The original Weyl-Titchmarsh functions and sectorial Schrödinger L-systems

This paper is a part of an ongoing project studying the realizations of the original Weyl-Titchmarsh function $m_{\infty}(z)$ and its linear-fractional transformation $m_{\alpha}(z)$ associated with a Schrödinger operator in $L_{2}[\ell,+\infty)$. In this project the Herglotz-Nevanlinna functions $(-m_{\infty}(z))$ and $(1/m_{\infty}(z))$ as well as $(-m_{\alpha}(z))$ and $(1/m_{\alpha}(z))$ are being realized as impedance functions of L-systems with a dissipative Schrödinger main operator $T_{h}$, ($\operatorname{Im}h>0$). For the sake of brevity we will refer to these L-systems as Schrödinger L-systems for the rest of the manuscript. The formal definition, exposition and discussions of general and Schrödinger L-systems are presented in Sections 2 and 4. We capitalize on the fact that all Schrödinger L-systems $\Theta_{\mu,h}$ form a two-parametric family whose members are uniquely defined by a real-valued parameter $\mu$ and a complex boundary value $h$ of the main dissipative operator.

The focus of this paper is set on the case when the realizing Schrödinger L-systems are based on non-negative symmetric Schrödinger operator with $(1,1)$ deficiency indices and have accretive state-space operator. It is known (see [2]) that in this case the impedance functions of such L-systems are Stieltjes. Here we study the situation when the realizing Schrödinger L-systems are also sectorial and the Weyl-Titchmarsh functions $(-m_{\alpha}(z))$ fall into sectorial classes $S^{\beta}$ and $S^{\beta_{1},\beta_{2}}$ of Stieltjes functions that are discussed in details in Section 3. Section 5 provides us with the general realization results (obtained in [7]) for the functions $(-m_{\infty}(z))$, $(1/m_{\infty}(z))$, and $(-m_{\alpha}(z))$. It is shown that $(-m_{\infty}(z))$, $(1/m_{\infty}(z))$, and $(-m_{\alpha}(z))$ can be realized as the impedance function of Schrödinger L-systems $\Theta_{0,i}$, $\Theta_{\infty,i}$, and $\Theta_{\tan\alpha,i}$, respectively.

The main results of the paper are contained in Section 6. Here we apply the realization theorems from Section 5 to Schrödinger L-systems that are based on non-negative symmetric Schrödinger operator to obtain additional properties. Utilizing the results presented in Section 4, we derive some new features of Schrödinger L-systems $\Theta_{\tan\alpha,i}$ whose impedance functions fall into particular sectorial classes $S^{\beta_{1},\beta_{2}}$ with $\beta_{1}$ and $\beta_{2}$ explicitly described. The results are given in terms of the parameter $\alpha$ that appears in the definition of the function $m_{\alpha}(z)$. Moreover, the knowledge of the limit value $m_{\infty}(-0)$ and the value of $\alpha$ allows us to find the angle of sectoriality of the main and state-space operators of the realizing L-system. This, in turn, leads to connections to Kato’s problem about sectorial extension of sectorial forms.

The paper is concluded with an example that illustrates main results and concepts. The present work is a further development of the theory of open physical systems conceived by M. Livs̆ic in [19].

For a pair of Hilbert spaces ${\mathcal{H}}_{1}$, ${\mathcal{H}}_{2}$ we denote by $[{\mathcal{H}}_{1},{\mathcal{H}}_{2}]$ the set of all bounded linear operators from ${\mathcal{H}}_{1}$ to ${\mathcal{H}}_{2}$. Let ${\dot{A}}$ be a closed, densely defined, symmetric operator in a Hilbert space ${\mathcal{H}}$ with inner product $(f,g),f,g\in{\mathcal{H}}$. Any non-symmetric operator $T$ in ${\mathcal{H}}$ such that ${\dot{A}}\subset T\subset{\dot{A}}^{*}$ is called a quasi-self-adjoint extension of ${\dot{A}}$.

