Friedrichs extensions for Sturm–Liouville operators with complex coefficients and their spectra

Abstract

In this paper, we study the Friedrichs extensions of Sturm–Liouville operators with complex coefficients according to the classification of B. M. Brown et al. [3]. We characterize the Friedrichs extensions both by boundary conditions at regular endpoint and asymptotic behaviours of elements in the maximal operator domains at singular endpoint. Some of spectral properties are also involved.

1. Introduction

Consider the Sturm–Liouville differential equation

(1.1)\begin{equation} \tau y:=\frac{1}{w}[-(py')'+qy]=\lambda y\ \text{on}\,[a, b), \end{equation}

where $p(x), q(x)$ are complex functions with $p(x)\neq 0$ a.e. in $[a, b)$, and $w(x)>0$ on $[a, b)$, $1/p, q, w$ are all locally integrable on $[a, b)$, $-\infty < a< b\le +\infty$, $\lambda$ is the so-called spectral parameter. The assumptions on $p, q, w$ ensure that $a$ is a regular endpoint of the equation $\tau y=\lambda y$, and $b$ is a singular endpoint, i.e., at least one of $b=\infty$ or $\int _a^b\left (w+\frac {1}{|p|}+|q|\right ){\rm d}x=\infty$ holds. However, we note that the regular endpoint $b$ is included in the analysis.

Spectral theory of differential operators is one of the hot research branches in differential equations (see the classical books [1, 5, 6, 9]). Among these researches, the Friedrichs extension plays an important role in the spectral analysis of differential operators. For a symmetric differential operator with real coefficients which is lower semi-bounded, the Friedrichs extension is a particular self-adjoint extension which preserves the lower bound of the given minimal differential operator. Suitable boundary value conditions are added on the endpoints to make the extension be the Friedrichs extension. Following this line, H. G. Kaper, M. K. Kwong and A. Zettl [13], M. Moller and A. Zettl [19], H. D. Niessen, A. Zettl [20] gave the characterization of the Friedrichs extensions for singular Sturm–Liouville differential operators, similar results are obtained by M. Moller and A. Zettl for singular $2n$ order differential operator, singular block operator matrices by A. Konstantinov and R. Mennicken [18], Schördinger operator by A. B. Keviczky, N. Saad and R. L. Hall [15], and for singular Hamiltonian operators with one singular endpoint by [10, 28–30] with the endpoint being limit-point case, limit-circle case and intermediate deficiency indices, respectively.

The theory for Sturm–Liouville theory for differential expressions of second order with complex potentials is currently of great interest as there are applied to so-called ${\mathcal {P}T}$ -symmetric quantum mechanics in theoretical physics (see [2, 3, 4, 21, 23] and references cited therein).

When $p(x)\equiv 1$ and $q(x)$ is a complex valued function with Im$q(x)$ being semi-bounded, A. R.  Sims in his seminal paper (see [23] for details) extended the famous limit point/limit circle classification of H. Weyl [27] to the case of complex coefficients. Since the restriction Im$q(x)$ being semi-bounded, it is too sharp to reduce its applications. The restriction was relaxed in the paper of B. M. Brown, D. K. R. McCormack, W. D. Evans, M. Plum [3]. However, the classification in [3] formally depends on the choice of rotation angles (or the half-planes). A new classification of equation (1.1) which is independent of the rotation angles (or the half-planes) was given by J. Qi, Z. Zheng and H. Sun [21]. Moreover, the J-self-adjoint realization was characterized by boundary conditions in [21].

Unlike Sturm–Liouville operators with real coefficients, the spectral theories of Sturm–Liouville operators with complex coefficients are not fully investigated. In this paper, we will give the Friedrichs extension without the restriction on $q(x)$, and $p(x)$ is an arbitrarily complex function. More concise, we obtain the Friedrichs extension domain not by imposing boundary condition on the singular end-point, but by asymptotic behaviours of elements in the maximal operator domains at singular endpoint. The spectral properties of the Friedrichs extension are also given.

This paper is organized as follows. In §2, we give some preliminaries on differential operators with complex coefficients, the characterizations of Friedrichs extensions under case I and case II are given separately in §3 and 4. Section 5 deals with the spectral properties of the Friedrichs extensions.

2. Preliminaries

In this paper, we consider the second order Sturm–Liouville equation

(2.1)\begin{equation} \tau y:=\frac{1}{w}[-(py')'+qy]=\lambda y\ \text{on}\,[a, b), \end{equation}

where $p(x), q(x)$ are complex functions with $p(x)\neq 0$ a.e. in $[a, b)$, and $w(x)>0$ on $[a, b)$, $1/p, q, w$ are all locally integrable on $[a, b)$, $-\infty < a< b\le +\infty$, $\lambda$ is the spectral parameter.

By $L_w^2[a, b)$, we mean the Hilbert space defined by

\[ L^2_w[a, b):=\{y:\,(a, b)\to\mathbb{C}\,{\rm is\ measurable}: \int_{a}^{b} w(x)|y(x)|^2\,{\rm d}x<\infty\} \]

with inner product $\langle y, z\rangle :=\int _{a}^{b}\bar {z}(x)w(x)y(x)\,{\rm d}x$ and the norm $\|y\|=(\langle y, y\rangle )^{1/2}$ for $y, z\in L^2_w[a, b)$. Here $w(x)$ is called the weight function. Similar Hilbert space can be defined by replacing $w(x)$ with other positive weight functions, such as $L^2_{|p|}[a, b)$ in §4.

2.1 Classification of (1.1) and corresponding operators

In paper [3], the hypothesis

(2.2)\begin{equation} \Omega=\overline{\rm co}\left\{\frac{q(x)}{w(x)}+rp(x):\,r>0, x\in(a, b)\right\}\neq\mathbb{C} \end{equation}

is introduced, where ${\overline {\rm co}}$ denotes the closed convex hull (i.e., the smallest closed convex set containing the exhibited set). Then for each point on the boundary $\partial \Omega$, there exists a line through this point such that each point of $\Omega$ either lies in the same side of this line or is on it. That is, there exists a supporting line through this point. Let $K$ be a point on $\partial \Omega$. Denote by $L$ an arbitrary supporting line touching $\Omega$ at $K$, which may be the tangent to $\Omega$ at $K$ if it exists. We then perform a transformation of the complex plane $z\mapsto z-K$ and a rotation through an appropriate angle $\theta \in (-\pi, \pi ]$, so that the image of $L$ now coincides with the new imaginary axis and the set $\Omega$ lies in the new right nonnegative half-plane. Therefore, for all $x\in (a, b)$ and $0 < r<\infty$,

(2.3)\begin{gather} {\rm Re}\left\{{\rm e}^{i\theta}\left[\frac{q(x)}{w(x)}+rp(x)-K\right]\right\}\geq 0. \end{gather}

(2.4)\begin{gather} {\rm Re}\left[{\rm e}^{i\theta} (\lambda-K)\right]\geq 0. \end{gather}

For convenience, we define all such admissible values of $K$ and $\theta$ by $\Pi$, i.e.,

\[ \Pi=\left\{(\theta, K):\,\theta\in(-\pi, \pi], K\in\partial\Omega, {\rm Re}\left\{{\rm e}^{i\theta}(\mu-K)\right\}\ge 0\ \text{for all} \mu\in\Omega\right\}, \]

and we define

(2.5)\begin{equation} E=\left\{\theta\in(-\pi, \pi]:\exists K\in\partial\Omega, (\theta, K)\in\Pi\right\}. \end{equation}

Note that for fixed $\theta _0\in E$, the $K$ such that $(\theta _0, K)\in \Pi$ may be not unique.

Since $\Omega \neq \mathbb {C}$ is convex and closed space, one sees that $\Pi, E\neq \emptyset$, and if we define

(2.6)\begin{equation} \Lambda_{\theta, \,K}=\left\{\mu\in\mathbb{C}:\,{\rm Re}\left\{{\rm e}^{i\theta}(\mu-K)\right\}<0\right\}, \end{equation}

then,

\[ \mathbb{C}\backslash \Omega=\bigcup_{(\theta, K)\in\Pi}\Lambda_{\theta, \,K}. \]

Note that each $\Lambda _{\theta, \,K}$ is a half plane. Then $\Lambda _{\theta _1, \,K_1}\cap \Lambda _{\theta _2, \,K_2}\neq \emptyset$ for $\theta _1\ne \theta _2({\rm mod}\,\pi )$.

Let $r\to 0$ and $r\to \infty$ in (2.3), respectively, we have the following lemma.

Lemma 2.1 For each $(\theta, K)\in \Pi$ and $\lambda \in \Lambda _{\theta, K}$ there exists $\delta _\lambda >0$ such that

(2.7)\begin{equation} {\rm Re}\left\{{\rm e}^{i\theta}(q-K\,w))\right\}\ge0, \ {\rm Re}\left\{{\rm e}^{i\theta}(q-\lambda\,w)\right\}\ge\delta_\lambda w, \ {\rm Re}\left\{{\rm e}^{i\theta}p\right\}\ge0 \end{equation}

on $[a, b)$.

With these definitions and the similar line of H. Weyl's in [27], B. M. Brown et al. [3] divided (1.1) into three cases with respect to the corresponding half-planes $\Lambda _{\theta, \,K}$ as follows.

Theorem 2.2 (see [[3], theorem 2.1])

For $\lambda \in \Lambda _{\theta, \,K}$, $(\theta, K)\in \Pi (\alpha )$ (where $\alpha$ is related to the boundary conditions on regular endpoint), the Weyl circles converge either to a limit-point $m(\lambda )$ or a limit-circle $C_b(\lambda )$. The following distinct cases are possible, the first two cases being sub-cases of the limit-point case.

