1 Introduction
Inner functions on
$\mathbb {C}_+$
are bounded analytic functions with unit modulus almost everywhere on the boundary
$\mathbb {R}$
. If an inner function extends to
$\mathbb {C}$
meromorphically, it is called a meromorphic inner function (MIF), which is usually denoted by
$\Theta $
.
MIFs appear in various fields such as the spectral theory of differential operators [Reference Baranov, Belov and Poltoratski1, Reference Makarov and Poltoratski8], Krein-de Branges theory of entire functions [Reference de Branges3, Reference Mitkovski and Poltoratski10, Reference Rupam22], functional model theory [Reference Nikolski, Hruščev, Khrushchëv, Peller and Peetre12, Reference Nikolski13], approximation theory [Reference Poltoratski16], Toeplitz operators [Reference Hartmann and Mitkovski4, Reference Makarov and Poltoratski9, Reference Poltoratski18], and nonlinear Fourier transform [Reference Poltoratski19]. MIFs also play a critical role in the Toeplitz approach to the uncertainty principle [Reference Makarov and Poltoratski8, Reference Poltoratski17], which was used in the study of problems of harmonic analysis [Reference Mitkovski and Poltoratski11, Reference Poltoratski14, Reference Poltoratski15].
Each MIF is represented by the Blaschke/singular factorization
$$ \begin{align*}\Theta(z) = Ce^{iaz}\prod_{n \in \mathbb{N}} c_n \frac{z-\omega_n}{z-\overline{\omega}_n}, \end{align*} $$
where C is a unimodular constant, a is a nonzero real constant,
$\{\omega _n\}_{n \in \mathbb {N}}$
is a discrete sequence (i.e., has no finite accumulation point) in
$\mathbb {C}_+$
satisfying the Blaschke condition
$\sum _{n \in \mathbb {N}}\mathrm {Im}\omega _n/(1+|\omega _n|^2) < \infty $
, and
$c_n = (i+\overline {\omega }_n)/(i+\omega _n)$
. MIFs are also represented as
$\Theta (x) = \exp (i\phi (x))$
on the real line, where
$\phi $
is an increasing real analytic function.
A complex-valued function is said to be real if it maps real numbers to real numbers on its domain. A meromorphic Herglotz function (MHF) m is a real meromorphic function with positive imaginary part on
$\mathbb {C}_+$
. It has negative imaginary part on
$\mathbb {C}_-$
through the relation
$m(\overline {z}) = \overline {m}(z)$
. There is a one-to-one correspondence between MIFs and MHFs given by the equations
$$ \begin{align*}\Theta = \frac{m_{\Theta}-i}{m_{\Theta}+i} \qquad \text{and} \qquad m_{\Theta} = i\frac{1+\Theta}{1-\Theta}. \end{align*} $$
Therefore MIFs can be described by parameters
$(b,c,\mu )$
via the Herglotz representation
$m_{\Theta }(z) = bz + c + iS\mu _{\Theta }(z)$
, where
$b \geq 0$
,
$c \in \mathbb {R}$
and
$$ \begin{align*}S\mu_{\Theta}(z) := \frac{1}{i\pi}\int\Big(\frac{1}{t-z}-\frac{t}{1+t^2}\Big)d\mu_{\Theta}(t) \end{align*} $$
is the Schwarz integral of the positive discrete Poisson-finite measure
$\mu _{\Theta }$
on
$\mathbb {R}$
, (i.e.,
$\int 1/(1+t^2) d\mu _{\Theta }(t) < \infty $
). The number
$\pi b$
is considered as the point mass of
$\mu _{\Theta }$
at infinity. This extended measure
$\mu _{\Theta }$
is called the Clark (or spectral) measure of
$\Theta $
. The spectrum of
$\Theta $
, denoted by
$\sigma (\Theta )$
, is the level set
$\{\Theta = 1\}$
, so
$\mu _{\Theta }$
is supported on
$\sigma (\Theta )$
or
$\sigma (\Theta ) \cup \{\infty \}$
. The point masses at
$t \in \sigma ({\Theta })$
are given by
$\mu _{\Theta }(t) = 2\pi /|\Theta '(t)|$
, and the point mass at infinity is nonzero if and only if
$\lim _{y \rightarrow +\infty }\Theta (iy) = 1$
and the limit
$\lim _{y \rightarrow +\infty }y^2\Theta '(iy)$
exists. All of these above-mentioned properties of MIFs can be found, for example, in [Reference Poltoratski17] and references therein.
The existence, uniqueness, interpolation, and other complex function theoretic problems for MIFs were studied in various papers [Reference Bénéteau, Condori, Liaw, Ross and Sola2, Reference Poltoratski17, Reference Poltoratski and Rupam20, Reference Rupam22]. In this paper, we consider the unique determination of the MIF
$\Theta $
from various spectral data and conditions depending fully or partially on
$\sigma (\Theta )$
,
$\sigma (-\Theta )$
and
$\{\mu _{\Theta }(t)\}_{t \in \sigma (\Theta )}$
. The two spectra
$\sigma (\Theta )$
and
$\sigma (-\Theta )$
are the sets of poles and zeros of
$m_{\Theta }$
, respectively. They are interlacing, discrete sequences in
$\mathbb {R}$
, so we denote them by
$\{a_n\}_{n \in \mathbb {Z}} = \sigma (\Theta ) := \{\Theta = 1\}$
and
$\{b_n\}_{n \in \mathbb {Z}} = \sigma (-\Theta ) := \{\Theta = -1\}$
indexed in increasing order. Throughout the paper,
$a_n < b_n < a_{n+1}$
is assumed whenever the spectrum is unbounded. We also introduce the disjoint intervals
$I_n := (a_n,b_n)$
. If
$\sigma (\Theta )$
is bounded below, then
$\{a_n\}$
,
$\{b_n\}$
, and
$\{I_n\}$
are indexed by
$\mathbb {N}$
.
MIFs (equivalently, MHFs) appear in inverse spectral theory. Inverse spectral problems aim to determine an operator from given spectral data. Classical results were given for Sturm-Liouville operators and can be found, for example, in [Reference Hatinoğlu6] and references therein.
Canonical Hamiltonian systems are the most general class of symmetric second-order operators, so the more classical second-order operators such as Schrödinger, Jacobi, Dirac, Sturm–Liouville operators and Krein strings can be transformed to canonical Hamiltonian systems [Reference Remling21]; for example, see Section 1.3 of [Reference Remling21] for rewriting the Schrödinger equation as a canonical Hamiltonian system.
A canonical Hamiltonian system is a
$2 \times 2$
differential equation system of the form
$$ \begin{align} J f'(x) = -zH(x)f(x), \end{align} $$
where
$d> 0$
,
$x \in (0,d)$
and
$$ \begin{align*}J:= {\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}}, \quad f(x) = {\begin{pmatrix} f_1(x) \\ f_2(x) \end{pmatrix}}. \end{align*} $$
We assume
$H(x) \in \mathbb {R}^{2\times 2}$
,
$H \in L_{\text {loc}}^1(0,d)$
and
$H(x) \geq 0$
a.e. on
$(0,d)$
. In this paper, we consider the limit circle case, which means
$\int _0^d \text {Tr} H(x) dx < \infty $
, on a finite interval (i.e.,
$0 < d < \infty $
). The self-adjoint realizations of a canonical Hamiltonian system in the limit circle case with separated boundary conditions are described by
$$ \begin{align} f_1(0)\sin\alpha-f_2(0)\cos\alpha = 0, \qquad f_1(d)\sin\beta-f_2(d)\cos\beta = 0, \end{align} $$
with
$\alpha , \beta \in [0,\pi )$
. Such a self-adjoint system has a discrete spectrum
$\sigma _{\alpha ,\beta }$
.
