Regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations

Abstract

This paper is concerned with regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations. It is shown that the kth eigenvalue of any given self-adjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular self-adjoint subspace extensions in the case that each endpoint is either regular or in the limit circle case. Furthermore, error estimates for the approximations of eigenvalues are given in this case. In addition, it is shown that isolated eigenvalues in every gap of the essential spectrum of any self-adjoint subspace extension are exactly the limits of eigenvalues of suitably chosen induced regular self-adjoint subspace extensions in the case that at least one endpoint is in the limit point case.

1 Introduction

Consider the following second-order symmetric linear difference equation:

$${-}\nabla\bigl(p(t)\triangle x(t)\bigr)+q(t)x(t)= \lambda w(t)x(t),\quad t\in I, \qquad(1.1_{\lambda}) $$

where I is the integer set $\{t\}_{t=a}^{b}$, a is a finite integer or −∞ and b is a finite integer or +∞; △ and ∇ are the forward and backward difference operators, respectively, i.e., $\triangle x(t)=x(t+1)-x(t)$, $\nabla x(t)=x(t)-x(t-1)$; $p(t)$ and $q(t)$ are all real-valued with $p(t)\neq0$ for $t\in I$, $p(a-1)\neq0$ if a is finite and $p(b+1)\neq0$ if b is finite; $w(t)>0$ for $t\in I$; and λ is a complex spectral parameter.

Spectral problems can be divided into two classifications. Those defined over finite closed intervals with well-behaved coefficients are called regular; otherwise they are called singular. Regular approximations of spectra of singular differential equations have been investigated widely and deeply, and some good results have been obtained, including spectral inclusion in general cases and spectral exactness in the case that each endpoint is either regular or in the limit circle case (briefly, l.c.c.) [1–9]. In particular, Stolz, Weidmann, and Teschl [5–9] got spectral exactness for isolated eigenvalues in essential spectral gaps. In addition, Brown et al. [3] constructed a sequence of regular problems for a given fourth-order singular symmetric differential operator and showed that the eigenvalues of the singular problem are exactly the limits of eigenvalues of this sequence in the case that each endpoint is either regular or in l.c.c.

In the present paper, we are wondering whether there are analogous results for singular symmetric difference equations. We shall study a similar problem for singular second-order symmetric linear difference equations. Note that for a symmetric linear difference equation, its minimal operator may not be densely defined, and its minimal and maximal operators may be multi-valued (cf. [10–12]). So it cannot be treated by the methods described in [3, 5–9], which are based on self-adjoint extensions of densely defined Hermitian operators.

This major difficulty can be overcome by using the theory of self-adjoint subspace extensions of Hermitian subspaces. This theory was developed by Coddington, Dijksma, de Snoo, and others (cf. [13–19]). The second author of the present paper extended the classical Glazman-Krein-Naimark (briefly, GKN) theory to Hermitian subspaces [11], and based on this, she with her coauthor Sun presented complete characterizations of self-adjoint extensions for second-order symmetric linear difference equation in both regular and singular cases [12]. Later, she studied some spectral properties of self-adjoint subspaces together with her coauthors Shao and Ren [20]. Recently, based on the above results, we studied the resolvent convergence and spectral approximations of sequences of self-adjoint subspaces [21].

Applying the results given in [12, 21], we studied regular approximations of spectra of singular second-order symmetric linear difference equations [22]. We constructed suitable induced regular self-adjoint subspace extensions and proved that the sequence of induced regular self-adjoint subspace extensions is both spectrally inclusive and exact for a given self-adjoint subspace extension in the case that each endpoint is either regular or in l.c.c., while, in general, it is only spectrally inclusive in the case that at least one endpoint is in the limit point case (briefly, l.p.c.). Here, we shall further investigate how to approximate the spectrum of singular second-order symmetric linear difference equations with eigenvalues of regular problems in the case that each endpoint is either regular or in l.c.c. Furthermore, we shall also give their error estimates. In addition, enlightened by Stolz, Weidmann, and Teschl’s work [5–9], we shall show the spectral exactness in an open interval laking essential spectral points in the case that at least one endpoint is in l.p.c.

In the study of regular approximation problems, the related induced regular self-adjoint extensions should be extended to the whole interval referred for the singular problems. This problem can easily be dealt with by ‘zero extension’ in the continuous case. But it is somewhat difficult in the discrete case. This difficulty was overcome in Section 3.2 in [22] and recalled in Section 2.3 in the present manuscript. So the method used in the present manuscript is not a trivial and direct generalization of that used for ODEs [3, 5–9]. Further, we shall remark that although the minimal operator is densely defined in the case that $a = -\infty$ and $b = +\infty$, the minimal operators of the induce regular problems that will be used to approximate the singular one are not densely defined, and so their self-adjoint extensions have to be characterized by the theory of subspaces. These self-adjoint extensions are multi-valued in general. Therefore, it is better for us to uniformly apply the theory of subspaces to study regular approximations in all the cases in the present paper.

