\frac{1}{\alpha+\beta
t}\left(\frac{\sin\lambda t}{\lambda}\right)^{2}\right].$

E.g., if $\alpha=\beta=1$, then

$h^{\mu}(t)=\sin^{2}\lambda t+\left[\cos\lambda t-\frac{\sin\lambda
t}{\lambda(1+t)}\right]^{2}$

(ii) We now turn to the computation of $g^{\mu}$ for

$\mu=\alpha+\pi\beta\delta_{\lambda}$

To compute $g^{\mu}$ we need

$l_{t}(0)=\frac{1}{2\pi
i}\int_{-t}^{t}\psi_{t}*(\sigma\hat{\mu})=\frac{\beta}{2i\sqrt{2\pi}}\int_{-t}^{t}\psi_{t}*(\sigma
e_{-\lambda})=$ $=\frac{\beta}{4i\pi\alpha}\int_{-t}^{t}1_{[-t,t]}*(\sigma
e_{-\lambda})-\frac{\beta^{2}c(t)}{4\alpha
i\sqrt{2\pi}}\int_{-t}^{t}(1_{[-t,t]}e_{-\lambda})*(\sigma e_{-\lambda}).$

The following formula can be established via direct calculations.

######  Lemma 7.6.

$\int_{-t}^{t}(1_{[-t,t]}e_{\lambda})*(\sigma
e_{-\mu})=\frac{4}{i(\lambda-\mu)}\left[{\text{\rm
sinc}}_{t}(\lambda)-\cos(\lambda-\mu)t~{}{\text{\rm sinc}}_{t}(\mu)\right].$

Using the lemma to continue the calculation of $l_{t}(0)$,

$l_{t}(0)=-\frac{\beta}{\alpha\lambda\pi}\left[t-\cos\lambda t\ {\text{\rm
sinc}}_{t}(\lambda)\right]-\frac{\beta^{2}c(t)}{\alpha\lambda\sqrt{2\pi}}\left[t\cos\lambda
t-{\text{\rm sinc}}_{t}(\lambda)\right]=$
$-\frac{\beta}{\alpha\lambda\pi}\left[t-\cos\lambda t\ {\text{\rm
sinc}}_{t}(\lambda)\right]-\sqrt{\frac{2}{\pi}}\frac{{\text{\rm
sinc}}_{t}(\lambda)}{\alpha+\beta
t}\frac{\beta^{2}}{\alpha\lambda\sqrt{2\pi}}\left[t\cos\lambda t-{\text{\rm
sinc}}_{t}(\lambda)\right]=$
$-\frac{\beta}{\alpha\pi\lambda}\left[t-\frac{\cos\lambda t\sin\lambda
t}{\lambda}+\frac{\beta t}{\alpha+\beta t}\frac{\cos\lambda t\sin\lambda
t}{\lambda}-\frac{\beta}{\alpha+\beta t}\frac{\sin^{2}\lambda
t}{\lambda^{2}}\right]=$
$-\frac{\beta}{\alpha\pi\lambda}\left[t-\frac{\alpha}{\alpha+\beta
t}\frac{\cos\lambda t\sin\lambda t}{\lambda}-\frac{\beta}{\alpha+\beta
t}\frac{\sin^{2}\lambda t}{\lambda^{2}}\right]$

Now,

$g^{\mu}(t)=\frac{d}{dt}l_{t}(0)=$
$-\frac{\beta}{\alpha\pi\lambda}\left[1-\frac{\alpha}{\alpha+\beta t}\cos
2\lambda t+\frac{\alpha\beta}{(\alpha+\beta t)^{2}}\frac{\cos\lambda
t\sin\lambda t}{\lambda}-\frac{\beta}{\alpha+\beta t}\frac{\sin 2\lambda
t}{\lambda}+\frac{\beta^{2}}{(\alpha+\beta t)^{2}}\frac{\sin^{2}\lambda
t}{\lambda^{2}}\right]=$