Let
$f = f (x, z)$
be a solution of
$Jf' = -zHf$
with the boundary condition
$f_1(d)\sin \beta -f_2(d)\cos \beta = 0$
. If we let u and v be solutions of
$Ju'= -zHu$
with initial conditions
$u_1(0) = \cos (\alpha )$
,
$u_2(0) = \sin (\alpha )$
and
$v_1(0) = -\sin (\alpha )$
,
$v_2(0) = \cos (\alpha )$
, then the solution u satisfies the boundary condition at
$x = 0$
, and
$f(x,z) = v(x,z) + m_{\alpha ,\beta }(z)u(x,z)$
is the unique solution of the form
$v + Mu$
that satisfies the boundary condition
$\beta $
at
$x = d$
. The complex function
$m_{\alpha ,\beta }$
is called the Weyl m-function, which is
$$ \begin{align} m_{\alpha,\beta}(z) := \frac{\cos(\alpha)f_1(0) + \sin(\alpha)f_2(0)}{-\sin(\alpha)f_1(0) + \cos(\alpha)f_2(0)}. \end{align} $$
The spectrum
$\sigma _{\alpha ,\beta } = \{a_n\}_n$
is the set of poles of
$m_{\alpha ,\beta }$
. The norming constant
$\gamma _{\alpha ,\beta }^{(n)}$
for
$a_n \in \sigma _{\alpha ,\beta }$
is defined as
$\gamma _{\alpha ,\beta }^{(n)} := 1/||u(\cdot ,a_n)||^2$
. The Weyl m-function
$m_{\alpha ,\beta }(z)$
is a MHF, so the corresponding MIF is defined as
$\Theta _{\alpha ,\beta } = (m_{\alpha ,\beta }-i)/(m_{\alpha ,\beta }+i)$
, which is called the Weyl inner function. The Clark (spectral) measure of
$\Theta _{\alpha ,\beta }$
is the discrete measure supported on
$\sigma _{\alpha ,\beta }$
with point masses given by the corresponding norming constants (i.e.,
$\mu _{\alpha ,\beta } = \sum _{n} \gamma _{\alpha ,\beta }^{(n)}\delta _{a_n}$
is the spectral measure). The MIF
$\Theta _{\alpha ,\beta }$
carries out spectral properties of the corresponding canonical Hamiltonian system through (1.3), so we will use results on MIFs in inverse spectral theory of canonical Hamiltonian systems.
The paper is organized as follows.
Section 2 includes the following uniqueness results for MIFs and canonical systems.
-
• In Theorem 2.1, we consider unique determination of
$\Theta $
from its spectrum, point masses (or equivalently, derivative values) at the spectrum and one of the constants
$\Theta (0)$
,
$\lim _{y \rightarrow +\infty }\Theta (iy)$
or
$\prod _{n \in \mathbb {Z}}a_n/b_n \neq 1$
with the condition that
$\{|I_n|/(1+\text {dist}(0,I_n))\}_{n \in \mathbb {Z}} \in l^1(\mathbb {Z})$
. -
• In Theorem 2.3, this condition is replaced by the condition
$(1+|x|)^{-1} \in L^1(\mu _{\Theta })$
and that
$\mu _{\Theta }$
has no point mass at infinity, so the Clark measure
$\mu _{\Theta }$
and one of the constants
$\Theta (0)$
,
$\lim _{y \rightarrow +\infty }\Theta (iy)$
or
$\prod _{n \in \mathbb {Z}}a_n/b_n \neq 1$
uniquely determine
$\Theta $
in this case. -
• Theorem 2.4 shows that if the spectrum
$\sigma (\Theta )$
is bounded from below and
$a_n < b_n$
, then
$\Theta $
is uniquely determined by the spectral data of Theorem 2.1 without any other condition. Its second part deals with the case
$a_n> b_n$
. -
• In Theorems 2.5 and 2.6, we prove that the knowledge of the derivative values from Theorems 2.1 and 2.4, respectively, can be partially replaced by the knowledge of the corresponding points from the second spectrum
$\sigma (-\Theta )$
. -
• Theorem 2.7 shows unique determination of a Hamiltonian from the spectral measure
$\mu _{\alpha _1,\beta }$
and boundary conditions
$\alpha _1, \alpha _2$
, and
$\beta $
with the condition that
$\{|I_n|/(1+\text {dist}(0,I_n))\}_{n \in \mathbb {Z}} \in l^1(\mathbb {Z})$
, where endpoints of
$I_n$
are given by the two spectra
$\sigma _{\alpha _1,\beta }$
and
$\sigma _{\alpha _2,\beta }$
. -
• In Theorem 2.8, we consider unique determination of a Hamiltonian from a spectral measure and boundary conditions in the case that the corresponding spectrum is bounded below.
-
• In Theorem 2.9 and Theorem 2.10, we prove that some missing norming constants from the spectral data of Theorems 2.7 and 2.8, respectively, can be compensated by the corresponding data from a second spectrum.
-
• As Corollary 2.11, we obtain a generalization of Borg-Levinson’s classical two-spectra theorem on Schrödinger operators to canonical Hamiltonian systems.
Section 3 includes proofs of the results for MIFs.
Section 4 includes proofs of the results for canonical Hamiltonian systems.
2 Results
We first obtain uniqueness theorems for meromporphic inner functions (MIFs) and then consider inverse spectral problems for canonical Hamiltonian systems as applications. However, let us start by fixing our notation. We denote MIFs by
$\Theta $
(or
$\Phi $
) and the corresponding MHFs by
$m_{\Theta }$
(or
$m_{\Phi }$
). Since the spectrum
$\sigma (\Theta )$
of a MIF
$\Theta $
is a discrete sequence in
$\mathbb {R}$
, we denote it by
$\{a_n\}_{n \in \mathbb {Z}}$
. Similarly, the second spectrum
$\sigma (-\Theta )$
is denoted by
$\{b_n\}_{n \in \mathbb {Z}}$
. Since
$\Theta = \exp (i\phi )$
on
$\mathbb {R}$
for an increasing real analytic function
$\phi $
and
$\{a_n\}_{n \in \mathbb {Z}}$
and
$\{b_n\}_{n \in \mathbb {Z}}$
are interlacing sequences, we get
$\{x\in \mathbb {R}~|~\mathrm {Im}\Theta (x)> 0\} = \{(a_n,b_n)\}$
. We denote the intervals
$(a_n,b_n)$
by
$I_n$
. A sequence of disjoint intervals
$\{J_n\}$
on
$\mathbb {R}$
is called short if
$$ \begin{align} \sum_{n}\frac{|J_n|^2}{1+\text{dist}^2(0,J_n)} < \infty, \end{align} $$
and long otherwise. Here,
$|J_n|$
and
$\text {dist}(0,J_n)$
denote the length of the interval
$J_n$
and its distance to the origin, respectively. Collections of short (and long) intervals appear in various areas of harmonic analysis (see [Reference Poltoratski17] and references therein). If we assume
$0 \notin \cup I_n$
, then condition (2.1) is nothing but
$\{|J_n|/\text {dist}(0,J_n)\} \in l^2$
. In some of our results, we use the condition that this sequence for
$\{I_n\}$
belongs to
$l^1$
; that is,
$$ \begin{align} \sum_{n}\frac{|I_n|}{1+\text{dist}(0,I_n)} < \infty. \end{align} $$
In our first theorem, with condition (2.2), we consider the unique determination of a MIF from its spectrum and derivative values on the spectrum.
Theorem 2.1 Let
$\Theta $
be a MIF,
$\sigma (\Theta ) = \{a_n\}_{n \in \mathbb {Z}}:= \{\Theta = 1\}$
,
$\sigma (-\Theta ) = \{b_n\}_{n \in \mathbb {Z}}:= \{\Theta = -1\}$
, and
$I_n := (a_n,b_n)$
. If
$$ \begin{align*}\sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\mathrm{dist}(0,I_n)} < \infty,\end{align*} $$
then the spectral data consisting of
-
•
$\{a_n\}_{n \in \mathbb {Z}}$
-
•
$\{\Theta '(a_n)\}_{n \in \mathbb {Z}}$
(or equivalently
$\{\mu _{\Theta }(a_n)\}_{n \in \mathbb {Z}}$
) and -
•
$L:=\displaystyle \lim _{y \rightarrow +\infty }\Theta (iy)$
or
$C:= \Theta (0)$
or
$p:=\prod _{n \in \mathbb {Z}}a_n/b_n \neq 1$
uniquely determine
$\Theta $
.