The rest of this paper is organized as follows. In Section 2, some basic concepts and fundamental results about subspaces in Hilbert spaces and second-order symmetric linear difference equations are introduced. In addition, the induced regular self-adjoint subspace extensions for any given self-adjoint subspace extension are introduced. In particular, a sufficient condition is given for spectral exactness of a sequence of self-adjoint subspaces in an open interval laking essential spectral points. It will play an important role in the study of regular approximations in the case that at least one endpoint is in l.p.c. In Section 3, regular approximations of isolated eigenvalues of equation (1.1) are studied in the case that each endpoints is either regular or in l.c.c. It is shown that the kth eigenvalue of the given self-adjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular self-adjoint subspace extensions. In addition, their error estimates are given. Spectral exactness in every gap of the essential spectrum of any self-adjoint subspace extension is obtained in the other three cases in Sections 4-6, separately.

2 Preliminaries

This section is divided into three parts. In Section 2.1, some basic concepts and fundamental results about subspaces are listed. In Section 2.2, the maximal, pre-minimal, and minimal subspaces corresponding to equation (1.1) are introduced. In Section 2.3, some results about self-adjoint subspace extensions of the minimal subspace and their induced self-adjoint restrictions given in [22] are recalled.

2.1 Some basic concepts and fundamental results about subspaces

By C, R, and N denote the sets of the complex numbers, real numbers, and positive integer numbers, respectively. Let X be a complex Hilbert space with inner product $\langle \cdot,\cdot\rangle$, and T a linear subspace (briefly, subspace) in the product space $X^{2}$ with the following induced inner product, still denoted by $\langle\cdot,\cdot\rangle$ without any confusion:

$$\bigl\langle (x,f), (y,g)\bigr\rangle =\langle x, y\rangle+\langle f, g\rangle, \quad (x,f), (y,g)\in X^{2}. $$

Denote the domain, range, and null space of T by $D(T)$, $R(T)$, and $N(T)$, respectively. Its adjoint subspace $T^{*}$ is defined by

$$T^{*}=\bigl\{ (y,g)\in X^{2}: \langle g,x\rangle=\langle y,f\rangle \mbox{ for all } (x,f)\in T\bigr\} . $$

Further, denote

$$T(x):=\bigl\{ f\in X: (x,f)\in T\bigr\} , \qquad T^{-1}:=\bigl\{ (f,x): (x,f)\in T\bigr\} . $$

It is evident that $T(0)=\{0\}$ if and only if T can uniquely determine a (singled-valued) linear operator from $D(T)$ into X whose graph is T. For convenience, a linear operator in X will always be identified with a subspace in $X^{2}$ via its graph.

Let T and S be two subspaces in $X^{2}$ and $\lambda\in{\mathbf {C}}$. Define

$$\begin{aligned}& \lambda T:=\bigl\{ (x,\lambda f): (x,f)\in T\bigr\} , \\& T+S:= \bigl\{ (x,f+g): (x,f)\in T, (x,g)\in S\bigr\} , \\& ST:=\bigl\{ (x,g)\in X^{2}: (x,f)\in T, (f,g)\in S \mbox{ for some } f \in X\bigr\} . \end{aligned}$$

It is evident that if T is closed, then $T-\lambda I_{id}$ is closed and $(T-\lambda I_{id})^{*}=T^{*}-{\bar{\lambda}}I_{id}$, where $I_{id}:=\{(x,x): x\in X\}$, briefly denoted by I without any confusion between it and the interval I.

Throughout the whole paper, denote the resolvent set, spectrum, point spectrum, essential spectrum, and discrete spectrum of T by $\rho(T)$, $\sigma(T)$, $\sigma_{p}(T)$, $\sigma_{e}(T)$, and $\sigma_{d}(T)$, respectively.

Definition 2.1

([21], Definition 5.1)

Let $\{T_{n}\}_{n=1}^{\infty}$ and T be subspaces in $X^{2}$.

• (1) The sequence $\{T_{n}\}_{n=1}^{\infty}$ is said to be spectrally inclusive for T if for any $\lambda\in\sigma(T)$, there exists a sequence $\{\lambda_{n}\}_{n=1}^{\infty}$, $\lambda_{n}\in\sigma(T_{n})$, such that $\lim_{n\to\infty} \lambda_{n}=\lambda$.

• (2) The sequence $\{T_{n}\}_{n=1}^{\infty}$ is said to be spectrally exact for T if it is spectrally inclusive and every limit point of any sequence $\{\lambda_{n}\}_{n=1}^{\infty}$ with $\lambda_{n}\in\sigma (T_{n})$ belongs to $\sigma(T)$.

• (3) The sequence $\{T_{n}\}_{n=1}^{\infty}$ is said to be spectrally exact for T in some set $\Omega\subset{\mathbf {R}}$ if the condition in (2) holds in Ω.