1. (I) there exists a unique solution $y$ of equation (1.1) satisfying

   (2.8)\begin{equation} \int_a^b\left[{\rm Re}\left\{{\rm e}^{i\theta}p\right\}|y'|^2 +{\rm Re}\left\{{\rm e}^{i\theta}(q-K\,w)\right\}|y|^2\right] +\int_a^bw\,|y|^2<\infty \end{equation}
   and this is the only one satisfying $y\in L^2_w[a, b)$;

2. (II) there exists a unique solution of equation (1.1) satisfying (2.8), but all solutions of (1.1) belong to $L^2_w[a, b)$.

3. (III) all solutions of (1.1) satisfy (2.8).

Remark 2.3 If $q(x)$ and $p(x)$ are real-valued, then $\Omega \subset \mathbb {R}$ and $(\theta, K)=(\pi /2, 0)\in \Pi$, and hence ${\rm Re}\left \{{\rm e}^{i\theta }p(x)\right \}={\rm Re}\left \{{\rm e}^{i\theta }(q(x)-K\,w)\right \}\equiv 0$. So case II is vacuous. This means that the classification mentioned above reduces to Weyl's limit-point, limit-circle classification.

Since $E$ is the set of rotation angles, in what follows, by a point $\theta \in E$, we denote the collection of points in the sense of $\theta$ module $\pi$. If $E$ has only one point, then the classification of Brown et al. in theorem 2.2 is independent of the choice of $(\theta, K)\in \Pi$. Using variation of parameters formula, we can verify that if all solutions of (1.1) belong to $L^2_w[a, b)$ for some $\lambda _0\in \mathbb {C}$, then it is true for all $\lambda \in \mathbb {C}$. This also means that case I is independent of the choice of $(\theta, K)\in \Pi$, too. However, if $E$ has more than one point, cases II and III depend on the choice of $(\theta, K)\in \Pi$ in general since the rotation angle $\theta$ lies in (2.8). The exact dependence of cases II and III on $(\theta, K)$ is given in [24].

Theorem 2.4 If there exists a $(\theta _0, K_0)\in \Pi$ such that (1.1) is in case II w.r.t. $\Lambda _{\theta _0, K_0}$, then (1.1) is in case II w.r.t. $\Lambda _{\theta, K}$ for all $(\theta, K)\in \Pi$ except for at most one $\theta _1\in E$ (in the sense of mod $\pi$) such that (1.1) is in case III w.r.t. $\Lambda _{\theta _1, K_1}$, where $(\theta _1, K_1)\in \Pi$.

Theorem 2.4 indicates that if there exist $\theta _1, \theta _2\in E$ such that $\theta _1\ne \theta _2$ (mod $\pi$) and (1.1) is in case III w.r.t. $\Lambda _{\theta _j, K_j}$ for $j=1, 2$, then (1.1) is in case III w.r.t. $\Lambda _{\theta, K}$ for all $(\theta, K)\in \Pi$.

In what follows, we always assume that $E$ has more than one point. Firstly, we prepare some properties of the set $E$.

Lemma 2.5 Let $E$ be defined as in (2.5).

1. (i) If $E$ has more than one point, then $E$ is a sub-interval of $(-\pi, \pi ]$.

2. (ii) If $E$ has more than one point, then for each $\lambda \in \mathbb {C}\setminus \Omega$, there exist $\theta _1, \theta _2\in E$ with $\theta _1<\theta _2$ such that for $\theta \in (\theta _1, \theta _2)\subset E$, $\lambda \in \Lambda _{\theta, \,K}$, where $(\theta, K)\in \Pi$.

Proof. (i) Let $\theta _1, \theta _2\in E$ with $\theta _1\ne \theta _2$ (mod $\pi$), $\theta _1<\theta _2$ and $K_1, K_2$ be the points on $\partial \Omega$ such that $(\theta _j, K_j)\in \Pi$, $j=1, 2$. We claim that $[\theta _1, \theta _2]\subset E$. For the case $K_1=K_2=K$, we prove that $(\theta, K)\in \Pi$ for all $\theta \in (\theta _1, \theta _2)$. Set

\[ \left\{\begin{array}{@{}l} \dfrac{q(x)}{w(x)}+rp(x)-K=R(x, r, K)\,{\rm e}^{i\gamma(x, r, K)}, \ R(x, r, K)=\left|\dfrac{q(x)}{w(x)}+rp(x)-K\right|, \\ \gamma_j(x, r, K)=\gamma(x, r, K)+\theta_j, \ r>0, \ x\in(a, b), \ j=1, 2. \end{array}\right. \]

It follows from the definition of $\Pi$ that for $j=1, 2$ and $r\ge 0$,

\[ {\rm Re}\left\{{\rm e}^{i\theta_j}\left(\frac{q(x)}{w(x)}+rp(x)-K\right)\right\}\ge 0 \]

on $(a, b)$, or equivalently, $\cos \gamma _j(x, r, K)\ge 0$. Without any confusion, we write $\gamma (x, r, K)$ (resp. $\gamma _j(x, r, K)$) as $\gamma$ (resp. $\gamma _j$). If we set

\[ J=\sin(\theta_2-\theta_1)\ne 0, \ J_1(x)=\left|\begin{array}{@{}cc@{}} \cos\gamma_1 & \sin\theta_1\\ \cos\gamma_2 & \sin\theta_2 \end{array}\right|, \ J_2(x)=\left|\begin{array}{@{}cc@{}} \cos\gamma_1 & \cos\theta_1\\ \cos\gamma_2 & \cos\theta_2 \end{array}\right|, \]

then $\cos \gamma$ and $\sin \gamma$ can be expressed as

(2.9)\begin{equation} \cos\gamma=\frac{J_1(x)}{J}, \ \sin\gamma=\frac{J_2(x)}{J} \end{equation}

by using the formulae $\cos \gamma _j=\cos \gamma \cos \theta _j-\sin \gamma \sin \theta _j$ for $j=1, 2$. This equality gives that

(2.10)\begin{align} \cos(\theta+\gamma)& =\cos\gamma\cos\theta-\sin\gamma\sin\theta =\frac{J_1}{J}\cos\theta-\frac{J_2}{J}\sin\theta\nonumber\\ & =\frac{1}{J}\left[\cos\gamma_1\sin(\theta_2-\theta) +\cos\gamma_2\sin(\theta-\theta_1)\right] \end{align}

for $\theta \in (\theta _1, \theta _2)$. Since $\cos \gamma _j\ge 0$ for $j=1, 2$ implies $-\pi /2\le \gamma _1, \gamma _2\le \pi /2({\rm mod}\ 2\pi )$, we have from $\pi \ge \theta _2>\theta _1>-\pi$, $\theta _2-\theta _1\ne \pi$ and $\theta _2-\theta _1=\gamma _2-\gamma _1$ that $0<\theta _2-\theta _1<\pi$. Consequently, for $\theta \in (\theta _1, \theta _2)$

\[ 0<\theta_2-\theta, \ \theta-\theta_1<\theta_2-\theta_1<\pi. \]

Therefore, each term in the right-hind side of (2.10) is nonnegative, so $\cos (\theta +\gamma )\ge 0$ on $(a, b)$. That is, for $r\ge 0$ and $x\in (a, b)$,

\[ {\rm Re}\left\{{\rm e}^{i\theta}\left(\frac{q(x)}{w(x)}+rp(x)-K\right)\right\}=R(x, r, K)\cos(\gamma(x, r, K)+\theta)\ge 0, \]

which implies $(\theta, K)\in \Pi$ for $\theta \in (\theta _1, \theta _2)$.

In case $K_1\ne K_2$, we choose $\mu _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. Then it holds that

\[ {\rm Re}\left\{{\rm e}^{i\theta_j}\left(\frac{q(x)}{w(x)} +rp(x)-\mu_0\right)\right\}=R(x, r, \mu_0)\cos(\gamma(x, r, \mu_0)+\theta_j)>0 \]

on $(a, b)$ for $j=1, 2$ and $r\ge 0$ by the definition of $\Lambda _{\theta, K}$, or $\cos (\gamma (x, r, \mu _0)+\theta _j)>0$. Then, the similar proof in (2.9) and (2.10) yields that

(2.11)\begin{equation} {\rm Re}\left\{{\rm e}^{i\theta}\left(\frac{q(x)}{w(x)} +rp(x)-\mu_0\right)\right\}>0, \ \theta\in(\theta_1, \theta_2). \end{equation}

Let $L$ be the line defined by

(2.12)\begin{equation} L=\{\lambda\in\mathbb{C}:\ {\rm Re}\left\{{\rm e}^{i\theta}(\lambda-\mu_0)\right\}=0\} \end{equation}

for fixed $\theta \in (\theta _1, \theta _2)$. One sees from (2.11), (2.12) and the definition of $\Omega$ that $L\subset \mathbb {C}\setminus \Omega$. Set $d={\rm dist}(L, \partial \Omega )$ and let $K\in \partial \Omega$ be a point such that $d={\rm dist}(K, L)$. Since

(2.13)\begin{equation} {\rm dist}(\mu, L)={\rm Re}\left\{{\rm e}^{i\theta}(\mu-\mu_0)\right\}, \ \mu\in\Omega, \end{equation}

we have that ${\rm Re}\left \{{\rm e}^{i\theta }(\mu -\mu _0)\right \}={\rm dist}(\mu, L)\ge {\rm dist}(K, L)={\rm Re}\left \{{\rm e}^{i\theta }(K-\mu _0)\right \}$ for $\mu \in \Omega$, hence ${\rm Re}\left \{{\rm e}^{i\theta }(\mu -K)\right \} ={\rm Re}\left \{{\rm e}^{i\theta }(\mu -\mu _0)\right \}-{\rm Re}\left \{{\rm e}^{i\theta }(K-\mu _0)\right \}\ge 0$, or

\[ {\rm Re}\left\{{\rm e}^{i\theta}\left(\frac{q(x)}{w(x)}+rp(x)-K\right)\right\}\ge0 \]

on $(a, b)$ for $r\ge 0$ and $\theta \in (\theta _1, \theta _2)$, which means $(\theta, K)\in \Pi$, or $\theta \in E$. This proves I of lemma 2.5.