Remark 2.2 The definition of MIF does not necessarily imply convergence of
$$ \begin{align*}p:=\prod_{n \in \mathbb{Z}}a_n/b_n.\end{align*} $$
However, the condition
$$ \begin{align*}\sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\mathrm{dist}(0,I_n)} < \infty\end{align*} $$
we assumed guarantees convergence of the infinite product p. The same thing happens in Theorems 2.3 and 2.5, as our assumptions on
$\mu _{\Theta }$
and
$\{I_n\}$
, respectively, imply convergence of p.
If the Clark measure has no point mass at infinity, summability condition (2.2) can be replaced by an integrability condition on the Clark measure.
Theorem 2.3 Let
$\Theta $
be a MIF,
$\mu _{\Theta }$
be its Clark measure,
$\sigma (\Theta ) = \{a_n\}_{n \in \mathbb {Z}}:= \{\Theta = 1\}$
, and
$\sigma (-\Theta ) = \{b_n\}_{n \in \mathbb {Z}}:= \{\Theta = -1\}$
. If
$$ \begin{align*}\int \frac{1}{1+|x|} d\mu_{\Theta}(x) < \infty\end{align*} $$
and
$\mu _{\Theta }$
has no point mass at
$\infty $
, then
$\mu _{\Theta }$
and
$L:=\lim _{y \rightarrow +\infty }\Theta (iy)$
or
$C:= \Theta (0)$
or
$p:=\prod _{n \in \mathbb {Z}}a_n/b_n \neq 1$
uniquely determine
$\Theta $
.
If the spectrum of a MIF is bounded below, then the summability condition of Theorem 2.1 is not required.
Theorem 2.4 Let
$\Theta $
be a MIF,
$\sigma (\Theta ) = \{a_n\}_{n \in \mathbb {N}}:= \{\Theta = 1\}$
, and
$\sigma (-\Theta ) = \{b_n\}_{n \in \mathbb {N}}:= \{\Theta = -1\}$
.
-
(1) If
$\sigma (\Theta )=\{a_n\}_{n \in \mathbb {N}}$
is bounded below and
$a_n < b_n$
, then the spectral data consisting of -
•
$\{a_n\}_{n \in \mathbb {N}}$
-
•
$\{\Theta '(a_n)\}_{n \in \mathbb {N}}$
and -
•
$C:= \Theta (0)$
$\Theta $
.
-
-
(2) If
$\sigma (\Theta )=\{a_n\}_{n \in \mathbb {N}}$
is bounded below and
$a_n> b_n$
, then the spectral data consisting of -
•
$\{b_n\}_{n \in \mathbb {N}}$
-
•
$\{\Theta '(b_n)\}_{n \in \mathbb {N}}$
and -
•
$C:= \Theta (0)$
$\Theta $
.
-
The next two theorems show that the missing part of the point masses of the Clark measure from the data of Theorems 2.1 and 2.4 can be compensated by the corresponding data from the second spectrum
$\sigma (-\Theta )$
.
Theorem 2.5 Let
$\Theta $
be a MIF,
$\sigma (\Theta ) = \{a_n\}_{n \in \mathbb {Z}}:= \{\Theta = 1\}$
,
$\sigma (-\Theta ) = \{b_n\}_{n \in \mathbb {Z}}:= \{\Theta = -1\}$
, and
$A \subseteq \mathbb {Z}$
. If
$$ \begin{align*} \sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\mathrm{dist}(0,I_n)} < \infty, \end{align*} $$
then the spectral data consisting of
-
•
$\{a_n\}_{n \in \mathbb {Z}}$
-
•
$\{b_n\}_{n \in \mathbb {Z}\setminus A}$
-
•
$\{\Theta ^{'}(a_n)\}_{n \in A}$
(or
$\{\mu (a_n)\}_{n \in A}$
) and -
•
$L:=\displaystyle \lim _{y \rightarrow +\infty }\Theta (iy)$
or
$C:= \Theta (0)$
or
$p:=\prod _{n \in A}a_n/b_n \neq 1$
uniquely determine
$\Theta $
.
Theorem 2.6 Let
$\Theta $
be a MIF,
$\sigma (\Theta ) = \{a_n\}_{n \in \mathbb {N}}:= \{\Theta = 1\}$
,
$\sigma (-\Theta ) = \{b_n\}_{n \in \mathbb {N}}:= \{\Theta = -1\}$
, and
$A \subseteq \mathbb {N}$
.
-
(1) If
$\{a_n\}_{n \in \mathbb {N}}$
is bounded below and
$a_n < b_n$
, then the spectral data consisting of -
•
$\{a_n\}_{n \in \mathbb {N}}$
-
•
$\{b_n\}_{n \in \mathbb {N}\setminus A}$
-
•
$\{\Theta ^{'}(a_n)\}_{n \in A}$
(or
$\{\mu (a_n)\}_{n \in A}$
) and -
•
$C:= \Theta (0)$
$\Theta $
.
-
-
(2) If
$\{a_n\}_{n \in \mathbb {N}}$
is bounded below and
$a_n> b_n$
, then the spectral data consisting of -
•
$\{a_n\}_{n \in \mathbb {N}\setminus A}$
-
•
$\{b_n\}_{n \in \mathbb {N}}$
-
•
$\{\Theta ^{'}(b_n)\}_{n \in A}$
and -
•
$C:= \Theta (0)$
$\Theta $
.
-
Next, we consider inverse spectral theorems on canonical systems as applications of the above-mentioned results. We follow the definitions and notations we introduced in Section 1.
Theorem 2.7 Let
$d> 0$
,
$\alpha _1,\alpha _2,\beta \in [0,\pi )$
,
$\alpha _1 \neq \alpha _2$
and let H be a trace normed canonical system on
$[0,d]$
. Also let
$\sigma _{\alpha _1,\beta } = \{a_n\}_{n \in \mathbb {Z}}$
,
$\sigma _{\alpha _2,\beta } = \{b_n\}_{n \in \mathbb {Z}}$
, and
$I_n = (a_n,b_n)$
. If
$$ \begin{align*}\sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\mathrm{dist}(0,I_n)} < \infty,\end{align*} $$
then the spectral measure
$\mu _{\alpha _1,\beta }$
and boundary conditions
$\alpha _1$
,
$\alpha _2$
, and
$\beta $
uniquely determine H.
Theorem 2.8 Let
$d> 0$
,
$\alpha ,\beta \in [0,\pi )$
and let H be a trace normed canonical Hamiltonian system on
$[0,d]$
. If
$\sigma _{\alpha ,\beta } = \{a_n\}_{n \in \mathbb {N}}$
is bounded below and
$a_n < b_n$
, then the spectral measure
$\mu _{\alpha ,\beta }$
and boundary conditions
$\alpha $
and
$\beta $
uniquely determine H.
The next two results show that the missing norming constants from the spectral data can be compensated by the corresponding data from a second spectrum.
Theorem 2.9 Let
$d> 0$
,
$\alpha _1,\alpha _2,\beta \in [0,\pi )$
,
$\alpha _1 \neq \alpha _2$
,
$A \subseteq \mathbb {Z}$
and let H be a trace normed canonical Hamiltonian system on
$[0,d]$
. Also let
$\sigma _{\alpha _1,\beta } = \{a_n\}_{n \in \mathbb {Z}}$
,
$\sigma _{\alpha _2,\beta } = \{b_n\}_{n \in \mathbb {Z}}$
, and
$I_n = (a_n,b_n)$
. If
$$ \begin{align*} \sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\mathrm{dist}(0,I_n)} < \infty, \end{align*} $$
then the spectral data consisting of
-
•
$\{a_n\}_{n \in \mathbb {Z}}$
-
•
$\{b_n\}_{n \in \mathbb {Z}\setminus A}$
-
•
$\{\gamma _{\alpha _1,\beta }^{(n)}\}_{n \in A}$
(or
$\{\mu _{\alpha _1,\beta }(a_n)\}_{n \in A}$
) -
•
$\alpha _1$
,
$\alpha _2$
and
$\beta $
uniquely determine H.