Consider the rigged Hilbert space (see [11], [2]) ${\mathcal{H}}_{+}\subset{\mathcal{H}}\subset{\mathcal{H}}_{-},$ where ${\mathcal{H}}_{+}=\text{\rm{Dom}}({\dot{A}}^{*})$ and

Let ${\mathcal{R}}$ be the Riesz-Berezansky operator ${\mathcal{R}}$ (see [11], [2]) which maps $\mathcal{H}_{-}$ onto $\mathcal{H}_{+}$ such that $(f,g)=(f,{\mathcal{R}}g)_{+}$ ($\forall f\in{\mathcal{H}}_{+}$, $g\in{\mathcal{H}}_{-}$) and $\|{\mathcal{R}}g\|_{+}=\|g\|_{-}$. Note that identifying the space conjugate to ${\mathcal{H}}_{\pm}$ with ${\mathcal{H}}_{\mp}$, we get that if ${\mathbb{A}}\in[{\mathcal{H}}_{+},{\mathcal{H}}_{-}]$, then ${\mathbb{A}}^{*}\in[{\mathcal{H}}_{+},{\mathcal{H}}_{-}].$ An operator ${\mathbb{A}}\in[{\mathcal{H}}_{+},{\mathcal{H}}_{-}]$ is called a self-adjoint bi-extension of a symmetric operator ${\dot{A}}$ if ${\mathbb{A}}={\mathbb{A}}^{*}$ and ${\mathbb{A}}\supset{\dot{A}}$. Let ${\mathbb{A}}$ be a self-adjoint bi-extension of ${\dot{A}}$ and let the operator $\hat{A}$ in ${\mathcal{H}}$ be defined as follows:

The operator $\hat{A}$ is called a quasi-kernel of a self-adjoint bi-extension ${\mathbb{A}}$ (see [26], [2, Section 2.1]). A self-adjoint bi-extension ${\mathbb{A}}$ of a symmetric operator ${\dot{A}}$ is called t-self-adjoint (see [2, Definition 4.3.1]) if its quasi-kernel $\hat{A}$ is self-adjoint operator in ${\mathcal{H}}$. An operator ${\mathbb{A}}\in[{\mathcal{H}}_{+},{\mathcal{H}}_{-}]$ is called a quasi-self-adjoint bi-extension of an operator $T$ if ${\mathbb{A}}\supset T\supset{\dot{A}}$ and ${\mathbb{A}}^{*}\supset T^{*}\supset{\dot{A}}.$ We will be mostly interested in the following type of quasi-self-adjoint bi-extensions. Let $T$ be a quasi-self-adjoint extension of ${\dot{A}}$ with nonempty resolvent set $\rho(T)$. A quasi-self-adjoint bi-extension ${\mathbb{A}}$ of an operator $T$ is called (see [2, Definition 3.3.5]) a ($*$)-extension of $T$ if ${\rm Re\,}{\mathbb{A}}$ is a t-self-adjoint bi-extension of ${\dot{A}}$. In what follows we assume that ${\dot{A}}$ has deficiency indices $(1,1)$. In this case it is known [2] that every quasi-self-adjoint extension $T$ of ${\dot{A}}$ admits $(*)$-extensions. The description of all $(*)$-extensions via Riesz-Berezansky operator ${\mathcal{R}}$ can be found in [2, Section 4.3].

Recall that a linear operator $T$ in a Hilbert space ${\mathcal{H}}$ is called accretive [17] if ${\rm Re\,}(Tf,f)\geq 0$ for all $f\in\text{\rm{Dom}}(T)$. We call an accretive operator $T$ $\beta$-sectorial [17] if there exists a value of $\beta\in(0,\pi/2)$ such that

We say that the angle of sectoriality $\beta$ is exact for a $\beta$-sectorial operator $T$ if

An accretive operator is called extremal accretive if it is not $\beta$-sectorial for any $\beta\in(0,\pi/2)$. A $(*)$-extension ${\mathbb{A}}$ of $T$ is called accretive if ${\rm Re\,}({\mathbb{A}}f,f)\geq 0$ for all $f\in{\mathcal{H}}_{+}$. This is equivalent to that the real part ${\rm Re\,}{\mathbb{A}}=({\mathbb{A}}+{\mathbb{A}}^{*})/2$ is a nonnegative t-self-adjoint bi-extension of ${\dot{A}}$.

The following definition is a “lite” version of the definition of L-system given for a scattering L-system with one-dimensional input-output space. It is tailored for the case when the symmetric operator of an L-system has deficiency indices $(1,1)$. The general definition of an L-system can be found in [2, Definition 6.3.4] (see also [10] for a non-canonical version).