(ii) For $\lambda _0\in \mathbb {C}\setminus \Omega$, choose $(\theta _0, K_0)\in \Pi$ and $\delta _0>0$ such that $\lambda _0\in \Lambda _{\theta _0, K_0}$ and

\[ {\rm Re}\left\{{\rm e}^{i\theta_0}(K_0-\lambda_0)\right\}=\delta_0>0. \]

Since $E$ has more than one point, we can choose $\widetilde {\theta }\in E$ such that $\widetilde {\theta }\ne \theta _0 ({\rm mod}\,\pi )$. Without loss of generality, we suppose that $\widetilde {\theta }>\theta _0$. It follows from the conclusion of (i) that $(\theta _0, \widetilde {\theta })\subset E$. For each $\theta \in (\theta _0, \widetilde {\theta })$, there exists a point $K(\theta )\in \partial \Omega$ such that $(\theta, K(\theta ))\in \Pi$. By the definition of $\Pi$ we have that

(2.14)\begin{equation} {\rm Re}\left\{{\rm e}^{i\theta_0}(K(\theta)-K_0)\right\}\ge0, \ {\rm Re}\left\{{\rm e}^{i\theta}(K_0-K(\theta))\right\}\ge0. \end{equation}

If we set $r(\theta )=|K(\theta )-K_0|$ and $K(\theta )-K_0=r(\theta )\,{\rm e}^{i\eta (\theta )}$, then (2.14) means that

\[ \cos(\theta_0+\eta(\theta))\ge 0, \ \cos(\theta+\eta(\theta))\le 0. \]

This together with $\theta >\theta _0$ gives that $\theta +\eta (\theta )\ge \pi /2\ge \theta _0+\eta (\theta )$ (mod $2\pi$), hence $\theta _0+\eta (\theta )\to \pi /2$ (mod $2\pi )$ as $\theta \to \theta _0+0$. We claim that $r(\theta )$ is bounded in a right-neighbourhood of $\theta _0$. Suppose on the contrary, there exists a sequence, say, $\{\theta _n\}$ such that $\theta _n\to \theta _0+0$ and $r_n=r(\theta _n)\to +\infty$ as $n\to \infty$. Choose $\eta _0\in (\theta _0, \widetilde {\theta })$ such that $\eta _0-\theta _0+\pi /2<\pi$ and a corresponding point $K(\eta _0)\in \partial \Omega$ such that $(\eta _0, K(\eta _0))\in \Pi$, we have that

\begin{align*} 0& \le\frac{1}{r_n}{\rm Re}\left\{{\rm e}^{i\eta_0} (K(\theta_n)-K(\eta_0))\right\} =\frac{1}{r_n}{\rm Re}\left\{{\rm e}^{i\eta_0} \left(K(\theta_n)-K_0-[K(\eta_0)-K_0]\right)\right\}\\ & ={\rm Re}\left\{{\rm e}^{i\eta_0}\left({\rm e}^{i\eta(\theta_n)} -[K(\eta_0)-K_0]/r_n\right)\right\}\to\cos(\eta_0-\theta_0+\pi/2)<0, \end{align*}

which is a contradiction. Since $r(\theta )$ is bounded and $\theta +\eta (\theta )\to \pi /2$ (mod $2\pi$) as $\theta \to \theta _0+0$, we have that ${\rm Re}\left \{{\rm e}^{i\theta }(K_0-K(\theta ))\right \} =-r(\theta )\cos (\theta +\eta (\theta ))\to 0$ as $\theta \to \theta _0+0$. Hence

\begin{align*} {\rm Re}\left\{{\rm e}^{i\theta}(\lambda_0-K(\theta))\right\} & ={\rm Re}\left\{{\rm e}^{i\theta}(\lambda_0-K_0)\right\} +{\rm Re}\left\{{\rm e}^{i\theta}(K_0-K(\theta))\right\}\\ & \to{\rm Re}\left\{{\rm e}^{i\theta_0}(\lambda_0-K_0)\right\}={-}\delta_0<0 \end{align*}

as $\theta \to \theta _0+0$. Therefore, there exists $\xi \in (\theta _0, \widetilde {\theta })$ such that for all $\theta \in (\theta _0, \xi )$, ${\rm Re}\left \{{\rm e}^{i\theta }(\lambda _0-K(\theta ))\right \}<0$. This means that $\lambda _0\in \Lambda _{\theta, \,K}$ for $\theta \in (\theta _0, \xi )$. This completes the proof.

Let $T$ be an operator in Hilbert space $H$, the numerical range of operator $T$ is denoted by $\Theta (T)$ as follows:

\[ \Theta(T):=\left\{\frac{(Tu, u)}{(u, u)}\, :\, 0\neq u\in D(T)\right\}. \]

Let $\sigma \in (0, \frac \pi 2)$ and let $S_{\sigma }$ denote the closed sector

\[ S_{\sigma}:=\{0\}\cup\{z\in \mathbb{C}: |\arg(z)|\leq \sigma\}, \]

in the right complex half-plane, cf. figure 1.

FIGURE 1. A sector $S_{\sigma }$.

An operator $T$ in Hilbert space $H$ is said to be accretive if the numerical range $\Theta (T)$ is a subset of the right half-plane, that is, if

\[ {\rm Re}(Tu, u)\geq 0 \mathrm{for\ all}\ u\in D(T). \]

If $T$ is closed, then def$(T-\zeta )=\mu$ is constant for Re$\zeta <0$. If $\mu =0$, the left open half-plane is contained in the resolvent set $P(T)$ with

(2.15)\begin{equation} \left.\begin{array}{l} (T+\lambda)^{{-}1}\in\mathcal{B}(H), \\ \|(T+\lambda)^{{-}1}\|\leq {\mathrm{Re}\lambda}^{{-}1}\\ \end{array}\right\}\ \mathrm{for\ Re}\ \lambda>0. \end{equation}

An operator $T$ satisfying (2.15) will be said to be m-accretive. An m-accretive operator $T$ is maximal accretive, in the sense that $T$ is accretive and has no proper accretive extension.

We shall say that $T$ is quasi-accretive if $T+\alpha$ is accretive for some scalar $\alpha$. This is equivalent to the condition that $\Theta (T)$ is contained in a half-plane of the form Re$\zeta \geq$const. In the same way we say that $T$ is quasi-m-accretive if $T+\alpha$ is m-accretive for some $\alpha$. Like an m-accretive operator, a quasi-m-accretive operator is maximal quasi-accretive and densely defined.

A linear operator $T$ in a Hilbert space $H$ is said to be sectorial with vertex $\gamma$ and semi-angle $\theta$ if the numerical range $\Theta (T-\gamma I)$ lies in a sector $S_{\sigma }$ for some $\gamma \in \mathbb {R}$, $T$ is said to be m-sectorial if it is sectorial and quasi-m-accretive.

Now, we turn to define operators by the formal differential operator $\tau$. For any compact interval $[\alpha, \beta ]\subset [a, b)$, using integration by parts, we obtain the so-called Green's formula.

\[ \int_{\alpha}^{\beta}\left[\overline{\psi}\tau\phi-\phi\overline{\tau^+\psi}\right]=[\phi, \psi](\beta)-[\phi, \psi](\alpha), \]

where $\tau ^+\psi =-(\overline {p}\psi ')'+\overline {q}\psi =\overline {\tau }\psi$, $[\phi, \psi ]=(\phi \overline {p\psi '}-\overline {\psi }p\phi ')(x)$. Since $\tau ^+=\overline {\tau }$, we know $\tau$ is formally J-symmetric. Set

\[ D_{\max}=\left\{y\in L^2_w[a, b): py'\in {\rm AC}_{\rm loc}[a, b), \,{\rm and}\,\,\tau y\in L^2_w[a, b)\right\}, \]

we define the ‘maximal operator’ $T_M=T_M(\tau )$ by

\begin{align*} T_M: D_{\max}& \rightarrow L^2_w[a, b)\\ y& \mapsto T_My=\tau y=\frac{1}{w}[-(py')'+qy]. \end{align*}

Similarly, we define

\begin{align*} D_{0}'& =\left\{y\in D_{\max}: y(a)=p(a)y'(a)=0,\ {\rm and}\right.\\ & \left. y=0\ {\rm outside\ a\ compact\ subset\ of}\,[a, b)\right\}, \end{align*}

$T|_{D_0'}=T_0'$ is called the pre-minimal operator, and $T_0'$ is closable, its closure, denoted by $T_0$, is called the minimal operator, its domain is denoted by $D_0$. By the book of Edmund and Evans [6, P144, theorem 10.7], we know that $\overline {D_0'}=L^2_w[a, b)$, and $T_0(\tau )^*=T_M(\overline {\tau })$, $T_0(\overline {\tau })=T_M(\tau )^*$, so $JT_0J\subset T_0^*$, where $J$ is the common conjugation, this means that $T_0$ is a J-symmetric operator. It's easy to see that for each $y\in D_0$,

\[ y(a)=p(a)y'(a)=0,\ {\rm and}\ [y, z](b)=0\ {\rm for\ all}\ z\in D_{\max}. \]

Lemma 2.6 Assume that $E$ has more than one point. We have that $T_0$ is a densely defined closed sectorial operator in $L^2_w[a, b)$.