Theorem 2.10 Let
$d> 0$
,
$\alpha _1,\alpha _2,\beta \in [0,\pi )$
,
$\alpha _1 \neq \alpha _2$
,
$A \subseteq \mathbb {N}$
and let H be a trace normed canonical Hamiltonian system on
$[0,d]$
. Also let
$\sigma _{\alpha _1,\beta } = \{a_n\}_{n \in \mathbb {N}}$
and
$\sigma _{\alpha _2,\beta } = \{b_n\}_{n \in \mathbb {N}}$
. If
$\sigma _{\alpha _1,\beta }$
is bounded below and
$a_n < b_n$
, then the spectral data consisting of
-
•
$\{a_n\}_{n \in \mathbb {N}}$
-
•
$\{b_n\}_{n \in \mathbb {N}\setminus A}$
-
•
$\{\gamma _{\alpha _1,\beta }^{(n)}\}_{n \in A}$
(or
$\{\mu _{\alpha _1,\beta }(a_n)\}_{n \in A}$
) -
•
$\alpha _1$
,
$\alpha _2$
and
$\beta $
uniquely determine H.
By letting
$A = \emptyset $
, we get a canonical Hamiltonian system version of Borg-Levinson’s classical two spectra theorem for Schrödinger operators. Note that the spectrum of a Schrödinger (Sturm-Liouville) operator on a finite interval is always bounded below.
Corollary 2.11 Let
$d> 0$
,
$\alpha _1,\alpha _2,\beta \in [0,\pi )$
and let H be a trace normed canonical Hamiltonian system on
$[0,d]$
. If
$\sigma _{\alpha _1,\beta } = \{a_n\}_{n \in \mathbb {N}}$
is bounded below and
$a_n < b_n$
, then the two spectra
$\sigma _{\alpha _1,\beta } = \{a_n\}_{n \in \mathbb {N}}$
,
$\sigma _{\alpha _2,\beta } = \{b_n\}_{n \in \mathbb {N}}$
and
$\alpha _1$
,
$\alpha _2$
and
$\beta $
uniquely determine H.
Remark 2.12 Schrödinger (Sturm-Liouville) operators on finite intervals are characterized by two spectra with specific asymptotics (see, for example, (2.4)–(2.7) in [Reference Hatinoğlu6]). Namely, two interlacing, discrete, bounded below subsets of the real-line satisfying these asymptotics correspond to two spectra of a Schrödinger operator on a finite interval and any pair of spectra of a Schrödinger operator on a finite interval satisfy these properties.
Corollary 2.11 shows that unique recovery from two spectra is not related with the special asymptotics of eigenvalues from Schrödinger (Sturm-Liouville) class if the given order relation between the two spectra is satisfied. It extends the two-spectra theorem (or Borg-Levinson’s theorem) to a broader class of canonical Hamiltonian systems on finite intervals with bounded below spectrum.
Remark 2.13 In this paper, we consider canonical Hamiltonian systems on finite intervals to guarantee existence of a discrete spectrum. In general, one can let
$L=\infty $
in the definition of canonical Hamiltonian systems (1.1). With the restriction of having a discrete spectrum, the inverse spectral theory results above may be obtained for canonical Hamiltonian systems on
$\mathbb {R}_{+}$
, since our results were obtained from the uniqueness results for MIF, which were considered in the general complex function theoretic framework. As a special case, similar versions of mixed spectral data results were obtained for semi-infinite Jacobi operators with discrete spectrum in [Reference Hatinoğlu5].
3 Proofs for meromorphic inner functions
To prove our uniqueness theorems, we use an infinite product representation result.
Lemma 3.1 ([Reference Levin7], Theorem VII.1)
The MHF
$m_{\Theta }$
corresponding to the MIF
$\Theta $
has the infinite product representation
$$ \begin{align} m_{\Theta}(z) = i\frac{1+\Theta(z)}{1-\Theta(z)} = c \prod_{n\in \mathbb{Z}}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1}, \end{align} $$
where
$c> 0$
,
$a_n < b_n < a_{n+1}$
, and the product converges normally on
$\displaystyle \mathbb {C} {\backslash } \cup _{n \in \mathbb {N}} a_n$
.
Note that if m is a MHF, then
$-1/m$
is also a MHF. Therefore, the roles of zeros and poles can be swapped in Lemma 3.1 by letting the coefficient
$c = m_{\Theta }(0)$
be negative.
Remark 3.2 If
$\sigma (\Theta )$
is bounded below and
$a_n < b_n$
, then representation (3.1) becomes
$$ \begin{align} m_{\Theta}(z) = c \prod_{n\in \mathbb{N}}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1}, \end{align} $$
where
$c = m_{\Theta }(0)> 0$
. If
$a_n> b_n$
, representation (3.2) is valid with
$c = m_{\Theta }(0) < 0$
.
Proof of Theorem 2.1
Let us note that
$$ \begin{align*}\sum_{n \in \mathbb{Z}}\frac{|b_n-a_n|}{1+|a_n|} \leq \sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\text{dist}(0,I_n)} < \infty ~ \text{and} ~ \sum_{n \in \mathbb{Z}}\frac{|b_n-a_n|}{1+|b_n|} \leq \sum_{n \in \mathbb{Z}}\frac{|I_n|}{1+\text{dist}(0,I_n)} < \infty. \end{align*} $$
Therefore,
$0 < \prod _{n \in \mathbb {Z}}|a_n|/|b_n| < \infty $
, and hence,
$$ \begin{align} l := \lim_{y \rightarrow +\infty}m_{\Theta}(iy) = \lim_{y \rightarrow +\infty}c \prod_{n \in \mathbb{Z}}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1} = c \prod_{n \in \mathbb{Z}}\frac{a_n}{b_n}, \end{align} $$
which implies
$L:=\lim _{y \rightarrow +\infty }\Theta (iy) = (l-i)/(l+i) \neq 1$
. Therefore,
$\mu _{\Theta }(\infty ) = 0$
; that is,
$$ \begin{align*}m_{\Theta}(z) = b + iS\mu_{\Theta} = b + \sum_{k \in \mathbb{Z}} \frac{\mu_{\Theta}(a_k)}{\pi}\left(\frac{1}{a_k-z} - \frac{a_k}{1+a_k^2}\right).\end{align*} $$
Now let’s observe that
$$ \begin{align*}l = b - \sum_{k \in \mathbb{Z}}\frac{\mu_{\Theta}(a_k)}{\pi}\frac{a_k}{1+a_k^2}.\end{align*} $$
Indeed, we just obtained that the limit of
$m_{\Theta }(iy)$
is finite as y goes to
$+\infty $
. Therefore,
$m_{\Theta }$
is bounded on
$i(\varepsilon ,\infty )$
for a fixed
$\varepsilon> 0$
and
$$ \begin{align*} l = \lim_{y \rightarrow +\infty} m_{\Theta}(iy) &= b + \frac{1}{\pi}\sum_{k \in \mathbb{Z}} \lim_{y \rightarrow +\infty} \mu_{\Theta}(a_k)\left(\frac{1}{a_k-iy} - \frac{a_k}{1+a_k^2}\right)\\ &= b - \frac{1}{\pi}\sum_{k \in \mathbb{Z}} \mu_{\Theta}(a_k)\frac{a_k}{1+a_k^2}, \end{align*} $$
which implies
$$ \begin{align} m_{\Theta}(z) = l + \frac{1}{\pi}\sum_{k \in \mathbb{Z}} \frac{\mu_{\Theta}(a_k)}{(a_k-z)}, \end{align} $$
so the only unknown on the right-hand side is l.
At this point, we consider three different cases of our spectral data given in the last item in the theorem statement.