$-\frac{\beta}{\alpha\pi\lambda}\left[1-\frac{\alpha}{\alpha+\beta t}\cos
2\lambda t-\frac{(\alpha\beta+2\beta^{2}t)}{(\alpha+\beta t)^{2}}\frac{\sin
2\lambda t}{2\lambda}+\frac{\beta^{2}}{(\alpha+\beta
t)^{2}}\frac{\sin^{2}\lambda t}{\lambda^{2}}\right].$

Let us simplify the above formula in the particular case
$\alpha=\beta=\lambda=1$:

$g^{\mu}(t)=-\frac{1}{\pi}\left[1-\frac{\cos
2t}{1+t}-\frac{(1+2t)}{(1+t)^{2}}\frac{\sin
2t}{2}+\frac{\sin^{2}t}{(1+t)^{2}}\right]=$ $\frac{(1+t)2\cos 2t+(1+2t)\sin
2t-(1-\cos 2t)}{2\pi(1+t)^{2}}-\frac{1}{\pi}=$ $\frac{2t(\cos 2t+\sin
2t)+3\cos 2t+\sin 2t-1}{2\pi(1+t)^{2}}-\frac{1}{\pi}=$ $\frac{(2t+1)(\cos
2t+\sin 2t)+2\cos 2t-1}{2\pi(1+t)^{2}}-\frac{1}{\pi}$

###  7.10. Adding multiple solitons

Let us solve the inverse problem for

$\mu=\alpha+\sum_{n}\pi\beta_{n}\delta_{\lambda_{n}}.$

For its Fourier transform we have

$\hat{\mu}=\sqrt{2\pi}\alpha\delta_{0}+\sum_{n}\frac{\pi\beta_{n}}{\sqrt{2\pi}}e_{-\lambda_{n}},$

where $e_{s}(t)=e^{ist}$.

As before, let $\psi=\psi_{t}$ be the Fourier transform of the reproducing
kernel $k_{t}(\cdot)=K_{t}(0,\cdot)\in{\mathcal{B}}_{t}$. Theorem 4.7 then
gives

$\hat{\mu}*\psi=1\textrm{ on }(-t,t)$

which can be rewritten as

$\sqrt{2\pi}\alpha\psi(x)+\sum_{n}c_{n}(t)\frac{\pi\beta_{n}}{\sqrt{2\pi}}e_{-\lambda_{n}}(x)=1$
(7.3)

where

$c_{n}(t)=\int_{\mathbb{R}}e_{\lambda_{n}}\psi=\sqrt{2\pi}\hat{\psi}(-\lambda_{n}).$

Solving (7.3) for $\psi$ we get

$\psi(x)=\frac{1}{\sqrt{2\pi}\alpha}-\sum_{n}\frac{\beta_{n}}{2\alpha}c_{n}(t)e_{-\lambda_{n}}(x).$
(7.4)

Let us recall the notation for the sinc function:

${\text{\rm sinc}}_{t}(x)=\frac{\sin(tx)}{x}.$

The reprokernel of $PW_{t}$ is $\mathring{K}_{t}(w,z)=\frac{1}{\pi}{\text{\rm
sinc}}_{t}(z-\bar{w})$.

Direct calculations give

######  Lemma 7.7.

$\int_{-t}^{t}e_{\lambda}=\frac{2\sin\lambda t}{\lambda}=2{\text{\rm
sinc}}_{t}(\lambda)$
$\int_{-t}^{t}e_{\lambda}e_{-\mu}=\frac{2\sin(\lambda-\mu)t}{\lambda-\mu}=2{\text{\rm
sinc}}_{t}(\lambda-\mu).$

Using (7.4),

$c_{n}(t)=\int_{-t}^{t}e_{\lambda_{n}}\psi_{t}=\int
e_{\lambda_{n}}\left[\frac{1}{\alpha\sqrt{2\pi}}-\sum_{k}\frac{\beta_{k}}{2\alpha}c_{k}(t)e_{-\lambda_{k}}\right]=$
$\frac{2}{\alpha\sqrt{2\pi}}{\text{\rm
sinc}}_{t}(\lambda_{n})-t\frac{\beta_{n}}{\alpha}c_{n}(t)-\sum_{k\neq
n}\frac{\beta_{k}}{\alpha}c_{k}(t){\text{\rm
sinc}}_{t}(\lambda_{n}-\lambda_{k}).$