Lemma 2.1

([21], Lemma 2.1)

Let T be a closed subspace in $X^{2}$. Then

$$\begin{aligned}& \rho\bigl(T^{-1}\bigr)\setminus\{0\}=\bigl\{ \lambda^{-1}: \lambda\in\rho(T)\textit{ with }\lambda\ne0\bigr\} , \\& \sigma\bigl(T^{-1}\bigr)\setminus\{0\}=\bigl\{ \lambda^{-1}: \lambda\in\sigma(T)\textit{ with }\lambda\ne0\bigr\} . \end{aligned}$$

Consequently, if $\rho(T)\ne\emptyset$, then

$$\sigma\bigl((\lambda_{0} I-T)^{-1}\bigr)\setminus\{0\}=\bigl\{ (\lambda_{0}-\lambda )^{-1}: \lambda\in\sigma(T)\bigr\} ,\quad \lambda_{0}\in\rho(T). $$

Let T and S be two subspaces in $X^{2}$. If $T\cap S=\{(0,0)\}$, denote

$$T\dot{+} S:=\bigl\{ (x+y,f+g): (x,f)\in T,(y,g)\in S\bigr\} . $$

Further, if T and S are orthogonal, denoted by $T\perp S$; that is, $\langle(x,f), (y,g)\rangle=0$ for any $(x,f)\in T$, $(y,g)\in S$, we denote

$$T\oplus S:=T\dot{+} S. $$

In addition, we introduce the following notation for convenience:

$$T\ominus S:=\bigl\{ (x,f)\in T: \langle x,y\rangle+\langle f,g\rangle=0 \mbox{ for all } (y,g)\in S\bigr\} . $$

In 1961, Arens [23] introduced the following important decomposition for a closed subspace T in $X^{2}$:

$$T=T_{s}\oplus T_{\infty}, $$

where

$$T_{\infty}:=\bigl\{ (0,f)\in X^{2}: (0,f)\in T\bigr\} ,\qquad T_{s} :=T\ominus T_{\infty}. $$

It can easily be verified that $T_{s}$ is an operator, and T is an operator if and only if $T=T_{s}$. $T_{s}$ and $T_{\infty}$ are called the operator and pure multi-valued parts of T, respectively. In addition,

$$ R(T_{\infty})=T(0),\qquad T_{\infty}=\{0\}\times T(0), \qquad R(T_{s})\subset T(0)^{\perp},\qquad D(T_{s})=D(T), $$

(2.1)

and $D(T_{s})=D(T)$ is dense in $T^{*}(0)^{\perp}$.

Throughout the present paper, the resolvent set and spectrum of $T_{s}$ and $T_{\infty}$ mean those of $T_{s}$ and $T_{\infty}$ restricted to $(T(0)^{\perp})^{2}$ and $T(0)^{2}$, respectively.

Lemma 2.2

([20], Proposition 2.1 and Theorems 2.1, 2.2, and 3.4)

Let T be a closed Hermitian subspace in $X^{2}$. Then

$$T_{\infty}=T\cap T(0)^{2}, \qquad T_{s}=T\cap \bigl(T(0)^{\perp}\bigr)^{2}, $$

$T_{s}$ is a closed Hermitian operator in $T(0)^{\perp}$, $T_{\infty}$ is a closed Hermitian subspace in $T(0)^{2}$,

$$\begin{aligned}& \rho(T)=\rho(T_{s}),\qquad \sigma(T)=\sigma(T_{s}), \qquad \sigma(T_{\infty})=\emptyset, \\& \sigma_{p}(T)=\sigma_{p}(T_{s}),\qquad \sigma_{e}(T)=\sigma_{e}(T_{s}), \qquad\sigma _{d}(T)=\sigma_{d}(T_{s}), \end{aligned}$$

and $N(T-\lambda I)=N(T_{s}-\lambda I)$ for every $\lambda\in\sigma_{p}(T)$.

Lemma 2.3

([17], p.26)

If T is a self-adjoint subspace in $X^{2}$, then $T_{\infty}$ and $T_{s}$ are self-adjoint subspaces in $T(0)^{2}$ and $(T(0)^{\perp})^{2}$, respectively.

To end this subsection, we shall briefly recall the concept of the spectral family of a self-adjoint subspace, which was introduced by Coddington and Dijksma in [15].

Let T be a self-adjoint subspace in $X^{2}$. By Lemma 2.3, $T_{s}$ is a self-adjoint operator in $T(0)^{\perp}$. Then $T_{s}$ has the following spectral resolution:

$$T_{s}= \int t \,dE_{s}(t), $$

where $\{E_{s}(t)\}_{t \in{\mathbf {R}} }$ is the spectral family of $T_{s}$ in $T(0)^{\perp}$. The spectral family of the subspace T is defined by

$$E(t)=E_{s}(t)\oplus O, \quad t \in{\mathbf {R}}, $$

where O is the zero operator defined on $T(0)$. It is obvious that for any $t\in{\mathbf {R}}$ and any $f\in X$,

$$ E(t)f=E_{s}(t)f_{1}, $$

(2.2)

where $f=f_{1}+f_{2}$ with $f_{1}\in T(0)^{\perp}$ and $f_{2}\in T(0)$.