If
$L = \lim _{y \rightarrow +\infty }\Theta (iy)$
is known, then L uniquely determines
$l = \lim _{y \rightarrow +\infty }m_{\Theta }(iy)$
, since
$L = (l-i)/(l+i)$
and
$(x-i)/(x+i)$
is an injective function. Therefore, by (3.4),
$m_{\Theta }$
is uniquely determined.
If
$c = m_{\Theta }(0)$
is known, then using (3.4), we get
$$ \begin{align*}l = c - \frac{1}{\pi}\sum_{k \in \mathbb{Z}} \frac{\mu_{\Theta}(a_k)}{a_k}, \end{align*} $$
so l is known, and hence,
$m_{\Theta }$
is uniquely determined.
If
$p=\prod _{n \in \mathbb {Z}}a_n/b_n \neq 1$
is known, in order to show uniqueness of l, let us consider another MIF
$\widetilde {\Theta }$
satisfying the following properties:
-
•
$\{\widetilde {\Theta }=1\} = \{a_n\}_{n \in \mathbb {Z}}$
, -
•
$\widetilde {\Theta }'(a_n)=\Theta '(a_n)$
for any
$n \in \mathbb {Z}$
, -
•
$\displaystyle \sum _{n \in \mathbb {Z}}\frac {|\widetilde {I}_n|}{1+\text {dist}(0,\widetilde {I}_n)} < \infty $
(and hence,
$\mu _{\widetilde {\Theta }}(\infty ) = 0$
),
where
$\widetilde {I}_n = (a_n,\widetilde {b}_n)$
and
$\{\widetilde {b}_n\}_{n \in \mathbb {Z}} := \sigma (-\Theta )$
. In other words, MIFs
$\Theta $
and
$\widetilde {\Theta }$
share the same Clark measure (i.e.,
$\mu _{\Theta } = \mu _{\widetilde {\Theta }}$
, and
$\{\widetilde {I}_n\}_{n}$
satisfy the summability condition), so
$$ \begin{align*} m_{\widetilde{\Theta}}(z) = i\frac{1+\widetilde{\Theta}(z)}{1-\widetilde{\Theta}(z)} = \widetilde{c} \prod_{n\in \mathbb{Z}}\left(\frac{z}{\widetilde{b}_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1} = \widetilde{l} + \frac{1}{\pi}\sum_{n \in \mathbb{Z}} \frac{\mu_{\Theta}(a_n)}{(a_n-z)}. \end{align*} $$
Let us recall from (3.3) that
$l = cp$
, so we have
$l/c = \widetilde {l}/\widetilde {c} = p$
. Also by assumption,
$p \neq 1$
, and we get
$l \neq c$
and
$\widetilde {l} \neq \widetilde {c}$
. Letting
$z=0$
in (3.4), we also get
$c - \widetilde {c} = l - \widetilde {l}$
. If we call this value x, then we have
$$ \begin{align*}\frac{l}{c} = \frac{l-x}{c-x}, \end{align*} $$
and hence,
$x(l-c) = 0$
, so
$x=0$
(i.e.,
$l=\widetilde {l}$
and
$m_{\Theta }$
is uniquely determined).
In all three cases,
$m_{\Theta }$
is uniquely determined from the given spectral data. Then
$\Theta $
is uniquely determined through the one-to-one correspondence between MHFs and MIFs, so we get the desired result.
Proof of Theorem 2.3
Recall that knowing
$\mu _{\Theta }$
means knowing
$\{a_n\}_{n}$
and
$\{\Theta '(a_n)\}_{n}$
. We also know that
$\mu _{\Theta }(\infty ) = 0$
, so the MHF corresponding to
$\Theta $
has the representation
$$ \begin{align} m_{\Theta}(z) = b + iS\mu_{\Theta} = b + \frac{1}{\pi}\sum_{n \in \mathbb{Z}} \mu_{\Theta}(a_n)\left(\frac{1}{a_n-z} - \frac{a_n}{1+a_n^2}\right). \end{align} $$
The condition
$ \int 1/(1+|x|) d\mu _{\Theta }(x) < \infty $
means
$\sum _{n \in \mathbb {Z}}\mu (a_n)/(1+|a_n|) < \infty $
, which implies that
$ \sum _{n \in \mathbb {Z}} a_n\mu (a_n)/(1+a_n^2) $
is convergent and
$ \sum _{n \in \mathbb {Z}}\mu (a_n)/(a_n - z) $
is uniformly bounded by
$\sum _{n \in \mathbb {Z}}\mu (a_n)/(1+|a_n|)$
on the set
$i(1,\infty )$
. These observations together with the representation (3.5) and Lemma 3.1 imply
$$ \begin{align*}c\prod_{n \in \mathbb{Z}}\frac{a_n}{b_n} = \lim_{y \rightarrow \infty} m_{\Theta}(iy) = b - \frac{1}{\pi}\sum_{n \in \mathbb{Z}} \mu_{\Theta}(a_n)\frac{a_n}{1+a_n^2} \end{align*} $$
(i.e.,
$\prod _{n \in \mathbb {Z}}a_n/b_n$
is convergent). Therefore,
$$ \begin{align*}m_{\Theta}(z) = l + \frac{1}{\pi}\sum_{n \in \mathbb{Z}} \frac{\mu_{\Theta}(a_n)}{a_n-z}, \end{align*} $$
so we get the representation (3.4) again. In order to show uniqueness of
$m_{\Theta }$
, we can follow the arguments (starting after (3.4)) we used in the proof of Theorem 2.1 in all three cases of given spectral data. Uniqueness of
$m_{\Theta }$
gives uniqueness of
$\Theta $
through the one-to-one correspondence between MHFs and MIFs, so we get the desired result.
Proof of Theorem 2.4
Let
$d> \max \{|a_1|,|b_1|\}$
. Since the MHF
$m_{\Theta }$
corresponding to
$\Theta $
satisfies the infinite product representation (3.2), by substituting
$z-d$
for z, we can keep the derivative values of
$\Theta $
the same and make
$\{a_n+d\}_{n \in \mathbb {N}}$
and
$\{b_n+d\}_{n \in \mathbb {N}}$
subsets of
$\mathbb {R}_+$
. Therefore, without loss of generality, we assume
$\{a_n\}_{n \in \mathbb {N}} \bigcup \{b_n\}_{n \in \mathbb {N}} \subset \mathbb {R}_+$
.