Proof. The denseness of $T_0$ follows from [6, theorem 10.5]. Since $E$ has more than one point, we choose $\theta _1, \theta _2\in E$ such that $\theta _1\neq \theta _2$. Choose $\lambda _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. By the definition of $T_0$, we see that for $u, v\in D(T_0)$

\begin{align*} \langle (T_0-\lambda)u, v\rangle & =\int_a^b\bar{v}w(T_0-\lambda_0)u \\ & = \int_a^b\bar{v}[-(pu')'+(q-\lambda_0 w)u]\\ & = \int_{\alpha}^{\beta}(pu'\overline{v'}+(q-\lambda_0 w)u\bar{v}), \end{align*}

Set $r(x)=|q(x)-\lambda _0w(x)|$ and

(2.16)\begin{equation} q(x)-\lambda_0w(x)=r(x)\,{\rm e}^{i\alpha(x)},\ \alpha_1(x)=\theta_1+\alpha(x), \ \alpha_2(x)=\theta_2+\alpha(x). \end{equation}

Then lemma 2.1 ensures that $\cos \alpha _1, \cos \alpha _2\geq 0$ on $[a, b)$. Since

\begin{align*} \sin^2(\theta_2-\theta_1)& =\sin^2(\alpha_2-\alpha_1)\\ & =\cos^2\alpha_2 +\cos^2\alpha_1-2\cos\alpha_2\cos\alpha_1\cos(\alpha_1-\alpha_2)\\ & \le\left(\cos\alpha_2+\cos\alpha_1\right)^2, \end{align*}

so we obtain

(2.17)\begin{equation} \cos\alpha_1+\cos\alpha_2\geq |\sin(\theta_1-\theta_2)|=\delta_0>0. \end{equation}

Set

(2.18)\begin{equation} p_{\theta}=[{\rm e}^{i\theta_1} +{\rm e}^{i\theta_2}]p, (q-\lambda_0 w)_{\theta}=[{\rm e}^{i\theta_1} +{\rm e}^{i\theta_2}](q-\lambda_0 w). \end{equation}

Then (2.17) implies

(2.19)\begin{equation} {\rm Re}\left\{(q-\lambda_0 w)_{\theta}\right\} \ge \delta_0|q-\lambda_0 w|. \end{equation}

Similarly, we have that

(2.20)\begin{equation} {\rm Re}\left\{p_{\theta}\right\} \ge \epsilon_0|p|, \end{equation}

where $\epsilon _0>0$ is a constant sufficiently small. Hence we have for $y\in D(T_0)$,

\begin{align*} & {\rm Re}\left\{({\rm e}^{i\theta_1}+{\rm e}^{i\theta_2})\langle (T_0-\lambda_0)\,y, y\rangle\right\}\\ & \quad= {\rm Re}\int^b_a p_{\theta}\,|y'|^2+{\rm Re}\int^b_a(q-\lambda_0w)_{\theta}\,|y|^2\\ & \quad\geq\int^b_a\left[\epsilon_0|p||y'|^2+\delta_0|q-\lambda_0 w||y|^2\right]. \end{align*}

Similarly we have that

(2.21)\begin{equation} {\rm Im}\left\langle ({\rm e}^{i\theta_1}+{\rm e}^{i\theta_2})(T_0-\lambda_0)\,y, y\right\rangle\leq \int^b_a\left[2|p|\,|y'|^2 +2|q-\lambda_0\,w|\,|y|^2\right]. \end{equation}

So $\tilde {T}=({\rm e}^{i\theta _1}+{\rm e}^{i\theta _2})(T_0-\lambda _0I)$ is a sectorial operator. Since $\tilde {T}$ is obtained by contraction and rotation of $T_0$, $T_0$ is a sectorial operator. It is obvious that the operator $T_0$ is a closed operator. This completes the proof.

2.2 Sesquilinear forms and Friedrichs extension

The sesquilinear forms in Hilbert space are closely related to the associated operators, bounded forms and bounded operators are equivalent, there is no such obvious relationship for unbounded forms and operators. However, there exists a closed theory on the relationship between semi-bounded symmetric forms and semi-bounded self-adjoint operators, this theory is extended to non-symmetric forms and operators within certain restrictions. Among these restrictions, the sectorial operators and sectorial forms are necessary.

A sesquilinear form $\textsf {t}[u, v]$ is defined for $u, v$ both belonging to a linear manifold $D$ of a Hilbert space $H$, $\textsf {t}[u, v]$ is complex-valued and linear in $u\in D$ for each fixed $v\in D$ and semilinear in $v\in D$ for each fixed $u\in D$. $D$ is called the domain of $\textsf {t}$ and is denoted by $D(\textsf {t})$; $\textsf {t}$ is densely defined if $D(\textsf {t})$ is dense in $H$; $\textsf {t}[u]=\textsf {t}[u, u]$ is called the quadratic form associated with $\textsf {t}[u, v]$; a sesquilinear form $\textsf {t}$ is said to be symmetric if $\textsf {t}[u, v]=\overline {\textsf {t}[v, u]}$.

For a sesquilinear form $\textsf {t}$, its numerical range is defined by

\[ \Theta(\textsf{t})=\{\textsf{t}(u)|u\in D(\textsf{t}), ||u||=1\}. \]

A nonsymmetric sesquilinear form $\textsf {t}$ is said to be a sectorially bounded form from the left (or simply sectorial) if $\Theta (\textsf {t})$ is a subset of a sectorial of the form

\[ |\arg (\zeta-\gamma)|\leq \theta, \quad 0\leq \theta<\frac{\pi}{2}, \gamma\ {\rm is\ real}, \]

$\gamma$ is called the vertex and $\theta$ is corresponding semi-angle of the form $\textsf {t}$.

If $T$ is a sectorial operator, then the form defined by

(2.22)\begin{equation} \textsf{t}[u, v]=\langle Tu, v\rangle\ {\rm with}\ D(\textsf{t})=D(T) \end{equation}

is also a sectorial form. By [14, theorem 1.27, P318], we know that a sectorial operator is form-closeable, i.e., the form $\textsf {t}$ defined above is closable, it closure is denoted by $\tilde {\textsf {t}}$. Particularly, If $T$ is a densely defined, sectorial operator, by the first representation theorem ([14, theorem 2.1, P322]), the closure $\tilde {\textsf {t}}$ of $\textsf {t}$ generated by (2.22), generates an m-sectorial operator $T_{\tilde {\textsf {t}}}$, $T_{\tilde {\textsf {t}}}$ is called the Friedrichs extension of $T$, we denote this operator by $T_F$ for convenience, i.e., the Friedrichs extension of a sectorial operator is the form extension of the corresponding sectorial form.

For the minimal operator $T_0$, by direct calculation, we deduce that the corresponding sesquilinear form is

\[ \textsf{s}[u, v]=\int_a^b[pu'\overline{v'}+qu\overline{v}], u, v\in D(\textsf{s})=D_0. \]

We know by lemma 2.6 that $T_0$ is a closed sectorial operator, so the Friedrichs extension exists. In the next two sections, we will characterize the Friedrichs extensions of $T_0$.

3. The Friedrichs extension under case I

Firstly, we give some spectral results on Hamiltonian differential system

(3.1)\begin{equation} \left\{\begin{array}{@{}l} u'=A\,u+B\,v+\xi W_2\,v, \\ v=C\,u-A^*\,v-\xi W_1\,u \end{array}\right.\ {\rm on}\ [a, b) \end{equation}

on the $\mathbb {C}^{2n}$ valued (column) functions $Y=(u^T, v^T)^T$, where $u, v$ are $\mathbb {C}^n$ valued functions, $u^T$ is the transpose of $u$, $A, B, C, W_1$ and $W_2$ are locally integrable, complex-valued $n\times n$ matrices on $[a, b)$, $B, C, W_1, W_2$ are Hermitian matrices and $W_1(t)>0, W_2(t)\ge 0$ on $[a, b)$, $\xi$ is the so-called spectral parameter. Assume that the definiteness condition (see, e.g., [1, chapter 9, p253]) holds:

\[ \int_a^{b} Y^* W Y >0\ \text{for each non-trivial solution $Y$ of (3.1)}, \]

where $W=\operatorname {diag}(W_1, W_2)$. Let $L^2_W:=L^2_W[a, b)$ be the space of Lebesgue measurable $2n$-dimensional functions $f$ satisfying $\int _a^{b}f^*(s)W(s)f(s)ds<\infty$. We say that (3.1) is in the limit point case at $b$ if there exists exactly $n$'s linearly independent solutions of (3.1) belong to $L^2_W$ for $\xi =\pm i$. Particularly, if $n=1$ and $A, B, C$ are real functions, then (3.1) is in the limit point case at $b$ if and only if there exists a unique solution of (3.1) belonging to $L^2_W$ for $\xi =i$ or $\xi =-i$.