The last equation can be rewritten as

$\left(\alpha+\beta_{n}t\right)c_{n}+\sum_{k\neq n}\ {\text{\rm
sinc}}_{t}(\lambda_{n}-\lambda_{k})\beta_{k}c_{k}=\frac{2}{\sqrt{2\pi}}\
{\text{\rm sinc}}_{t}(\lambda_{n}).$ (7.5)

Let us introduce the following two matrices,

$B=\left(\begin{array}[]{cccccc}\beta_{1}&0&\dots&0&0&0\\\
0&\beta_{2}&\dots&0&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&\dots&\beta_{n-2}&0&0\\\ 0&0&\dots&0&\beta_{n-1}&0\\\
0&0&\dots&0&0&\beta_{n}\end{array}\right),$

and

$S_{t}=\left({\text{\rm sinc}}_{t}(\lambda_{j}-\lambda_{k})\right)_{1\leq
j,k\leq n},$

and two vectors

$L_{t}=\sqrt{\frac{2}{\pi}}\left(\begin{array}[]{c}{\text{\rm
sinc}}_{t}(\lambda_{1})\\\ {\text{\rm sinc}}_{t}(\lambda_{2})\\\ \vdots\\\
{\text{\rm sinc}}_{t}(\lambda_{n})\end{array}\right),\ {\rm and}\
C(t)=\left(\begin{array}[]{c}c_{1}(t)\\\ c_{2}(t)\\\ \vdots\\\
c_{n}(t)\end{array}\right).$

Our calculations above lead to the following

######  Theorem 7.2.

$\left(\alpha+S_{t}B\right)^{-1}L_{t}=C(t).$

It follows that $h_{11}=h^{\mu}$ can now be calculated via the following
formula, which concludes the solution in the case when $\mu$ is even.

######  Corollary 7.5.

$h^{\mu}(t)=\frac{1}{\pi\alpha}-\frac{1}{\sqrt{2\pi}\alpha}\frac{d}{dt}<B(\alpha+S_{t}B)^{-1}L_{t},L_{t}>.$

In the general case, the off-diagonal term of ${\mathcal{H}}$, $g^{\mu}$, can
be calculated using the formulas of Section 4.7. These calculations will be
presented elsewhere.

###  7.11. Dual solitons

Let us, once more, consider a measure $\nu=\alpha+\beta\pi\delta_{0}$. The
dual measure $\mu=\tilde{\nu}$ is

$\mu=\frac{\alpha
x^{2}}{\alpha^{2}x^{2}+\beta^{2}}=\frac{1}{\alpha}\left(1-\frac{\beta^{2}}{\alpha^{2}x^{2}+\beta^{2}}\right)=$
$\frac{1}{\alpha}\left(1-\frac{1}{1+\frac{\alpha^{2}}{\beta^{2}}x^{2}}\right).$

Let us carry on with the case

$\alpha=\beta=1,\ \mu=1-\frac{1}{1+x^{2}}.$

Since

$\widehat{\frac{1}{1+x^{2}}}=\frac{\sqrt{2\pi}}{2}e^{-|y|}\text{ and
}\widehat{1}=\sqrt{2\pi}\delta_{0},$

the convolution equation from Theorem 4.7 becomes

$\psi-\frac{1}{2}(e^{-x}F(x)+e^{x}G(x))=\frac{1}{\sqrt{2\pi}},$ (7.6)

where

$F(x)=\int_{-t}^{x}\psi(y)e^{y}dy\text{ and
}G(x)=\int_{x}^{t}\psi(y)e^{-y}dy.$ (7.7)

Note that since $\psi$ is even, $G(x)=F(-x)$. Next notice that

$F^{\prime}(x)=\psi e^{x}\text{ and }G^{\prime}(x)=-\psi e^{-x}.$ (7.8)

In particular,

$F(-t)=0,\ \psi(0)=F^{\prime}(0)\text{ and from \eqref{e07}
}F^{\prime}(0)-F(0)=\frac{1}{\sqrt{2\pi}}.$ (7.9)