Operators $T$ and ${\mathbb{A}}$ are called a main and state-space operators respectively of the system $\Theta$, and $K$ is a channel operator. It is easy to see that the operator ${\mathbb{A}}$ of the system (3) is such that $\operatorname{Im}{\mathbb{A}}=(\cdot,\chi)\chi$, $\chi\in{\mathcal{H}}_{-}$ and pick $Kc=c\cdot\chi$, $c\in{\mathbb{C}}$ (see [2]). A system $\Theta$ in (3) is called minimal if the operator ${\dot{A}}$ is a prime operator in ${\mathcal{H}}$, i.e., there exists no non-trivial reducing invariant subspace of ${\mathcal{H}}$ on which it induces a self-adjoint operator. Minimal L-systems of the form (3) with one-dimensional input-output space were also considered in [6].

We associate with an L-system $\Theta$ the function

which is called the transfer function of the L-system $\Theta$. We also consider the function

that is called the impedance function of an L-system $\Theta$ of the form (3). The transfer function $W_{\Theta}(z)$ of the L-system $\Theta$ and function $V_{\Theta}(z)$ of the form (5) are connected by the following relations valid for $\operatorname{Im}z\neq 0$, $z\in\rho(T)$,

An L-system $\Theta$ of the form (3) is called an accretive L-system ([9], [14]) if its state-space operator operator ${\mathbb{A}}$ is accretive, that is ${\rm Re\,}({\mathbb{A}}f,f)\geq 0$ for all $f\in{\mathcal{H}}_{+}$. An accretive L-system is called sectorial if the operator ${\mathbb{A}}$ is sectorial, i.e., satisfies (2) for some $\beta\in(0,\pi/2)$ and all $f\in{\mathcal{H}}_{+}$.

A scalar function $V(z)$ is called the Herglotz-Nevanlinna function if it is holomorphic on ${{\mathbb{C}}\setminus{\mathbb{R}}}$, symmetric with respect to the real axis, i.e., $V(z)^{*}=V(\bar{z})$, $z\in{{\mathbb{C}}\setminus{\mathbb{R}}}$, and if it satisfies the positivity condition $\operatorname{Im}V(z)\geq 0$, $z\in{\mathbb{C}}_{+}$. The class of all Herglotz-Nevanlinna functions, that can be realized as impedance functions of L-systems, and connections with Weyl-Titchmarsh functions can be found in [2], [6], [13], [15] and references therein. The following definition can be found in [16]. A scalar Herglotz-Nevanlinna function $V(z)$ is a Stieltjes function if it is holomorphic in $\operatorname{Ext}[0,+\infty)$ and

It is known [16] that a Stieltjes function $V(z)$ admits the following integral representation

where $\gamma\geq 0$ and $G(t)$ is a non-decreasing on $[0,+\infty)$ function such that $\int^{\infty}_{0}\frac{dG(t)}{1+t}<\infty.$ We are going to focus on the class $S_{0}(R)$ (see [9], [14], [2]) of scalar Stieltjes functions such that the measure $G(t)$ in representation (7) is of unbounded variation. It was shown in [2] (see also [9]) that such a function $V(z)$ can be realized as the impedance function of an accretive L-system $\Theta$ of the form (3) with a densely defined symmetric operator if and only if it belongs to the class $S_{0}(R)$.

Now we are going to consider sectorial subclasses of scalar Stieltjes functions introduced in [1]. Let $\beta\in(0,\frac{\pi}{2})$. Sectorial subclasses $S^{\beta}$ of Stieltjes functions are defined as follows: a scalar Stieltjes function $V(z)$ belongs to $S^{\beta}$ if

for an arbitrary sequences of complex numbers $\{z_{k}\}$, ($\operatorname{Im}z_{k}>0$) and $\{h_{k}\}$, ($k=1,...,n$). For $0<\beta_{1}<\beta_{2}<\frac{\pi}{2}$, we have

where $S$ denotes the class of all Stieltjes functions (which corresponds to the case $\beta=\frac{\pi}{2}$). Let $\Theta$ be a minimal L-system of the form (3) with a densely defined non-negative symmetric operator ${\dot{A}}$. Then (see [2]) the impedance function $V_{\Theta}(z)$ defined by (5) belongs to the class $S^{\beta}$ if and only if the operator ${\mathbb{A}}$ of the L-system $\Theta$ is $\beta$-sectorial.