The following result weakens the condition (5.7) of Theorem 5.3 in [21]. It will be useful in studying spectral exactness in every gap of the essential spectrum of any self-adjoint subspace extension in Sections 4-6.

Lemma 2.4

Let $\{T_{n}\}_{n=1}^{\infty}$ and T be self-adjoint subspaces in $X^{2}$, and let $E(T_{n},\lambda)$ and $E(T,\lambda)$ be spectral families of $T_{n}$ and T, respectively. Assume that $I_{0}\subset{\mathbf {R}}$ is an open interval and satisfies

$$ I_{0} \cap\sigma_{e}(T)=\emptyset, \qquad I_{0} \cap\sigma_{d}(T)\ne\emptyset. $$

(2.3)

Let $\gamma\in I_{0}$. If for any given $\alpha,\beta\in I_{0}\cap\rho (T)$ with $\alpha<\gamma\leq\beta$, there exists an integer $n_{0} \ge1$ such that for all $n\geq n_{0}$,

$$ \dim R\bigl\{ E(T_{n},\beta)-E(T_{n},\alpha) \bigr\} =\dim R\bigl\{ E(T,\beta)-E(T,\alpha)\bigr\} , $$

(2.4)

then $\{T_{n}\}_{n=1}^{\infty}$ is spectrally exact for T in $I_{0}$.

Proof

By Theorem 5.3 in [21], it suffices to show that (2.4) holds for all $\alpha,\beta\in I_{0}\cap \rho(T)$ with $\alpha<\beta$. Fix any $\alpha,\beta\in I_{0}\cap\rho(T)$ with $\alpha<\beta$. The following discussions are divided into three cases.

Case 1. $\gamma\in(\alpha,\beta]$. Obviously, (2.4) holds in this case.

Case 2. $\gamma\leq\alpha$. By (2.3), there exists $\epsilon>0$ such that $\gamma-\epsilon\in I_{0}\cap\rho(T)$. By (2.4), there exists $n_{0}\ge1$ such that for all $n\ge n_{0}$,

$$ \begin{aligned} &\dim R\bigl\{ E(T_{n}, \beta)-E(T_{n},\gamma-\epsilon)\bigr\} =\dim R\bigl\{ E(T,\beta )-E(T, \gamma-\epsilon)\bigr\} , \\ &\dim R\bigl\{ E(T_{n},\alpha)-E(T_{n},\gamma-\epsilon) \bigr\} = \dim R\bigl\{ E(T,\alpha )-E(T,\gamma-\epsilon)\bigr\} . \end{aligned} $$

(2.5)

Note that

$$\begin{aligned}& E(T_{n},\beta)-E(T_{n},\alpha)=\bigl\{ E(T_{n}, \beta)-E(T_{n},\gamma-\epsilon)\bigr\} -\bigl\{ E(T_{n}, \alpha)-E(T_{n},\gamma-\epsilon)\bigr\} , \\& E(T,\beta)-E(T,\alpha)=\bigl\{ E(T,\beta)-E(T,\gamma-\epsilon)\bigr\} -\bigl\{ E(T, \alpha )-E(T,\gamma-\epsilon)\bigr\} ; \end{aligned}$$

that is,

$$ \begin{aligned} &E\bigl(T_{n},(\alpha, \beta]\bigr)=E\bigl(T_{n},( \gamma-\epsilon,\beta]\bigr)-E\bigl(T_{n},(\gamma -\epsilon,\alpha]\bigr), \\ &E\bigl(T,(\alpha, \beta]\bigr)=E\bigl(T,(\gamma-\epsilon,\beta]\bigr) -E\bigl(T,(\gamma-\epsilon , \alpha]\bigr). \end{aligned} $$

(2.6)

Since $E(T_{n},(\alpha, \beta])$ and $E(T,(\alpha, \beta])$ are orthogonal projections and

$$\begin{aligned}& R\bigl\{ E\bigl(T_{n},(\gamma-\epsilon,\alpha]\bigr)\bigr\} \subset R\bigl\{ E\bigl(T_{n},(\gamma-\epsilon ,\beta]\bigr)\bigr\} , \\& R\bigl\{ E\bigl(T,(\gamma-\epsilon,\alpha]\bigr)\bigr\} \subset R\bigl\{ E\bigl(T,(\gamma-\epsilon ,\beta]\bigr)\bigr\} , \end{aligned}$$

by (c) of Theorem 4.30 in [24] we have

$$\begin{aligned}& R\bigl\{ E\bigl(T_{n},(\alpha, \beta]\bigr)\bigr\} =R\bigl\{ E\bigl(T_{n},( \gamma-\epsilon,\beta]\bigr)\bigr\} \ominus R\bigl\{ E\bigl(T_{n},(\gamma-\epsilon, \alpha]\bigr)\bigr\} , \\& R\bigl\{ E\bigl(T,(\alpha, \beta]\bigr)\bigr\} = R\bigl\{ E\bigl(T,(\gamma-\epsilon,\beta]\bigr)\bigr\} \ominus R\bigl\{ E\bigl(T,(\gamma-\epsilon,\alpha]\bigr)\bigr\} . \end{aligned}$$