Let $D$ be the maximal domain associated to (3.1), i.e. $(u^T, v^T)^T\in D$ if and only if $(u^T, v^T)^T\in AC_{\text {loc}}\cap L^2_W$ and there exists an element $(f^T, g^T)^T\in L^2_W$ such that

(3.2)\begin{equation} \left\{\begin{array}{@{}l} u'=A\,u+B\,v+W_2\,g, \\ v'=C\,u-A^*\,v-W_1\,f. \end{array}\right. \end{equation}

It is well known (cf. [11, 16]) that (3.1) is in the limit point case at $b$ if and only if

(3.3)\begin{equation} Y_1^*(x)JY_2(x)\to 0,\ {\rm as}\ x\to b, \ J=\begin{pmatrix} 0 & -I_n\\ I_n & 0 \end{pmatrix} \end{equation}

for all $Y_1, Y_2\in D$, and for each $\xi \in \mathbb {C}$ with Im$\xi \ne 0$ there exists a Green function $G(t, s, \xi )$ such that for $F=(f^T, g^T)^T\in L^2_W$, (3.2) has an $L^2_W$-solution $Y$ given by

(3.4)\begin{equation} Y=\begin{pmatrix} u\\ v \end{pmatrix}=\int^b_a G({\cdot}, s, \xi)W(s)F(s)\,{\rm d}s. \end{equation}

Now we give the asymptotic behaviours of elements of $\mathcal {D}_{\max }$ under case I.

Theorem 3.1 Assume that $E$ has more than one point, and (1.1) is in case I. For each $y\in D_{\max }$, we obtain

\[ y\in L^2_{|q|},\ {\rm and}\ y'\in L^2_{|p|}, \]

and for all $y_1, y_2\in D_{\max }$,

(3.5)\begin{equation} p(x)\,y_1(x)\,y'_2(x)\to 0,\ {\rm as}\ x\to b. \end{equation}

Proof. Since $E$ has more than one point, we choose $(\theta _1, K_1), (\theta _2, K_2)\in \Pi$ (with $\theta _1\neq \theta _2({\rm mod}\,\pi )$) and $\lambda _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. Then by the definition of $\Lambda _{\theta, K}$, we obtain

(3.6)\begin{equation} {\rm Re}\left\{{\rm e}^{i\theta_j}(q-\lambda_0\,w)\right\}\ge\delta_j w, \ {\rm Re}\left\{{\rm e}^{i\theta_j}p\right\}\ge 0,\ j=1, 2 \end{equation}

for some $\delta _1, \delta _2>0$. Set

(3.7)\begin{equation} \begin{aligned} & r(x)=|q(x)-\lambda_0\,w(x)|, \ q(x)-\lambda_0\,w(x)=r(x)\,{\rm e}^{i\alpha(x)}, \ \alpha_j(x)=\theta_j+\alpha(x), \\ & \widetilde{r}(x)=|p(x)|, \ p(x)=\widetilde{r}(x)\,{\rm e}^{i\beta(x)}, \ \beta_j(x)=\theta_j+\beta(x),\ j=1, 2. \end{aligned} \end{equation}

For $j=1$, Let $y$ be a solution of (1.1) with $\lambda =\lambda _0$. Set $u=y$, $v=-i\,{\rm e}^{i\theta _1}py'$, then (1.1) is transformed into the Hamiltonian differential system

(3.8)\begin{equation} u'=B\,v+\xi w_2\,v,\quad v'=C\,u-\xi w_1\,u \end{equation}

with the new spectral parameter $\xi =i$, where

(3.9)\begin{equation} \begin{aligned} & C(x)=r(x)\sin\alpha_1(x), \ w_1(x)=r(x)\cos\alpha_1(x), \\ & B(x)=\sin\beta_1(x)/\widetilde{r}(x), \ w_2(x)={\cos\beta_1}(x)/{\widetilde{r}(x)}. \end{aligned} \end{equation}

This is the Hamiltonian differential system (3.1) with $n=1$, $A(x)\equiv 0$ and $\xi =i$. Clearly, $w_1={\rm Re}\left \{{\rm e}^{i\theta _1}(q-\lambda _0\,w)\right \}\ge \delta _0\,w>0$, $w_2=\frac {{\rm Re}\left \{{\rm e}^{i\theta _1}p(t)\right \}}{r^2_2}\ge 0$ by (3.6). We note further that the coefficients of the Hamiltonian system (3.8) are real functions. It is easy to verify that the definiteness condition holds for the system (3.8). Therefore, (1.1) is in case I or II w.r.t. $(\theta _1, K_1)\in \Pi$ if and only if (3.8) is in the limit point case at $b$.

Let $D(\theta _1)$ be the maximal domain associated to (3.8), i.e., $(u, v)^T\in D(\theta _1)$ if and only if $(u, v)^T\in AC_{\text {loc}}\cap L^2_W$ and there exists an element $(f, g)^T\in L^2_W$ such that

(3.10)\begin{equation} u'=B\,v+w_2\,g, \quad v'=C\,u-w_1\,f. \end{equation}

For $y_0\in D(\tau )$, set

(3.11)\begin{equation} f_0=\tau y_0,\ g_0=f_0-\lambda_0 y_0. \end{equation}

Then $y_0$ satisfies

(3.12)\begin{equation} (\tau-\lambda_0)y={w}^{{-}1}[-(py')'+(q-\lambda_0\,w)y]=g_0. \end{equation}

Set $u_0=y_0$, $v_0=-i\,{\rm e}^{i\theta _1}py'_0$. Then $(u_0, v_0)$ satisfies

(3.13)\begin{equation} u'=B\,v+i w_2\,v, \ v'=C\,u-i w_1\,u-w_1f_1, \ f_1=\frac{w}{w_1}\left({-}i\,{\rm e}^{i\theta_1}g_0\right). \end{equation}

Conversely, if $(u, v)$ satisfies (3.13), then $y=u$ solves (3.12).

Let $g_0$ be given in (3.11). Consider the equation (3.13), we get from (3.4) that (3.13) has a solution $(u_1, v_1)^T$ such that $u_1\in L^2_{w_1}, v_1\in L^2_{w_2}$ and $v_1=-i\,{\rm e}^{i\theta _1}p\,u'_1$. Set $y_1=u_1$. Then $y_1$ satisfies (3.12), hence $(\tau -\lambda _0)(y_0-y_1)=0$. Note that $y_1=u_1\in L^2_{w_1}$ and $w_1\ge \delta w$ implies that $y_1\in L^2_w$. Thus, $y_1-y_0$ is an $L^2_w$-solution of $\tau y=\lambda _0 y$. Since $\tau$ is in case I w.r.t. $(\theta _1, K_1)$, it follows from (2.8) that $y_1-y_0\in L^2_{w_1}$ and $v_1-v_0\in L^2_{w_2}$. This together with $y_1\in L^2_{w_1}$ and $v_1\in L^2_{w_2}$ gives that $y_0\in L^2_{w_1}$ and $v_0\in L^2_{w_2}$. In fact, we have proved that for $y\in D(\tau )$,

(3.14)\begin{equation} \int^b_a |q-\lambda_0 w| \cos\alpha_1 |y|^2<\infty,\ \int_a^b |p| \cos\beta_1 |y'|^2<\infty, \end{equation}

where $\alpha _1$ and $\beta _1$ are defined in (3.7). Recall that $g_0\in L^2_w$, or $-i\,{\rm e}^{i\theta _1}g_0\in L^2_w$ and $w_1\ge \delta w$ implies $f_1\in L^2_{w_1}$. It follows (3.13) that $(y_0, v_0)$ satisfies (3.10) with $f=i y_0+f_1$ and $g=i v_0$. This yields that

(3.15)\begin{equation} y\in D(\tau)\quad \Longrightarrow (y, v)^T\in D(\theta_1)\ {\rm with}\ v={-}i\,{\rm e}^{i\theta_1}p y'. \end{equation}

Note that $Y\in D(\theta _1)$ if and only if $\overline {Y}\in D(\theta _1)$. Then for $\overline {y}\in D(\tau )$, we have from (3.15) that $(\overline {y}, \overline {v})^T\in D(\theta _1)$ with $\overline {v}=-i\,{\rm e}^{i\theta _1}p \overline {y}'$, hence

(3.16)\begin{equation} \overline{y}\in D(\tau) \Longrightarrow (y, v)^T\in D(\theta_1)\ {\rm with}\ v=i\,{\rm e}^{{-}i\theta_1}\overline{p} y'. \end{equation}

Let $y_1, \overline {y}_2\in D(\tau )$. Since (3.8) is in the limit point case at $b$ and $(y_j, v_j)^T\in D(\theta _1)$ for $j=1, 2$ with $v_1=-i\,{\rm e}^{i\theta _1}p y'_1$ and $v_2=i\,{\rm e}^{-i\theta _1}\overline {p} y'_2$ by (3.15) and (3.16), respectively, we get from (3.3) that

(3.17)\begin{equation} (\overline{y}_2, \overline{v}_2)\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}\begin{pmatrix}y_1\\v_1\end{pmatrix} =i\,{\rm e}^{i\theta_1}\,p (y_1'\,\overline{y}_2-y_1\,\overline{y}'_2)\to 0 \end{equation}

as $x\to b$. Furthermore, for $y_1, y_2\in D(\tau )$, since $(y_j, v_j)^T\in D(\theta _1)$ by (3.15) with $v_j=-ie^{i\theta _1}p y'_j$, $j=1, 2$, (3.3) also gives that

\[ (\overline{y}_2, \overline{v}_2)\begin{pmatrix} 0 & -1\\ 1 & 0\end{pmatrix}\begin{pmatrix}y_1\\v_1\end{pmatrix} =i\,{\rm e}^{{-}i\theta_1}(\overline{p}\,\overline{y}_2'\,y_1+{\rm e}^{2i\theta_1} p\,y'_1\,\overline{y}_2)\to 0 \]

as $x\to b$, or

(3.18)\begin{equation} \overline{p}\,\overline{y}_2'\,y_1+{\rm e}^{2i\theta_1} p\,y'_1\,\overline{y}_2\to 0, \ y_1, y_2\in D(\tau) \end{equation}

as $x\to b$. Similarly, for $j=2$, the above methods also give that (3.18) holds for $y_1, y_2\in D(\tau )$ with $\theta _1$ replaced by $\theta _2$. Therefore, we have

(3.19)\begin{equation} \left({\rm e}^{2i\theta_1}-{\rm e}^{2i\theta_0}\right) p\,y_1'\,\overline{y}_2 \to 0 \end{equation}

as $x\to b$. This clearly gives (3.5) for $y_1, y_2\in D(\tau )$ since $\theta _1\ne \theta _2$ (mod $\pi$).