Let $0\leq\beta_{1}<\frac{\pi}{2}$, $0<\beta_{2}\leq\frac{\pi}{2}$, and $\beta_{1}\leq\beta_{2}$. We say that a scalar Stieltjes function $V(z)$ belongs to the class $S^{\,\beta_{1},\beta_{2}}$ if

The following connection between the classes $S^{\,\beta}$ and $S^{\,\beta_{1},\beta_{2}}$ can be found in [2]. Let $\Theta$ be an L-system of the form (3) with a densely defined non-negative symmetric operator ${\dot{A}}$ with deficiency numbers $(1,1).$ Let also ${\mathbb{A}}$ be an $\beta$-sectorial $(*)$-extension of $T$. Then the impedance function $V_{\Theta}(z)$ defined by (5) belongs to the class $S^{\beta_{1},\beta_{2}}$, $\tan\beta_{2}\leq\tan\beta$. Moreover, the main operator $T$ is $(\beta_{2}-\beta_{1})$-sectorial with the exact angle of sectoriality $(\beta_{2}-\beta_{1})$. In the case when $\beta$ is the exact angle of sectoriality of the operator $T$ we have that $V_{\Theta}(z)\in S^{0,\beta}$ (see [2]). It also follows that under this set of assumptions, the impedance function $V_{\Theta}(z)$ is such that $\gamma=0$ in representation (7).

Now let $\Theta$ be an L-system of the form (3), where ${\mathbb{A}}$ is a $(*)$-extension of $T$ and ${\dot{A}}$ is a closed densely defined non-negative symmetric operator with deficiency numbers $(1,1).$ It was proved in [2] that if the impedance function $V_{\Theta}(z)$ belongs to the class $S^{\beta_{1},\beta_{2}}$ and $\beta_{2}\neq\pi/2$, then ${\mathbb{A}}$ is $\beta$-sectorial, where

Under the above set of conditions on L-system $\Theta$, it is shown in [2] that ${\mathbb{A}}$ is $\beta$-sectorial $(*)$-extension of an $\beta$-sectorial operator $T$ with the exact angle $\beta\in(0,\pi/2)$ if and only if $V_{\Theta}(z)\in S^{0,\beta}$. Moreover, the angle $\beta$ can be found via the formula

where $G(t)$ is the measure from integral representation (7) of $V_{\Theta}(z)$.

Let ${\mathcal{H}}=L_{2}[\ell,+\infty)$, $\ell\geq 0$, and $l(y)=-y^{\prime\prime}+q(x)y$, where $q$ is a real locally summable on $[\ell,+\infty)$ function. Suppose that the symmetric operator

has deficiency indices (1,1). Let $D^{*}$ be the set of functions locally absolutely continuous together with their first derivatives such that $l(y)\in L_{2}[\ell,+\infty)$. Consider ${\mathcal{H}}_{+}=\text{\rm{Dom}}({\dot{A}}^{*})=D^{*}$ with the scalar product

Let ${\mathcal{H}}_{+}\subset L_{2}[\ell,+\infty)\subset{\mathcal{H}}_{-}$ be the corresponding triplet of Hilbert spaces. Consider the operators

where $\operatorname{Im}h>0$. Let ${\dot{A}}$ be a symmetric operator of the form (12) with deficiency indices (1,1), generated by the differential operation $l(y)=-y^{\prime\prime}+q(x)y$. Let also $\varphi_{k}(x,\lambda)(k=1,2)$ be the solutions of the following Cauchy problems:

It is well known [20], [18] that there exists a function $m_{\infty}(\lambda)$ introduced by H. Weyl [27], [28] for which

belongs to $L_{2}[\ell,+\infty)$. The function $m_{\infty}(\lambda)$ is not a Herglotz-Nevanlinna function but $(-m_{\infty}(\lambda))$ and $(1/m_{\infty}(\lambda))$ are.

Now we shall construct an L-system based on a non-self-adjoint Schrödinger operator $T_{h}$ with $\operatorname{Im}h>0$. It was shown in [4], [2] that the set of all ($*$)-extensions of a non-self-adjoint Schrödinger operator $T_{h}$ of the form (13) in $L_{2}[\ell,+\infty)$ can be represented in the form

Moreover, the formulas (14) establish a one-to-one correspondence between the set of all ($*$)-extensions of a Schrödinger operator $T_{h}$ of the form (13) and all real numbers $\mu\in[-\infty,+\infty]$. One can easily check that the ($*$)-extension ${\mathbb{A}}$ in (14) of the non-self-adjoint dissipative Schrödinger operator $T_{h}$, ($\operatorname{Im}h>0$) of the form (13) satisfies the condition

where

and $\delta(x-\ell),\delta^{\prime}(x-\ell)$ are the delta-function and its derivative at the point $\ell$, respectively. Furthermore,

where $y\in{\mathcal{H}}_{+}$, $g\in{\mathcal{H}}_{-}$, and ${\mathcal{H}}_{+}\subset L_{2}[\ell,+\infty)\subset{\mathcal{H}}_{-}$ is the triplet of Hilbert spaces discussed above.