Now we are ready to prove part
$(1)$
. We know that
$$ \begin{align} m_{\Theta}(z) = az + b + iS\mu_{\Theta} = az + b + \frac{1}{\pi}\sum_{n \in \mathbb{N}} \mu_{\Theta}(a_n)\left(\frac{1}{a_n-z} - \frac{a_n}{1+a_n^2}\right), \end{align} $$
where
$a \geq 0$
,
$b \in \mathbb {R}$
and
$\mu _{\Theta }(a_n) \geq 0$
for any
$n \in \mathbb {N}$
. First, let’s show that
$a=0$
. Note that in part
$(1)$
, we assume
$a_n < b_n$
, so
$m_{\Theta }$
satisfies the infinite product representation (3.2) with positive
$c = m_{\Theta }(0)$
. Then partial products of
$m_{\Theta }$
are represented as
$$ \begin{align*}m_{\Theta}^{(N)}(z) := c\prod_{n=1}^{N}\frac{a_n}{a_n-z}\frac{b_n-z}{b_n} = \sum_{n=1}^N \frac{\alpha_{n,N}}{a_n-z} + c\prod_{n=1}^{N}\frac{a_n}{b_n}, \end{align*} $$
where
$c> 0$
and
$\alpha _{n,N}> 0$
for any
$n \in \{1,\cdots ,N\}$
,
$N \in \mathbb {N}$
. Let us also note that
$\alpha _{n,N}$
converges to
$\mu _{\Theta }(a_n)/\pi $
for any fixed n and
$\{\mu _{\Theta }(a_n)/a_n^2\}_{n \in \mathbb {N}} \in l^1(\mathbb {N})$
. Then for
$K < N$
, let us consider the difference
$$ \begin{align*}m_{\Theta}^{(N)}(z) - \sum_{n=1}^K \alpha_{n,N}\left(\frac{1}{a_n-z} - \frac{a_n}{1+a_n^2}\right) = \sum_{n=K+1}^{N}\frac{\alpha_{n,N}}{a_n-z} + c\prod_{n=1}^{N}\frac{a_n}{b_n} + \sum_{n=1}^K \alpha_{n,N}\frac{a_n}{1+a_n^2}. \end{align*} $$
Note that for any
$z < 0$
,
$K < N$
and
$N \in \mathbb {N}$
, the right-hand side is positive since
$\alpha _{n,N}$
, c,
$a_n$
,
$b_n$
are positive numbers, so the left-hand side is positive for any
$z < 0$
,
$K < N$
, and
$N \in \mathbb {N}$
. First, letting N tend to infinity, we get
$$ \begin{align*}m_{\Theta}(z) - \frac{1}{\pi}\sum_{n=1}^K \mu_{\Theta}(a_n)\left(\frac{1}{a_n-z} - \frac{a_n}{1+a_n^2}\right), \end{align*} $$
and then letting K tend to infinity, we get
$$ \begin{align*}m_{\Theta}(z) - \frac{1}{\pi}\sum_{n\in \mathbb{N}} \mu_{\Theta}(a_n)\left(\frac{1}{a_n-z} - \frac{a_n}{1+a_n^2}\right). \end{align*} $$
We observed that this difference should be nonnegative for
$z<0$
. However, by (3.6), it is nothing but
$az + b$
with
$a\geq 0$
, which is possible only if
$a=0$
; that is,
$$ \begin{align} m_{\Theta}(z) = b + \frac{1}{\pi}\sum_{n \in \mathbb{N}} \mu(a_n)\left(\frac{1}{a_n-z} - \frac{a_n}{1+a_n^2}\right). \end{align} $$
Therefore,
$$ \begin{align*}b = c - \frac{1}{\pi}\sum_{n \in \mathbb{N}} \mu(a_n)\left( \frac{1}{a_n+a_n^3}\right), \end{align*} $$
so b and hence
$m_{\Theta }$
is uniquely determined. Using the one-to-one correspondence between MHFs and MIFs, we get the desired result of part
$(1)$
.
For part
$(2)$
, let’s observe that
$-\Theta $
is a MIF and its MHF is
$-1/m_{\Theta }$
. Using
$-1/m_{\Theta }$
as our MHF allows us to swap the roles of
$\{a_n\}$
and
$\{b_n\}$
. Also note that
$a_n> b_n$
and
$m_{\Theta } < 0$
in part
$(2)$
. Therefore,
$-1/m_{\Theta }$
(or
$-\Theta $
) with the spectral data of part
$(2)$
falls to the setting of part
$(1)$
, so we follow the proof of part
$(1)$
and obtain uniqueness of
$-1/m_{\Theta }$
. Using the one-to-one correspondence between MHFs and MIFs, we get the desired result of part
$(2)$
.
Remark 3.3 In the proof of Theorem 2.4, we showed that there is no point mass at infinity. However, it does not necessarily imply that
$L:=\lim _{y\rightarrow +\infty }\Theta (iy) \neq 1$
(or equivalently,
$l:=\lim _{y\rightarrow +\infty } m_{\Theta }(iy) < \infty $
). If we know that this limit condition is satisfied, then we can replace c in the spectral data with L by following the arguments we used in the proof of Theorem 2.1. The same applies to
$p:=\prod _{n \in \mathbb {Z}}a_n/b_n$
, if it is convergent and not equal to
$1$
.
Proof of Theorem 2.5
From Lemma 3.1, without loss of generality, we can assume that
$m_{\Theta }$
has the representation
$$ \begin{align*} m_{\Theta}(z) = c \prod_{n\in \mathbb{N}}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1}. \end{align*} $$
Note that for any
$k\in A$
, we know
$$ \begin{align} \frac{-\mu_{\Theta}(a_k)}{\pi} = \mathrm{Res}(m_{\Theta},a_k) = c(b_k - a_k)\frac{a_k}{b_k} \prod_{n\in \mathbb{N},n\neq k}\left(\frac{a_k}{b_{n}}-1\right)\left(\frac{a_k}{a_n}-1\right)^{-1}. \end{align} $$
Let
$m_{\Theta }(z) = f(z)g(z)$
, where f and g are two infinite products defined as
$$ \begin{align*} f(z) := c \prod_{n\in A}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1}, \qquad g(z) := \prod_{n\in\mathbb{N} {\backslash} A}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1}. \end{align*} $$
Let us observe that at any point of
$\{a_n\}_{n \in A}$
, the infinite product
$$ \begin{align} g(z)=\prod_{n\in\mathbb{N} {\backslash} A}\left(\frac{z}{b_{n}}-1\right)\left(\frac{z}{a_n}-1\right)^{-1} \end{align} $$
is also known. Then conditions (3.8) and (3.9) imply that for any
$k\in A$
, we know
$$ \begin{align*} \mathrm{Res}(f,a_k) = \frac{\mathrm{Res}(m_{\Theta},a_k)}{g(a_k)}. \end{align*} $$
Since real zeros and poles of f are simple and interlacing, f is a MHF. If
$\Phi $
is the MIF corresponding to f, then
$\{\Phi = 1\} = \{a_n\}_{n \in A}$
,
$\{\Phi = -1\} = \{b_n\}_{n \in A}$
, and our spectral data become
$\{a_n\}_{n \in A}$
,
$\{\Phi '(a_n)\}_{n \in A}$
, and
$c = f(0)$
,
$\lim _{y \rightarrow +\infty }\Phi (iy)$
or
$\prod _{n \in A}a_n/b_n$
. The convergence condition (2.2) implies
$0 < \prod _{n \in \mathbb {N}}|a_n|/|b_n| < \infty $
and hence
$0 < \prod _{n \in A}|a_n|/|b_n| < \infty $
. Therefore, applying the proof of Theorem 2.1, we determine
$\Phi $
and hence
$\{b_n\}_{n \in A}$
uniquely. This means unique recovery of
$m_{\Theta }$
and then unique recovery of
$\Theta $
through the one-to-one correspondence between MHFs and MIFs.
Proof of Theorem 2.6
For part
$(1)$
, following arguments from the proof of Theorem 2.5, we convert our problem to the problem of unique determination of the MIF
$\Phi $
from the spectral data
$\{a_n\}_{n \in A}$
and
$\{\Phi '(a_n)\}_{n \in A}$
, where
$\{\Phi = 1\} = \{a_n\}_{n \in A}$
,
$\{\Phi = -1\} = \{b_n\}_{n \in A}$
. Since
$\{a_n\}_{n \in A}$
is bounded below, by Theorem 2.4, part
$(1)$
, these spectral data uniquely determine
$\Phi $
and hence
$\{b_n\}_{n \in A}$
. This means unique recovery of
$m_{\Theta }$
and then unique recovery of
$\Theta $
through the one-to-one correspondence between MHFs and MIFs.
For part
$(2)$
, considering the MHF
$-1/m_{\Theta }$
and following arguments from the proof of Theorem 2.5, we convert our problem to the problem of unique determination of the MIF
$\Phi $
from the spectral data
$\{b_n\}_{n \in A}$
and
$\{\Phi '(b_n)\}_{n \in A}$
, where
$\{\Phi = 1\} = \{b_n\}_{n \in A}$
,
$\{\Phi = -1\} = \{a_n\}_{n \in A}$
. Since
$\{a_n\}_{n \in A}$
is bounded below, by Theorem 2.4, part
$(2)$
, these spectral data uniquely determine
$\Phi $
and hence
$\{a_n\}_{n \in A}$
. This means unique recovery of
$-1/m_{\Theta }$
and then unique recovery of
$\Theta $
through the one-to-one correspondence between MHFs and MIFs.