Finally, we prove $y\in L^2_{|q|}$ and $y'\in L^2_{|p|}$ for $y\in D(\tau )$. In fact, the similar proof as in (3.14) gives that for $y\in D(\tau )$,

(3.20)\begin{equation} \int_a^b |q-\lambda_0 w| \cos\alpha_2 |y|^2<\infty, \ \int_a^b |p| \cos\beta_2 |y'|^2<\infty. \end{equation}

Therefore, (3.14), (3.20) and (2.17) together yield that $\int _a^b |q-\lambda _0 w||y|^2<\infty$ and $\int _a^b |p||y'|^2<\infty$, or $y\in L^2_{|q|}$ and $y'\in L^2_{|p|}$ since $y\in L^2_w$. This completes the proof.

Theorem 3.2 Assume $E$ has more than one point, and (1.1) is in case I. Then the J-self-adjoint extension $T$ defined by

(3.21)\begin{equation} \begin{aligned} T: D(T) & \to L_w^2[a, b)\\ y & \mapsto Ty=\tau y=\frac{1}{w}[-(py')'+qy] \end{aligned} \end{equation}

is the Friedrichs extension of $T_0$, where $D(T)=\{y\in D_{\max }: y(a)=0\}.$

Proof. Since $E$ has more than one point, we choose $(\theta _1, K_1), (\theta _2, K_2)\in \Pi$ (with $\theta _1\neq \theta _2({\rm mod}\,\pi )$) and $\lambda _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. Since (1.1) is in case I, we know that $T$ is a J-self-adjoint extension of $T_0$ by theorem 4.4 in paper [3]. Next, we define

(3.22)\begin{equation} \textsf{t}[u, v]=\int_a^b[pu'\overline{v'}+qu\overline{v}], u, v\in D(\textsf{t}), \end{equation}

(3.23)\begin{align} D(\textsf{t})=\left\{u\in AC_{\rm loc}[a, b): u(a)=0, u\in L_w^2[a, b)\cap L_{|q-\lambda_0w|}^2[a, b), u'\in L_{|p|}^2[a, b) \right\}. \end{align}

Since (1.1) is in case I, we see by theorem 3.1 that $D(T_0)\subseteq D(\textsf {t})$, so $\textsf {t}$ is densely defined. Similar method as in lemma 2.6 can deduce $\textsf {t}$ as a sectorial operator. Now we turn to prove that $\textsf {t}$ is closed.

Suppose that $y_n\in D(\textsf {t})$, $y_n\to y$ in $L_w^2[a, b)$ and

\[ \textsf{t}[y_n-y_m]=\int_a^b[p|y_n'-y_m'|^2+q|y_n-y_m|^2]\to 0 \]

for $n, m \to \infty$. Let $y_{nm}=y_n-y_m$ for convenience. Since $\textsf {t}[y_{nm}]\to 0$, we obtain $({\rm e}^{i\theta _1}+{\rm e}^{i\theta _2})\textsf {t}[y_{nm}]\to 0$, and

\[ \lambda_0\int_a^bw|y_{nm}|^2\leq 2|\lambda_0|\int_a^bw\left(|y_{n}-y|^2+|y_{m}-y|^2\right)\to 0. \]

So we obtain that

(3.24)\begin{equation} \int_a^b[p_{\theta}|y_{nm}'|^2+(q-\lambda_0w)_{\theta}|y_{nm}|^2]\to 0. \end{equation}

Since ${\rm Re}\left \{(q-\lambda _0 w)_{\theta }\right \} \ge \delta _0|q-\lambda _0 w|$ and ${\rm Re}\left \{p_{\theta }\right \} \ge \epsilon _0|p|$ for some constants $\delta _0, \epsilon _0>0$ by (2.19) and (2.20), and noticing $\textsf {t}$ is a sectorial operator, we obtain

\[ \epsilon_0\int_a^b|p|y_{nm}'|^2+\delta_0\int_a^b|q-\lambda_0w||y_{nm}|^2\to 0. \]

So

\[ \int_a^b|p|y_{nm}'|^2\to 0, \int_a^b|q-\lambda_0w||y_{nm}|^2\to 0 \ {\rm as}\ n, m\to\infty. \]

This shows that $\{y_n'\}$ and $\{y_n\}$ are Cauchy sequences in $L^2_{|p|}[a, b)$ and $L^2_{|q-\lambda _0w|}[a, b)$, respectively. We may assume $y_n'\to z$ in $L^2_{|p|}[a, b)$. For each fixed $x\geq a$,

\begin{align*} |y_{nm}(x)|& =\left|\int_a^xy_{nm}'(s)\,{\rm d}s\right|\\ & \leq \left(\int_a^x\frac{1}{|p(s)|}\,{\rm d}s\right)^{1/2}\left(\int_a^x|p(s)||y_{nm}'(s)|\,{\rm d}s\right)^{1/2}\\ & \leq \left(\int_a^x\frac{1}{|p(s)|}\,{\rm d}s\right)^{1/2}\left(\int_a^b|p(s)||y_{nm}'(s)|\,{\rm d}s\right)^{1/2}\to 0 \end{align*}

as $n, m\to \infty$, since $y_n\to y$ in $L^2_{w}[a, b)$, we obtain $y_n(x)\to y(x)$ point-wise for $x\geq a$ as $n\to \infty$, hence $y(0)=\lim _{n\to \infty }y_n(0)=0$, and $y'(x)=z(x)$, so $y\in D(\textsf {t})$, and $\textsf {t}$ is a closed sesquilinear form.

By the first representation theorem [14, P322, theorem 2.1], there exists an m-sectorial operator $T=T_{\textsf {t}}$ such that $D(T)\subseteq D(\textsf {t})$, $\textsf {t}[u, v]=\langle Tu, v\rangle$ for $u, v\in D(T)$, and $D(T)$ is a core of $\textsf {t}$. Since $T=T_{\textsf {t}}$, so $T_F=T_{\textsf {t}}$ is the Friedrichs extension of $T_0$. We prove $D(T_F)=\{y\in D_{\max }: y(a)=0\}:=\tilde {D}$ to complete the proof. For each $y\in \tilde {D}$, then $y\in D_{\max }$, by theorem 3.1, $y'\in L^2_{|p|}[a, b)$ and $y\in L^2_{|q|}[a, b)$, this together with $y\in L^2_{w}[a, b)$ implies

\[ \int_a^b|q-\lambda_0w|||y|^2\leq \int_a^b|q|||y|^2+|\lambda_0|\int_a^bw|y|^2<\infty, \]

so $y\in L^2_{|q-\lambda _0w|}[a, b)$, thus we obtain $y\in T_F$. Conversely, for each $y\in D(T_F)$, set $T_Fy=f$. By the definition of $T_F$, we obtain $\textsf {t}[y, v]=\langle T_Fy, v\rangle =\langle f, v\rangle$ for all $v\in D(\textsf {t})$, or

(3.25)\begin{equation} \int_a^bwf\overline{v}=\int_a^b\left[py'\overline{v'}+qy\overline{v}\right]. \end{equation}

Set $z'(x)=w(x)f(x)-q(x)y(x)$ and for arbitrary $d< b$,

\[ A_d=\{v:\ v'\in L^2[a, d], \ v(a)=v(d)=0 \}. \]

Then for $v\in A_d$

(3.26)\begin{equation} \int^d_a z'\,\overline{v}=z\,\overline{v}\big{|}^d_a-\int^d_a z\,\overline{v'}={-}\int^d_a z\,\overline{v}'. \end{equation}

If we set

\[ v_0(x)=\left\{\begin{array}{@{}ll} v(x), & x\in[a, d], \\ 0, & x\in [d, b), \end{array}\right. \]

then $v_0\in D(\textsf {t})$, hence (3.25) and (3.26) imply that

(3.27)\begin{equation} \int^d_a (py'+z)\overline{v'}=0. \end{equation}

Note that

\[ \{v':\ v\in A_d \}=\{f:\ f\in L^2[a, d], \ \int^d_a f=0 \}, \]

which is the set of orthogonal to $1$ in $L^2[a, d]$. Therefore, $py'+z=C_d$, where $C_d$ is a constant, hence $z'(x)=-(py')'(x)$ a.e. $x\in [a, d]$, or $-(py')'+qy=wf$ on $[a, d]$. By the arbitrary of $d$ we know that

(3.28)\begin{equation} -(py')'+qy=wf\ {\rm on}\ [a, b). \end{equation}

This means $y\in \tilde {D}$. So $D(T_F)= \tilde {D}$, this completes the proof.

