Uniqueness theorems for meromorphic inner functions and canonical systems

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 spectral theory of differential operators [BBP17, MP05], Krein-de Branges theory of entire functions [B68, MP010, R17], functional model theory [N86, N02], approximation theory [P015], Toeplitz operators [HM16, MP10, P18] and non-linear Fourier transform [P21]. MIFs also play a critical role in the Toeplitz approach to the uncertainty principle [MP05, P15], which was used in the study of problems of harmonic analysis [MP15, P12, P13].

Each MIF is represented by the Blaschke/singular factorization

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

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

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 $\mu_{\Theta}(t)=2\pi/|\Theta^{\prime}(t)|$ and the point mass at infinity is non-zero if and only if $\lim_{y\rightarrow+\infty}\Theta(iy)=1$ and the limit $\lim_{y\rightarrow+\infty}y^{2}\Theta^{\prime}(iy)$ exists. All of these above mentioned properties of MIFs can be found e.g in [P15] and references therein.

Existence, uniqueness, interpolation and other complex function theoretic problems for MIFs were studied in various papers [BCLRS16, P15, PR16, R17]. In this paper we consider unique determination of the MIF $\Theta$ from several 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 on $\mathbb{R}$, 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. 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 e.g. in [HAT] 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 [REM18], e.g. see Section 1.3 of [REM18] 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

where $L>0$, $x\in(0,L)$ and

We assume $H(x)\in\mathbb{R}^{2\times 2}$, $H\in L_{\text{loc}}^{1}(0,L)$ and $H(x)\geq 0$ a.e. on $(0,L)$. In this paper we consider the limit circle case, whixh means $\int_{0}^{L}\text{Tr}H(x)dx<\infty$, on a finite interval, i.e. $0<L<\infty$. The self-adjoint realizations of a canonical Hamiltonian system in the limit circle case with separated boundary conditions are described by

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 $Jf^{\prime}=-zHf$ with the boundary condition $f_{1}(L)\sin\beta-f_{2}(L)\cos\beta=0$. If we let $u$ and $v$ be solutions of $Ju^{\prime}=-zHu$ with the 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 $x=L$. The complex function $m_{\alpha,\beta}$ is called the Weyl $m$-function, which is

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 system.

The paper is organized as follows.

Section 2 includes 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 2.2, 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 2.3 shows that if $\sigma(\Theta)$ is bounded 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 2.4 and 2.5, we prove that the knowledge of the derivative values from Theorems 2.1 and 2.3 respectively, can be partially replaced by the knowledge of the corresponding points from the second spectrum $\sigma(-\Theta)$.

• •

  Theorem 2.6 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 2.7 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 2.8 and Theorem 2.9 we prove that some missing norming constants from the spectral data of Theorems 2.6 and 2.7 respectively, can be compensated by the corresponding data from a second spectrum.

• •

  As Corollary 2.10 we obtain a generalization of Borg-Levison’s classical two-spectra theorem on Schrödinger operators to canonical Hamiltonian systems.

Section 3 includes proofs of results for MIFs.

Section 4 includes proofs of results for canonical Hamiltonian systems.

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 notations. 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

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 [P15] 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 $l^{1}$, i.e.

In our first theorem, with condition (2.2), we consider unique determination of a MIF from its spectrum and derivative values on the spectrum.

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.

If the spectrum of a MIF is bounded below, then summability condition of Theorem 2.1 is not required.

Next two theorems show that missing part of point masses of the Clark measure from the data of Theorems 2.1 and 2.3 can be compensated by the corresponding data from the second spectrum $\sigma(-\Theta)$.

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.

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.

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.

To prove our uniqueness theorems, we use an infinite product representation result.

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.