4 Applications to inverse spectral theory of canonical systems
In this section, we prove our inverse spectral theorems on canonical Hamiltonian systems. Recall that we introduced these differential systems in (1.1) and discussed some of their properties in the limit circle case (i.e.,
$\int _0^d \text {Tr} H(x) dx < \infty $
) for
$0 < d < \infty $
in Section 1. Since we are in the limit circle case, we can normalize our Hamiltonian systems by letting the trace be identically one. Also, note that throughout this section, d will be arbitrary but fixed.
In order to obtain our results, we need another definition. The transfer matrix
$T(z)$
is the
$2\times 2$
matrix solution of (1.1) with the initial condition identity matrix at
$x=0$
. Some properties of transfer matrices are given in Theorem 1.2 in [Reference Remling21]. Following definition 4.3 in [Reference Remling21], we call collection of all matrices with these properties
$TM$
– namely, the set of matrix functions
$T:\mathbb {C} \rightarrow \mathrm {SL}(2,\mathbb {C})$
such that T is entire,
$T(0) = I_2$
,
$\overline {T(\overline {z})} = T(z)$
and if
$\mathrm {Im}z \geq 0$
, then
$i(T^{*}(z)JT(z) - J) \geq 0$
. We denote any
$T \in TM$
by
$$ \begin{align*}T(z) := {\begin{pmatrix} A(z) & B(z) \\ C(z) & D(z) \end{pmatrix}}. \end{align*} $$
Note that the transfer matrix of any trace normed canonical system on
$[0,d]$
satisfies
$C'(0) - B'(0) = d$
and
$C'(0) - B'(0) \geq 0$
for any
$T \in TM$
(see page 106 in [Reference Remling21] for explanations). Therefore, if we define the disjoint subset
$$ \begin{align} TM(d) := \{ T \in TM~|~ C'(0) - B'(0) = d\}, \end{align} $$
then
$TM = \cup _{d \geq 0} TM(d)$
. The following result shows that
$TM$
characterizes all transfer matrices on finite intervals.
Theorem 4.1 ([Reference Remling21], Theorem 5.2)
Let
$d \geq 0$
. For every
$T \in TM(d)$
, there is a unique trace normed canonical Hamiltonian system H on
$[0, d]$
such that T is the transfer matrix of H.
Next, we focus on connections between m-functions and transfer matrices. The entries of T appear in m-functions with Dirichlet-Neumann and Dirichlet-Dirichlet boundary conditions – namely,
$-B/A = m_{0,\pi /2}$
and
$-D/C = m_{0,0}$
(page 86, [Reference Remling21]). This allows us to obtain unique recovery of transfer matrices from m-functions.
Proposition 4.2 Let
$d> 0$
. Then the Weyl m-function
$m_{0,\pi /2}$
uniquely determines the transfer matrix
$d \in TM(d)$
.
Proof Let
$T,\widetilde {T} \in TM(d)$
share the same
$m_{0,\pi /2}$
(i.e.,
$-B(z)/A(z) = -\widetilde {B}(z)/\widetilde {A}(z)$
). By Theorem 4.22 in [Reference Remling21],
$$ \begin{align} {\begin{pmatrix} \widetilde{A}(z) & \widetilde{B}(z) \\ \widetilde{C}(z) & \widetilde{D}(z) \end{pmatrix}} = {\begin{pmatrix} 1 & 0 \\ az & 1 \end{pmatrix}}{\begin{pmatrix} A(z) & B(z) \\ C(z) & D(z) \end{pmatrix}} = {\begin{pmatrix} A(z) & B(z) \\ azA+C(z) & azB+D(z) \end{pmatrix}} \end{align} $$
for some
$a \in \mathbb {R}$
. Since
$T,\widetilde {T} \in TM(d)$
, we also know that
$C'(0) - B'(0) = \widetilde {C}'(0) - \widetilde {B}'(0) = d$
and
$T(0) = \widetilde {T}(0) = I_2$
. Therefore, by (4.2),
$d = \widetilde {C}'(0) - \widetilde {B}'(0) = aA(0) + C'(0) - B'(0) = a + d$
, and hence,
$T = \widetilde {T}$
.
In order to consider general boundary conditions, we introduce another notation. Again following [Reference Remling21], let
$$ \begin{align*}R_{\alpha} := {\begin{pmatrix} \cos\alpha& -\sin\alpha \\ \sin\alpha & \cos\alpha \end{pmatrix}}. \end{align*} $$
Note that
$R_{\alpha }$
is a unitary matrix,
$R^{-1}_{\alpha } = R_{-\alpha }$
and
$\mathrm {det}R_{\alpha } = 1$
. If f is a single variable, then by
$R_{\alpha }f$
we mean division of the first entry of the
$2\times 1$
vector
$R_{\alpha }(f,1)^{\mathrm {T}}$
by its second entry. For example,
$m_{\alpha ,\beta } = R_{-\alpha }m_{0,\beta }$
. We will use the same notation for the transfer matrices.
Now we are ready to prove our inverse spectral results. Let’s start with the proof of Theorem 2.8, since the proof of Theorem 2.7 will require handling both spectra in the same MHF and hence introducing generalized m-functions and R-matrices.
Proof of Theorem 2.8
In order to use Theorem 2.4, first let’s show that
$m_{\alpha ,\beta }(0)$
solely depends on
$\alpha $
and
$\beta $
. We discussed the identity
$m_{0,\beta } = R_{\alpha }m_{\alpha ,\beta }$
. Also recalling the identity
$m_{0,\beta }(z) = T^{-1}(z)\cot \beta $
(
$(3.4)$
in [Reference Remling21]), we get
$$ \begin{align*}m_{\alpha,\beta}(0) = R_{-\alpha}m_{0,\beta}(0) = R_{-\alpha}\cot\beta = \frac{\cos\alpha\cos\beta + \sin\alpha\sin\beta}{-\sin\alpha\cos\beta + \cos\alpha\sin\beta} = \cot(\beta-\alpha). \end{align*} $$
Therefore, by Theorem 2.4, part
$(1)$
, the spectral measure
$\mu _{\alpha ,\beta }$
and boundary conditions
$\alpha $
and
$\beta $
uniquely determine the Weyl inner function
$\Theta _{\alpha ,\beta }$
and hence the Weyl m-function
$m_{\alpha ,\beta }$
since there is a one-to-one correspondence between MIFs and MHFs. By the identity
$m_{0,\beta } = R_{\alpha }m_{\alpha ,\beta }$
, the m-function
$m_{\alpha ,\beta }$
and the boundary condition
$\alpha $
uniquely determine
$m_{0,\beta }$
. We still need to pass to
$\pi /2$
from general
$\beta $
in order to use Proposition 4.2. For this, we use a transformation of H – namely,
$H_{\gamma } := R_{\gamma }^{\mathrm {T}}HR_{\gamma }$
. If
$m^{(\gamma )}$
denotes the m-function of
$H_{\gamma }$
, then
$m^{(\gamma )}_{0,\beta -\gamma }(z) = R_{-\gamma }m_{0,\beta }(z)$
(see Theorem 3.20 and following explanation on page 57 in [Reference Remling21]). By letting
$\gamma = \beta - \pi /2$
, we can uniquely determine
$m^{(\gamma )}_{0,\pi /2}$
from the knowledge of
$m_{0,\beta }$
and
$\beta $
. Now by Proposition 4.2, we obtain the transfer matrix of
$H_{\beta - \pi /2}$
and then by Theorem 4.1 the Hamiltonian
$H_{\beta - \pi /2}$
uniquely. Finally, recalling that
$H = R_{\gamma }H_{\gamma }R_{\gamma }^{\mathrm {T}}$
, we get unique determination of H from uniqueness of
$H_{\beta -\pi /2}$
and
$\beta $
.