4. The Friedrichs extension under case II

When $\tau$ is not in case I, the J-self-adjoint extension $T_F$ is comparatively complicated. Even for the case where the corresponding formal differential operator is symmetric but not in the limit-point case, the characterization of Friedrichs extensions is hard to be obtained. This problem has been studied by a lot of authors both for formal symmetric arbitrary order differential operator and Hamiltonian differential operators [7, 8, 25, 26, 31] and Hamiltonian differential systems [12, 16, 17] and references therein. When $\tau$ is not in case I, by operator theory we know that such domain is a restriction of the maximal domain. We should impose some restrictions on the elements of the maximal domain to instruct the domain of a operator realization. The standard method is to choose suitable number independent elements from the maximal domain to construct the corresponding boundary conditions at end-points of the interval considered. The similar result is also valid for the case where $p(x)$, $q(x)$ are complex valued (see [3, theorem 4.1]).

We will not follow the same line as above to give the Friedrichs extensions associated to $\tau$. The following Friedrichs extension is given by the restriction of the maximal domain on a suitable Hilbert space. The main result of this section is also a generalization of the corresponding result in [22] to some extent.

Theorem 4.1 Assume $E$ has more than one point, and (1.1) is in case II. For each $y\in D_{\max }$ with $y'\in L^2_{|p|}[a, b)$, we obtain

\[ y\in L^2_{|q|}[a, b), \]

and for all $y_1, y_2\in D_{\max }$ with $y_1', y_2'\in L^2_{|p|}[a, b)$,

(4.1)\begin{equation} p(x)\,y_1(x)\,y'_2(x)\to 0,\ {\rm as}\ x\to b. \end{equation}

Proof. Similar to the proof of theorem 3.1, we choose $(\theta _1, K_1), (\theta _2, K_2)\in \Pi$ (with $\theta _1\neq \theta _2({\rm mod}\,\pi )$) and $\lambda _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. Since (1.1) is in case II (with $\lambda$ replaced by $\lambda _0$), we obtain the Hamiltonian system (3.8) is in the limit point case at $b$.

Let $D(\theta _1)$ denote the maximal domain associated to (3.8), we obtain that $y\in D_{\max }$ with $y'\in L^2_{|p|}[a, b)$ yields $(y, v)^T\in D(\theta _1)$, where$v=-ie^{i\theta _1}p y'$, and $\overline {y}\in D_{\max }$ with $\overline {y}'\in L^2_{|p|}[a, b)$ yields $(y, v)^T\in D(\theta _1)$, where$v=ie^{-i\theta _1}\overline {p} y'$. Then as $x\to b$, we obtain

\[ \overline{p}\,\overline{y}_2'\,y_1+{\rm e}^{2i\theta_1} p\,y'_1\,\overline{y}_2\to 0\quad {\rm for\ all}\ y_1, y_2\in D_{\max}\quad{\rm with}\ y_1', y_2'\in L^2_{|p|}[a, b). \]

Similar method implies

\[ \overline{p}\,\overline{y}_2'\,y_1+{\rm e}^{2i\theta_2} p\,y'_1\,\overline{y}_2\to 0\quad {\rm for\ all}\ y_1, y_2\in D_{\max}\quad{\rm with}\ y_1', y_2'\in L^2_{|p|}[a, b). \]

The above two formula imply (4.1) holds, so we complete the first part of theorem 4.1. Similar argument as theorem 3.1 deduce $y\in L^2_{|q|}[a, b)$ for all $y\in D_{\max }$ with $y'\in L^2_{|p|}$. This completes the proof.

Theorem 4.2 Suppose that $E$ has more than one point and (1.1) is in case II. Then the J-self-adjoint extension $T$ defined by

(4.2)\begin{equation} \begin{aligned} T: D(T) & \to L_w^2[a, b)\\ y & \mapsto Ty=\tau y=\frac{1}{w}[-(py')'+qy] \end{aligned} \end{equation}

is the Friedrichs extension of $T_0$, where $D(T)=\{y\in D_{\max }: y(a)=0, y'\in L^2_{|p|}[a, b)\}.$

Proof. Since $E$ has more than one point, we choose $(\theta _1, K_1), (\theta _2, K_2)\in \Pi$ (with $\theta _1\neq \theta _2({\rm mod}\,\pi )$) and $\lambda _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. We define the sesquliniear form $\textsf {t}$ as (3.22), then the similar method implies $\textsf {t}$ is a densely defined closed sectorial operator. By the first representation theorem [14, P322, theorem 2.1], the m-sectorial operator $T=T_{\textsf {t}}$ defined by $\textsf {t}[u, v]=\langle Tu, v\rangle$ for $u, v\in D(T)$ is the Friedrichs extension of $T_0$. Now, using the similar method as theorem 3.2, we obtain

\[ D(T_F)=\{y\in D_{\max}: y(a)=0, y'\in L_{|p|}^2[a, b)\}:=\tilde{D}. \]

Finally, we turn to prove $T_F$ is a J-self-adjoint operator by three steps.

1. (1) $JT_{F}J\subset T^*_{F}$. It is equivalent to prove that

(4.3)\begin{equation} \overline{y}_0\in \tilde{D} \Longrightarrow y_0\in D(T^*_{F}), J T_{F} J y_0=T^*_{F} y_0. \end{equation}

Note that for $Jy_0=\overline {y}_0\in \tilde {D}$ and all $y\in \tilde {D}$

\begin{align*} \langle T_{F} y, y_0\rangle- \langle y, J T_{F} J y_0\rangle& =\int_a^{b} \left(\overline{y_0}w\tau y-y w\overline{\tau^+ y_0}\right)\\ & =\int_a^{b} \left(\overline{y_0}\left[-(py')'+qy\right]-y\left[ \overline{-(\overline{p}y'_0)'+\overline{q}y_0}\right]\right)\\ & =\int_a^{b}\left(y\overline{(\overline{p}y'_0)'}-\overline{y_0}(py')'\right) =\int_a^{b}\left(y(p\overline{y'_0})'-\overline{y_0}(py')'\right)\\ & =p\left(y\overline{y'_0}-\overline{y_0}y'\right)\big|^b_a. \end{align*}

This equality together with the boundary condition $y(a)=\overline {y_0}(a)=0$ gives

(4.4)\begin{equation} \langle T_{F} y, y_0\rangle- \langle y, J T_{F} J y_0\rangle=p\left(y\overline{y'_0}-\overline{y_0}y'\right)(b), \end{equation}

(here we denote $\lim _{x\to b}p(y\overline {y'_0}-\overline {y_0}y')(x)$ by $p(y\overline {y'_0}-\overline {y_0}y')(b)$ since this limit always exists). Since $E$ has more than one point and (1.1) is in case II, we know from theorem 4.1 that $p(y\overline {y'_0}-\overline {y_0}y')(b)=0$, and hence we have $y_0\in D(T^*_{F})$ and $J T_{F} J y_0=T^*_{F} y_0$ by (4.4). This proves that $T_{F}\subset J T^*_{F} J$.

1. (2) $T_{F}$ is closed operator. Since for $u, v\in D(\alpha )$, we find that

\[ \langle (T_{F}-\lambda_0)\,u, v\rangle=\int_a^{b}\left[ p\,u'\,\overline{v}'+(q-\lambda_0 w)\,u\,\overline{v}\right]. \]

Suppose $y_n\in \tilde {D}$ such that $y_n\to y_0$ and $T_{F} y_n\to f_0$, or

\[ y_n\to y_0, \ (T_{F}-\lambda_0 I)y_n\to g_0=f_0-\lambda_0 y_0 \quad{\rm in}\ L^2_w[a, b). \]

Let $y_{nm}=y_n-y_m$, as (3.24), we have that $\{y_n\}$ and $\{y'_n\}$ are Cauchy sequences in $L^2_{|q|}[a, b)$ and $L^2_{|p|}[a, b)$, respectively. Since $y_n\to y$ in $L^2_w[a, b)$ as $n\to \infty$ and $\{y_n'\}$ is convergent in $L^2_{|p|}[a, b)$, we see that,

\[ y_n(x)= \int^x_a\frac{1}{p} py_n'\to y_0(x) \]

pointwise in $x$ as $n\to \infty$ on $[a, b)$. It follows from $\tau y_n= T_{F}y_n=f_n$ that

\[ y_n(x)=\int^x_a\frac{1}{p}\left(py_n'(a)+\int^s_a[w\,f_n-q\,y_n]\right). \]

Note that the convergence of each sequence in the above equality except for $\{py_n'(a)\}$ implies $\{py_n'(a)\}$ is convergent, and hence by letting $n\to \infty$ we have that

\[ y(x)=\int^x_a\frac{1}{p}\left(\xi+\int^s_a[w\,f-q\,y]\right), \]

where $py'_n(a)\to \xi$ as $n\to \infty$. This means that $\tau y=f$ and $y(a)=0$, and hence $y\in \tilde {D}$ and $T_{F}y=f$. This proves the closedness of $T_{F}$.

1. (3) $T_{F}$ is a $J$-self-adjoint operator. By the definition of an $m$-sectorial operator we know def $T_F=0$. So

\[ {\text{def}}(T_{F}-\lambda_0\,I)={\text{def}}\,T_F=0, \]

where $\lambda _0\in \Lambda _{\theta _1, K_1}\cap \Lambda _{\theta _2, K_2}$. Then it follows from [6, p115, theorem 5.5] that $T_{F}$ is a $J$-self-adjoint extension of $T_0$.