It was also shown in [2] that the quasi-kernel $\hat{A}_{\xi}$ of ${\rm Re\,}{\mathbb{A}}_{\mu,h}$ is given by

Let $E={\mathbb{C}}$, $K_{\mu,h}{c}=cg_{\mu,h},\;(c\in{\mathbb{C}})$. It is clear that

and $\operatorname{Im}{\mathbb{A}}_{\mu,h}=K_{\mu,h}K^{*}_{\mu,h}.$ Therefore, the array

is an L-system with the main operator $T_{h}$, ($\operatorname{Im}h>0$) of the form (13), the state-space operator ${\mathbb{A}}_{\mu,h}$ of the form (14), and with the channel operator $K_{\mu,h}$ of the form (17). It was established in [4], [2] that the transfer and impedance functions of $\Theta_{\mu,h}$ are

and

It was shown in [2, Section 10.2] that if the parameters $\mu$ and $\xi$ are related via (16), then the two L-systems $\Theta_{\mu,h}$ and $\Theta_{\xi,h}$ of the form (18) have the following property

It is known [18], [20] that the original Weyl-Titchmarsh function $m_{\infty}(z)$ has a property that $(-m_{\infty}(z))$ is a Herglotz-Nevanlinna function. The question whether $(-m_{\infty}(z))$ can be realized as the impedance function of a Schrödinger L-system is answered in the following theorem that was proved in [7].

A similar result for the function $1/m_{\infty}(z)$ was also proved in [7].

We note that both L-systems $\Theta_{0,i}$ and $\Theta_{\infty,i}$ obtained in Theorems 2 and 3 share the same main operator

Now we recall the definition of Weyl-Titchmarsh functions $m_{\alpha}(z)$. Let ${\dot{A}}$ be a symmetric operator of the form (12) with deficiency indices (1,1), generated by the differential operation $l(y)=-y^{\prime\prime}+q(x)y$. Let also $\varphi_{\alpha}(x,{z})$ and $\theta_{\alpha}(x,{z})$ be the solutions of the following Cauchy problems:

It is known [12], [20], [21] that there exists an analytic in ${\mathbb{C}}_{\pm}$ function $m_{\alpha}({z})$ for which

belongs to $L_{2}[\ell,+\infty)$. It is easy to see that if $\alpha=\pi$, then $m_{\pi}({z})=m_{\infty}({z})$. The functions $m_{\alpha}({z})$ and $m_{\infty}(z)$ are connected (see [12], [21]) by

We know [20], [21] that for any real $\alpha$ the function $-m_{\alpha}({z})$ is a Herglotz-Nevanlinna function. Also, modifying (24) slightly we obtain

The following realization theorem (see [7]) for Herglotz-Nevanlinna functions $-m_{\alpha}(z)$ is similar to Theorem 2.

We note that when $\alpha=\pi$ we obtain $\mu_{\alpha}=0$, $m_{\pi}(z)=m_{\infty}(z)$, and the realizing Schrödinger L-system $\Theta_{0,i}$ is thoroughly described in [7, Section 5]. If $\alpha=\pi/2$, then we get $\mu_{\alpha}=\infty$, $-m_{\alpha}(z)=1/m_{\infty}(z)$, and the realizing Schrödinger L-system is $\Theta_{\infty,i}$ (see [7, Section 5]). Assuming that $\alpha\in(0,\pi]$ and neither $\alpha=\pi$ nor $\alpha=\pi/2$ we give the description of a Schrödinger L-system $\Theta_{\mu_{\alpha},i}$ realizing $-m_{\alpha}(z)$ as follows.

where

$K_{\tan\alpha,i}\,{c}=c\,g_{\tan\alpha,i}$, $(c\in{\mathbb{C}})$ and

Also,

The realization theorem for Herglotz-Nevanlinna functions $1/m_{\alpha}(z)$ is similar to Theorem 3 and can be found in [7].

Now let us assume that ${\dot{A}}$ is a non-negative (i.e., $({\dot{A}}f,f)\geq 0$ for all $f\in\text{\rm{Dom}}({\dot{A}})$) symmetric operator of the form (12) with deficiency indices (1,1), generated by the differential operation $l(y)=-y^{\prime\prime}+q(x)y$. The following theorem takes place.