It follows that

$$\begin{aligned}& \dim R\bigl\{ E\bigl(T_{n},(\alpha, \beta]\bigr)\bigr\} = \dim R\bigl\{ E\bigl(T_{n},(\gamma-\epsilon ,\beta]\bigr)\bigr\} - \dim R\bigl\{ E\bigl(T_{n},(\gamma-\epsilon,\alpha]\bigr)\bigr\} , \\& \dim R\bigl\{ E\bigl(T,(\alpha, \beta]\bigr)\bigr\} = \dim R\bigl\{ E\bigl(T,(\gamma-\epsilon, \beta ]\bigr)\bigr\} - \dim R\bigl\{ E\bigl(T,(\gamma-\epsilon,\alpha]\bigr)\bigr\} . \end{aligned}$$

This, together with (2.5), shows that (2.4) holds in this case.

Case 3. $\gamma>\beta$. With a similar argument for Case 2, one can easily show that (2.4) holds in Case 3. This completes the proof. □

2.2 Maximal, pre-minimal and minimal subspaces

In this subsection, we first introduce the concepts of maximal, pre-minimal and minimal subspaces corresponding to (1.1) and then briefly recall their properties.

Since a, b may be finite or infinite, we give the following convention for briefness in the sequent expressions: $a-1$ means −∞ in the case of $a=-\infty$; $b+1$ means +∞ in the case of $b=+\infty$.

Denote

$$l_{w}^{2}(I):=\Biggl\{ x=\bigl\{ x(t)\bigr\} _{t=a-1}^{b+1}\subset\mathbf{C}: \sum ^{b}_{t=a} w(t) \bigl|x(t)\bigr|^{2}< +\infty\Biggr\} . $$

Then $l_{w}^{2}(I)$ is a Hilbert space with the inner product

$$\langle x,y\rangle:=\sum^{b}_{t=a} w(t){ \bar{y}}(t)x(t), $$

where $x=y$ in $l_{w}^{2}(I)$ if and only if $\|x-y\|=0$, i.e., $x(t)=y(t)$, $t\in I$, while $\|\cdot\|$ is the induced norm.

The natural difference operator corresponding to (1.1) is denoted by

$$\mathcal{L}(x) (t):=-\nabla\bigl(p(t)\triangle x(t)\bigr)+q(t)x(t), \quad t\in I. $$

Now, we introduce the corresponding maximal, pre-minimal, and minimal subspaces corresponding to (1.1) in the interval I. Let

$$\begin{aligned}& H:=\bigl\{ (x,f)\in\bigl(l_{w}^{2}(I)\bigr)^{2}: \mathcal{L}(x) (t)=w(t)f(t), t\in I\bigr\} , \\& \begin{aligned}[b] H_{00}:={}&\bigl\{ (x,f)\in H: \mbox{there exist two integers } t_{0}, t_{1}\in I\mbox{ with } t_{0}< t_{1} \\ &{}\mbox{such that } x(t)=0 \mbox{ for } t\leq t_{0} \mbox{ and } t \geq t_{1}\bigr\} , \end{aligned} \end{aligned}$$

where H and $H_{00}$ are called the maximal and pre-minimal subspaces corresponding to $\mathcal{L}$ or (1.1), respectively. The subspace $H_{0}:={\bar{H}_{00}}$ is called the minimal subspace corresponding to $\mathcal{L}$ or (1.1). By Corollary 3.1 and Theorem 3.3 in [12], $H_{0}$ is a closed densely defined Hermitian operator in $l_{w}^{2}(I)$ in the case that $a=-\infty$ and $b=+\infty$, and a closed non-densely defined Hermitian operator in $l_{w}^{2}(I)$ in the other case that at least one of a and b is finite. In addition, $H\subset{H_{0}}^{*}$ and $H={H_{0}}^{*}$ in the sense of the norm $\|\cdot\|$.

In addition, we take the notation for convenience:

$$ (x,y) (t)=p(t)\bigl[\bigl(\triangle\bar{y}(t)\bigr)x(t)-\bar{y}(t) \triangle x(t)\bigr],\quad t=\{t\}_{t=a-1}^{b}. $$

(2.7)

2.3 Self-adjoint subspace extensions and their induced self-adjoint restrictions

In this subsection, we recall the results about self-adjoint subspace extensions of $H_{0}$ and their induced regular self-adjoint subspace extensions, i.e., induced self-adjoint restrictions constructed in [22].