Proof of Theorem 2.7
In order to handle both spectra in the same MHF, let’s introduce generalized m-functions and R-matrices:
$$ \begin{align*}m_{\alpha_1,\alpha_2,\beta}(z) := \frac{-\sin(\alpha_2)f_1(0) + \cos(\alpha_2)f_2(0)}{-\sin(\alpha_1)f_1(0) + \cos(\alpha_1)f_2(0)}, \qquad R_{\alpha_1,\alpha_2} := {\begin{pmatrix} -\sin\alpha_2& \cos\alpha_2 \\ -\sin\alpha_1 & \cos\alpha_1 \end{pmatrix}}.\end{align*} $$
Note that
$\det R_{\alpha _1,\alpha _2} = \sin (\alpha _1 - \alpha _2)$
, so
$R_{\alpha _1,\alpha _2}$
is invertible. Also
$m_{\alpha _1,\alpha _2,\beta } = R_{\alpha _1,\alpha _2} m_{0,\beta }$
, and hence,
$m_{\alpha _1,\alpha _2,\beta } = R_{\alpha _1,\alpha _2}R^{-1}_{\alpha _1}m_{\alpha _1,\beta }$
. Moreover,
$\sigma _{\alpha _1,\beta }$
and
$\sigma _{\alpha _2,\beta }$
are sets of poles and zeros of
$m_{\alpha _1,\alpha _2,\beta }$
, respectively. Another critical observation is that
$m_{\alpha _1,\alpha _2,\beta }$
is a MHF since
$m_{0,\beta }$
is a MHF. Therefore, we can introduce corresponding MIF
$\Theta _{\alpha _1,\alpha _2,\beta }$
and spectral measure
$\mu _{\alpha _1,\alpha _2,\beta } = \sum \gamma _{\alpha _1,\alpha _2,\beta }^{(n)}\delta _{a_n}$
. Keeping this notation in mind, we need to pass from
$\mu _{\alpha _1,\beta }$
to
$\mu _{\alpha _1,\alpha _2,\beta }$
. We can do this using two observations: first, both measures are supported on
$\sigma _{\alpha _1,\beta }$
, and second, the point masses or norming constants are related by the identity
$\gamma ^{(n)}_{\alpha _1,\alpha _2,\beta } = \sin (\alpha _1 - \alpha _2)\gamma ^{(n)}_{\alpha _1,\beta }$
, which is also valid for the point masses at infinity. The first observation follows from the fact that the
$m_{\alpha _1,\alpha _2,\beta }$
and
$m_{\alpha _1,\beta }$
share the same set of poles
$\sigma _{\alpha _1,\beta }$
. Let’s prove the second observation. For simplicity, we use the following notation:
$s_k := \sin (\alpha _k)$
and
$c_k := \cos (\alpha _k)$
. We know that
$a_n$
is a pole for both m-functions, so
$$ \begin{align*}\mathrm{Res}(m_{\alpha_1,\beta},z=a_n) = \big(c_1f_1(0) + s_1f_2(0)\big)\Big|_{z=a_n}\lim_{z \rightarrow a_n}\frac{z-a_n}{-s_1f_1(0) + c_1f_2(0)}, \end{align*} $$
and similarly,
$$ \begin{align*}\mathrm{Res}(m_{\alpha_1,\alpha_2,\beta},z=a_n) = \big(-s_2f_1(0) + c_2f_2(0)\big)\Big|_{z=a_n}\lim_{z \rightarrow a_n}\frac{z-a_n}{-s_1f_1(0) + c_1f_2(0)}. \end{align*} $$
Therefore,
$$ \begin{align*}\frac{\mathrm{Res}(m_{\alpha_1,\beta},z=a_n)}{\mathrm{Res}(m_{\alpha_1,\alpha_2,\beta},z=a_n)} = \frac{c_1\frac{f_1(0)}{f_2(0)}\Big|_{z=a_n} + s_1}{-s_2\frac{f_1(0)}{f_2(0)}\Big|_{z=a_n} + c_2}.\end{align*} $$
Since
$a_n$
is a pole, at
$z=a_n$
,
$-s_1f_1(0)+c_1f_2(0) = 0$
(i.e.,
$(f_1(0)/f_2(0))|_{z=a_n} = c_1/s_1$
). Hence,
$$ \begin{align*}\frac{\mathrm{Res}(m_{\alpha_1,\beta},z=a_n)}{\mathrm{Res}(m_{\alpha_1,\alpha_2,\beta},z=a_n)} = \frac{\frac{c_1^2}{s_1}+s_1}{-s_2\frac{c_1}{s_1}+c_2} = \frac{c_1^2+s_1^2}{-s_2c_1+s_1c_2} = \frac{1}{\sin(\alpha_1-\alpha_2)} \end{align*} $$
if
$s_1 \neq 0$
and
$c_1 \neq 0$
. One can check other cases similarly and get
$$ \begin{align*}\frac{\mathrm{Res}(m_{\alpha_1,\beta},z=a_n)}{\mathrm{Res}(m_{\alpha_1,\alpha_2,\beta},z=a_n)} = -\frac{c_1}{s_2} \qquad \text{and} \qquad \frac{\mathrm{Res}(m_{\alpha_1,\beta},z=a_n)}{\mathrm{Res}(m_{\alpha_1,\alpha_2,\beta},z=a_n)} = -\frac{s_1}{c_2} \end{align*} $$
for the cases
$s_1 = 0$
and
$c_1 = 0$
, respectively. In all three cases,
$\mathrm {Res}(m_{\alpha _1,\beta },z=a_n)$
is given in terms of
$\mathrm {Res}(m_{\alpha _1,\alpha _2,\beta },z=a_n)$
,
$\alpha _1$
and
$\alpha _2$
for any n. Finally recalling that the residue of a MHF at a pole is
$-1/\pi $
times the corresponding point mass by Herglotz representation (3.6), we get unique determination of
$\mu _{\alpha _1,\alpha _2,\beta }$
from
$\mu _{\alpha _1,\beta }$
,
$\alpha _1$
, and
$\alpha _2$
. The point mass at infinity can be handled similarly by comparing residues of
$m_{\alpha _1,\alpha _2,\beta }(1/z)$
and
$m_{\alpha _1,\beta }(1/z)$
at 0.
Now, by the identity
$m_{\alpha _1,\alpha _2,\beta }(0) = R_{\alpha _1,\alpha _2}\cot (\beta )$
and Theorem 2.1, the spectral measure
$\mu _{\alpha _1,\alpha _2,\beta }$
and boundary conditions
$\alpha _1$
,
$\alpha _2$
, and
$\beta $
uniquely determine the inner function
$\Theta _{\alpha _1,\alpha _2,\beta }$
and hence the m-function
$m_{\alpha _1,\alpha _2,\beta }$
. We know that
$m_{\alpha _1,\beta } = R_{\alpha _1}R^{-1}_{\alpha _1,\alpha _2}m_{\alpha _1,\alpha _2,\beta }$
, so
$m_{\alpha _1,\beta }$
is uniquely determined. Then we can follow the same steps we used in the proof of Theorem 2.8, starting at the unique determination of
$m_{\alpha ,\beta }$
step, and get the desired result.
Proof of Theorem 2.9
By Theorem 2.5 and the arguments we used at the beginning of the proof of Theorem 2.7, the given spectral data uniquely determine the MIF
$\Theta _{\alpha _1,\alpha _2,\beta }$
and hence the MHF
$m_{\alpha _1,\alpha _2,\beta }$
. Then we can follow the same steps we used in the proof of Theorem 2.7 and get the desired result.
Proof of Theorem 2.10
By Theorem 2.6 and the arguments we used at the beginning of the proof of Theorem 2.8, the given spectral data uniquely determine the MIF
$\Theta _{\alpha _1,\alpha _2,\beta }$
and hence the MHF
$m_{\alpha _1,\alpha _2,\beta }$
. Then we can follow the same steps we used in the proof of Theorem 2.7 and get the desired result.
Acknowledgements
Part of this work was conducted at Georgia Institute of Technology, where the author was a postdoc of Svetlana Jitomirskaya. The author thanks funding from the NSF DMS-2052899, DMS-2155211, and Simons 681675. The author also acknowledges the support of the NSF through grant DMS–2052519.
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