Remark 4.3 If $E$ has more than one point and (1.1) is in case III for at least two $\theta _1\neq \theta _2({\rm mod}\,\pi )$, then the Hamiltonian system (3.8) is in the limit circle case at $b$, by classical Friedrichs characterization of the Dirichlet boundary conditions at regular endpoint and principal solutions conditions at the singular endpoint as in paper [10], we can obtain the Friedrichs extension of (1.1) in case III. Here we omit the details.

5. Spectral properties of the Friedrichs extensions

In this section, we give some of the properties of the Friedrichs extensions and their applications.

Since $T_F$ is an $m$-sectorial extension of $T_0$, we know that $T_F$ has the smallest form-domain (that is, the domain of the associated form $\textsf {t}$ is contained in the domain of any other $J$-self-adjoint extension operator $T$), and $T_F$ is the only $m-$sectorial extension of $S$ with domain contained in $D(\textsf {t})$. The next theorem characterizes the spectral properties of $T_F$.

Theorem 5.1 Let $\sigma (T_F)$ denote the spectrum of the Friedrichs extension operator $T_F$. Then we have $\sigma (T_F)\subseteq \Omega.$

Proof. Since each spectral point of given operator lies in the numerical range of the operator, we turn to prove the numerical range of $T_F$

\[ \Theta(T_F)=\{\langle T_Fu, u\rangle: u\in D(T_F), ||u||=1\} \]

is contained in $\Omega$. For all $\lambda \in \Theta (T_F)$ there exists $u\in D(T_F), ||u||=1$ such that

\[ \lambda=\langle T_Fu, u \rangle=\textsf{t}[u, u]=\int_a^b[p|u'|^2+q|u|^2]. \]

First, we assume $u\neq 0$ in any subinterval of $[a, b)$. Then

\begin{align*} \lambda& =\int_a^b\left(p(x)\frac{|u'(x)|^2}{w(x)|u(x)|^2}+\frac{q(x)}{w(x)}\right)w(x)|u(x)|^2\,{\rm d}x\\ & :=\int_a^b\left(p(x)r_1(x)+\frac{q(x)}{w(x)}\right)w(x)|u(x)|^2\,{\rm d}x, \end{align*}

where $\displaystyle {r_1(x)=\frac {|u'(x)|^2}{w(x)|u(x)|^2}\in (0, \infty )}$. For arbitrary $\epsilon >0$, there exists $T'\in (a, b)$ such that

\begin{align*} & \left|\int_a^{T'}\left(p(x)r_1(x)+\frac{q(x)}{w(x)}\right)w(x)|u(x)|^2\,{\rm d}x-\lambda\right|<\frac{\epsilon}{4}, \\ & \left|\int_{T'}^b\left(p(x)r_1(x)+\frac{q(x)}{w(x)}\right)w(x)|u(x)|^2\,{\rm d}x\right|<\frac{\epsilon}{4}. \end{align*}

Since $||u||=1$ implies $\int _a^bw(x)|u(x)|^2\,{\rm d}x=1$, there exists $T^{''}\in (a, b)$ such that

\[ \int_{T^{''}}^bw(x)|u(x)|^2\,{\rm d}x<\frac{\epsilon}{4}\left|p(T')r_1(T')+\frac{q(T')}{w(T')}\right|^{{-}1}. \]

Let $T=\max \{T', T^{''}\}$ and $\displaystyle {\lambda _T=\int _a^{T}(p(x)r_1(x)+\frac {q(x)}{w(x)})w(x)|u(x)|^2\,{\rm d}x}$. Then

\begin{align*} & |\lambda_{T}-\lambda|<\frac{\epsilon}{2}, \\ & \int_{T}^bw(x)|u(x)|^2\,{\rm d}x<\frac{\epsilon}{4}\left[p(T')r_1(T')+\frac{q(T')}{w(T')}\right]^{{-}1}. \end{align*}

Now we consider the constant $\lambda _{T}$, by the definition of integrand, for $\epsilon >0$, there exists $\delta >0$, and a partition $a=x_0< x_1< x_2<\cdots < x_n=T$,

\[ \left| \sum_{i=1}^n\left[p(\xi_i)r_1(\xi_i)+\frac{q(\xi_i)}{w(\xi_i)}\right] \Delta\left(\int_{x_{i-1}}^{x_i}w(x)|u(x)|^2\,{\rm d}x\right)-\lambda_{T}\right|<\frac{\epsilon}{4} \]

provided $\displaystyle {\Delta (T)=\max _{1\leq i\leq n}\{x_i-x_{i-1}\}<\delta }$, where $\xi _i\in (x_{i-1}, x_i)$. Since $\int _a^bw(x)|u(x)|^2 {\rm d}x=1$, we obtain

\begin{align*} \rho(\Delta)& :=\sum_{i=1}^n\left[p(\xi_i)r_1(\xi_i)+\frac{q(\xi_i)}{w(\xi_i)}\right] \Delta\left(\int_{x_{i-1}}^{x_i}w(x)|u(x)|^2\,{\rm d}x\right)\\ & \quad+\left[p(T')r_1(T')+\frac{q(T')}{w(T')}\right]\int_{T}^bw(x)|u(x)|^2\,{\rm d}x\\ & \in {\rm co}\left\{\frac{q(x)}{w(x)}+rp(x):\,r>0, x\in(a, b)\right\}. \end{align*}

So we obtain

\begin{align*} |\lambda-\rho(\Delta)|& \leq |\lambda-\lambda_T|+\left|\lambda_{T}- \sum_{i=1}^n\left[p(\xi_i)r_1(\xi_i)+\frac{q(\xi_i)}{w(\xi_i)}\right] \Delta\left(\int_{x_{i-1}}^{x_i}w(x)|u(x)|^2\,{\rm d}x\right)\right|\\ & \quad+\left| \sum_{i=1}^n\left[p(\xi_i)r_1(\xi_i)+\frac{q(\xi_i)}{w(\xi_i)}\right] \Delta\left(\int_{x_{i-1}}^{x_i}w(x)|u(x)|^2\,{\rm d}x\right)-\rho(\Delta)\right|\\ & < \frac{\epsilon}{2}+\frac{\epsilon}{4}+\left[p(T')r_1(T')+\frac{q(T')}{w(T')}\right]\int_{T}^bw(x)|u(x)|^2\,{\rm d}x\\ & = \frac{3\epsilon}{4}+\left|p(T')r_1(T')+\frac{q(T')}{w(T')}\right|\int_{T}^bw(x)|u(x)|^2\,{\rm d}x<\epsilon, \end{align*}

which implies

\[ \lambda\in \overline{\rm co}\left\{\frac{q(x)}{w(x)}+rp(x):\,r>0, x\in(a, b)\right\}=\Omega. \]

For the case where $u(t)\equiv 0$ in some sub-interval of $[a, b)$, we define $V=\bigcup _{i=1}^n(a_i, b_i)$, where $n$ is finite or $n=\infty$, $(a_i, b_i)$ are disjoint open intervals such that $u(x)\equiv 0$ for $x\in (a_i, b_i)$, $1\leq i\leq n$. We assume further that $a< a_i< b_i< a_{i+1}$ and the set $\{a_i\}$ has no finite accumulation point if $b=\infty$, and the only possible finite accumulation point is $b$ if $b<\infty$.

Let $\tau =\tau (x)=m([a, x]-V)$, where $m$ denotes the usual linear Lebesgue measure, and let $B=\tau (b)$, $A_i=\tau (a_i)$, $i=1, 2, \cdots, n$. $[a, B)$ is obtained from $[a, b)$ by shrinking each interval $(a_i, b_i)$ to its left endpoint.

Let $D_I$ denote the set of piecewise continuous functions on $I$. We construct a class of transformation $L_V: D_{[a, b)}\to D_{[a, B)}$ as follows: Let $f\in D_{[a, b)}$. Then $F=L_V(f)$ is defined by

\[ F(\tau)=f(x)\ {\rm if}\ \tau =\tau(x), \tau \neq A_i,\ {\rm and}\ F(A_i)=f(a_i). \]

The function $F$ is obtained from $f$ by collapsing each interval $(a_i, b_i)$ to a point. For all $\lambda \in \Theta (T_F)$, we obtain

\begin{align*} \lambda& =\int_a^b[p(x)|u'(x)|^2+q(x)|u(x)|^2]\,{\rm d}x\\ & = \int_a^B\left[(L_Vp)(\tau)|(L_Vu)'(\tau)|^2+(L_Vq)(\tau)|(L_Vu)(\tau)|^2\right]{\rm d}\tau. \end{align*}

The remainder is similar as above, so we omit the details. In both cases, we have proved $\Theta (T_F)\subseteq \Omega$. This completes the proof.

Corollary 5.2 Let $\sigma (T_F)$ denote the spectrum of the Friedrichs extension operator $T_F$. Then for all $(\theta _0, K_0)\in \Pi$, we have

\begin{align*} & \inf\left\{\Re\left\{{\rm e}^{i\theta}\langle (T_0-K_0)y, y\rangle: y\in D(T_0), ||y||=1\right\}\right\}\\ & \quad =\min\left\{\Re\left\{{\rm e}^{i\theta}(\mu-K_0):\mu\in\sigma(T_F)\right\}\right\}. \end{align*}