Let $I_{r}=\{t\}_{t=a_{r}}^{b_{r}}$, where $-\infty< a_{r}+1< b_{r}-1<+\infty$, $a_{r+1}\leq a_{r}< b_{r}\leq b_{r+1}$, $r\in\mathbf{N}$, and $a_{r}\rightarrow a$, $b_{r}\rightarrow b$ as $r\rightarrow\infty$. That is, $\lim_{r\to\infty}I_{r}=I$. If a (resp. b) is finite, take $a_{r}=a$ (resp. $b_{r}=b$). For convenience, by $H^{r}$ and $H^{r}_{0}$ denote the corresponding maximal and minimal subspaces to equation (1.1) or $\mathcal{L}$ on $I_{r}$, respectively. Noting that all the coefficient functions p and q and weight function w in (1.1) are real-valued, one has $d_{+}(H_{0})=d_{-}(H_{0})$, where $d_{\pm}(H_{0})$ are the positive and negative defect indices of $H_{0}$. Consequently, $H_{0}$ has self-adjoint subspace extensions by [14].

Let $\varphi_{1}(\cdot,\lambda)$ and $\varphi_{2}(\cdot,\lambda)$ be two linearly independent solutions of (1.1 _λ ) with $\lambda\in\mathbf{R}$ satisfying the following initial conditions:

$$\begin{aligned} \begin{aligned} &\varphi_{1}(d_{0}-1, \lambda)=0, \qquad p(d_{0}-1)\triangle\varphi_{1}(d_{0}-1, \lambda )=-1, \\ &\varphi_{2}(d_{0}-1,\lambda)=1, \qquad p(d_{0}-1) \triangle\varphi_{2}(d_{0}-1,\lambda)=0, \end{aligned} \end{aligned}$$

(2.8)

where $d_{0}\in I$ is any fixed.

In the case that $I=[a,+\infty)$ (resp. $I=(-\infty,b]$), $\mathcal{L}$ is regular at a (resp. b) and either in l.c.c. or l.p.c. at $t=+\infty$ (resp. $t=-\infty$). In the case that $I=(-\infty,+\infty)$, $\mathcal{L}$ is either in l.c.c. or l.p.c. at each endpoint. Consequently, the following discussions are divided into the five cases due to different expressions of their self-adjoint subspace extensions.

Case 1. One endpoint is regular and the other in l.c.c.

Without loss of generality, we only consider the case that $\mathcal{L}$ is regular at a and in l.c.c. at $t=+\infty$. Take $d_{0}=a$ in (2.8) in this case.

Suppose that $H_{1}$ is any fixed self-adjoint subspace extension of $H_{0}$. Then, by (3.5) in [22], we have

$$ H_{1}= \bigl\{ (x,f)\in H: (x,\hat{y}_{j}) (a-1)-(x,\hat{y}_{j}) (+\infty)=0, j=1,2 \bigr\} , $$

(2.9)

where

$$\begin{aligned} &\hat{y}_{j}(a-1)=-m_{2j}, \qquad p(a-1) \triangle\hat{y}_{j}(a-1)=m_{1j},\qquad \hat{y}_{j}(t)=u_{j}(t), \quad t\geq c_{0}, \\ &JM^{*}= \begin{pmatrix} -m_{21} & -m_{22} \\ m_{11} & m_{12} \end{pmatrix},\qquad N=-(n_{ij})_{2\times2}, \qquad u_{j}:=\sum^{2}_{k=1} \bar{n}_{jk}\varphi_{k}, \quad j=1,2, \end{aligned}$$

(2.10)

while

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

matrices $M, N \in\mathbf{C}^{2\times2}$ satisfying $\operatorname{rank}(M,N)=2$ and $MJM^{*}=NJN^{*}$, and $c_{0}> a+1$ is any fixed integer.

Let $a_{r}=a$ and $b_{r}>c_{0}$. According to (3.7) in [22], an induced self-adjoint restriction of $H_{1}$ on $I_{r}$ can be given by

$$ H_{1,r}= \bigl\{ (x,f)\in H^{r}: (x, \hat{y}_{j}) (a-1)-(x,\hat{y}_{j}) (b_{r})=0, j=1,2 \bigr\} . $$

(2.11)

Case 2. One endpoint is regular and the other in l.p.c.

Without loss of generality, we only consider the case that $\mathcal{L}$ is regular at a and in l.p.c. at $t=+\infty$. Still take $d_{0}=a$ in (2.8) in this case.

Suppose that $H_{1}$ is any fixed self-adjoint subspace extension of $H_{0}$. Then, by (3.9) in [22], we have

$$ H_{1}=\bigl\{ (x,f)\in H: (x,\hat{y}) (a-1)=0\bigr\} , $$

(2.12)

where

$$ \hat{y}(a-1)=-m_{2},\qquad p(a-1)\triangle \hat{y}(a-1)=m_{1}, \qquad \hat {y}(t)=0, \quad t\geq c_{0}, $$

(2.13)

with $M=(m_{1},m_{2})\in\mathbf{R}^{1\times2}$ and $M\neq0$, and $c_{0}> a+1$ is any fixed integer.

Let $a_{r}=a$ and $b_{r}>c_{0}$. According to the discussion for (3.12) in [22], an induced self-adjoint restriction of $H_{1}$ on $I_{r}$ can be given by

$$ H_{1,r}=\bigl\{ (x,f)\in H^{r}: (x,\hat{y}) (a-1)=0,(x,u) (b_{r})=0\bigr\} , $$

(2.14)

where

$$ u=n_{1}\varphi_{1}(\cdot, \lambda)+n_{2}\varphi_{2}(\cdot,\lambda), $$

(2.15)

with $N=(n_{1},n_{2})\in\mathbf{R}^{1\times2}$ and $N\neq0$.

Case 3. Both endpoints are in l.c.c.

Suppose that $H_{1}$ is any fixed self-adjoint subspace extension of $H_{0}$. Then, by (4.4) in [22], we have

$$ H_{1}= \bigl\{ (x,f)\in H: (x,\hat{y}_{j}) (- \infty)-(x,\hat{y}_{j}) (+\infty )=0, j=1,2 \bigr\} , $$

(2.16)

where

$$\begin{aligned} \hat{y}_{j}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} u_{j}, & t\leq d_{0}-1, \\ v_{j}, & t\geq d_{0}+1, \end{array}\displaystyle \right .\quad u_{j}:=\sum^{2}_{k=1} \bar{m}_{jk}\varphi_{k}(\cdot,\lambda ),\quad v_{j}:=\sum^{2}_{k=1} \bar{n}_{jk}\varphi_{k}(\cdot,\lambda),\quad j=1,2, \end{aligned}$$

(2.17)

with matrices $M=(m_{jk})_{2\times2}$ and $N=(n_{jk})_{2\times2}$ satisfying $\operatorname{rank}(M,N)=2$ and $MJM^{*}=NJN^{*}$.

Let $a_{r}< d_{0}-1$, $b_{r}>d_{0}$. Based on the discussion for (4.6) in [22], an induced self-adjoint restriction of $H_{1}$ on $I_{r}$ can be given by

$$ H_{1,r}= \bigl\{ (x,f)\in H^{r}: (x, \hat{y}_{j}) (a_{r}-1)-(x,\hat {y}_{j}) (b_{r})=0, j=1,2 \bigr\} . $$

(2.18)

Case 4. One endpoint is in l.c.c. and the other in l.p.c.

Without loss of generality, we only consider the case that $\mathcal{L}$ is in l.c.c. at $t=-\infty$ and l.p.c. at $t=+\infty$.

Suppose that $H_{1}$ is any fixed self-adjoint subspace extension of $H_{0}$. According to the discussion for (4.8) in [22], we have

$$ H_{1}=\bigl\{ (x,f)\in H: (x,\hat{y}) (-\infty)=0\bigr\} , $$

(2.19)

where

$$\begin{aligned} \hat{y}(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} m_{1}\varphi_{1}(t,\lambda)+m_{2}\varphi_{2}(t,\lambda), & t\leq d_{0}-1,\\ 0, & t\geq d_{0}+1, \end{array}\displaystyle \right . \end{aligned}$$

(2.20)

while $M=(m_{1},m_{2})\in\mathbf{R}^{1\times2}$ with $M\neq0$.

Let $a_{r}< d_{0}-1$, $b_{r}>d_{0}$. By the discussion for (4.11) in [22], an induced self-adjoint restriction of $H_{1}$ on $I_{r}$ can be given by

$$ H_{1,r}=\bigl\{ (x,f)\in H^{r}: (x,\hat{y}) (a_{r}-1)=0,(x,u) (b_{r})=0\bigr\} , $$

(2.21)

where

$$ u=n_{1}\varphi_{1}(\cdot, \lambda)+n_{2}\varphi_{2}(\cdot,\lambda), $$

(2.22)

while $N=(n_{1},n_{2})\in\mathbf{R}^{1\times2}$ with $N\neq0$.

Case 5. Both endpoints are in l.p.c.

In this case that $\mathcal{L}$ is in l.p.c. at both endpoints $t=\pm \infty$, $H_{1}=H_{0}$ is the unique self-adjoint subspace extension of $H_{0}$.

Let $a_{r}< d_{0}-1$, $b_{r}>d_{0}$. By the discussion for (4.12) in [22], an induced self-adjoint restriction of $H_{1}$ on $I_{r}$ can be given by

$$ H_{1,r}=\bigl\{ (x,f)\in H^{r}: (x,u) (a_{r}-1)=0,(x,u) (b_{r})=0\bigr\} , $$

(2.23)

where

$$\begin{aligned} u=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} m_{1}\varphi_{1}(\cdot,\lambda)+m_{2}\varphi_{2}(\cdot,\lambda), & t\leq d_{0}-1, \\ n_{1}\varphi_{1}(\cdot,\lambda)+n_{2}\varphi_{2}(\cdot,\lambda), & t\geq d_{0}+1, \end{array}\displaystyle \right . \end{aligned}$$

(2.24)

while $M=(m_{1},m_{2})\in\mathbf{R}^{1\times2}$ with $M \neq0$ and $N=(n_{1},n_{2})\in\mathbf{R}^{1\times2}$ with $N \neq0$.

Remark 2.1

By Theorem 6.1 in [12], each self-adjoint subspace extension $H_{1}$ of $H_{0}$ is a self-adjoint operator in the case that $I=(-\infty, +\infty)$; that is, $H_{1}$ can define a single-valued self-adjoint operator in $l_{w}^{2}(-\infty,+\infty)$ whose graph is $H_{1}$.

To end this section, we consider extensions of the induced self-adjoint restrictions from $I_{r}$ to I.

Note that $H_{1}$, $H_{1,r}$ are self-adjoint subspaces in $(l_{w}^{2}(I))^{2}$ and $(l_{w}^{2}(I_{r}))^{2}$, respectively. It is difficult to study the convergence of $H_{1,r}$ to $H_{1}$ in some sense since $l_{w}^{2}(I)$ and $l_{w}^{2}(I_{r})$ are different spaces. In order to overcome this problem, we extended $l_{w}^{2}(I_{r})$ and $H_{1,r}$ to $\tilde{l}_{w}^{2}(I_{r})$ and $\tilde{H}_{1,r}$, separately, in [22]. Now, we recall them for convenience.

In the case that $I=[a, +\infty)$,

$$\begin{aligned} &\tilde{l}_{w}^{2}(I_{r}):=\bigl\{ f \in l_{w}^{2}(I):f(t)=0, t\geq b_{r}+1\bigr\} , \\ &\tilde{H}_{1,r}:=\bigl\{ (\tilde{x},\tilde{f})\in\bigl( \tilde{l}_{w}^{2}(I_{r})\bigr)^{2}: \mbox{there exists } (x,f)\in H_{1,r} \mbox{ such that} \\ &\hphantom{\tilde{H}_{1,r}:=}{}\tilde{x}(t)=x(t),\tilde{f}(t)=f(t), a-1\leq t\leq b_{r}\bigr\} . \end{aligned}$$

(2.25)

In the case that $I=(-\infty, +\infty)$,

$$\begin{aligned} &\tilde{l}_{w}^{2}(I_{r}):=\bigl\{ f \in l_{w}^{2}(I):f(t)=0, t\leq a_{r}-1 \mbox{ and } t\geq b_{r}+1\bigr\} , \\ &\tilde{H}_{1,r}:=\bigl\{ (\tilde{x},\tilde{f})\in\bigl( \tilde{l}_{w}^{2}(I_{r})\bigr)^{2}: \mbox{there exists } (x,f)\in H_{1,r} \mbox{ such that} \\ &\hphantom{\tilde{H}_{1,r}:=}{}\tilde{x}(t)=x(t),\tilde{f}(t)=f(t), a_{r}\leq t\leq b_{r}\bigr\} . \end{aligned}$$

(2.26)

Let $P_{r}$ be the orthogonal projection from $l_{w}^{2}(I)$ onto $\tilde {l}_{w}^{2}(I_{r})$. Define

$$ H_{1,r}':=\tilde{H}_{1,r}G(P_{r}). $$

(2.27)

Lemma 2.5

([22], Lemmas 3.1, 3.2, 3.3 and 4.1)

$\tilde{H}_{1,r}$ and $H_{1,r}'$ are self-adjoint subspaces in $(\tilde {l}_{w}^{2}(I_{r}))^{2}$ and $(l_{w}^{2}(I))^{2}$, respectively, $D(H_{1,r}')=D(\tilde{H}_{1,r})\oplus(\tilde{l}_{w}^{2}(I_{r}))^{\bot}$, $\sigma(\tilde{H}_{1,r})=\sigma(H_{1,r})$, and $\sigma(H_{1,r}')=\sigma (\tilde{H}_{1,r})\cup\{0\}=\sigma(H_{1,r})\cup\{0\}$.

The following result can be directly derived from (2.25)-(2.27).

Lemma 2.6

$H_{1,r}'(0)=\tilde{H}_{1,r}(0)=\{\tilde{f}\in\tilde{l}_{w}^{2}(I_{r}) :\textit{ there exists }f\in H_{1,r}(0)\textit{ such that }\tilde{f}(t) =f(t)\mbox{ }\textit{for }t\in I_{r}\}$.

3 One endpoint is regular or in l.c.c. and the other in l.c.c

In this section, we study regular approximations of isolated eigenvalues of (1.1) in Cases 1 and 3. Without loss of generality, we only consider the case that $\mathcal{L}$ is regular or in l.c.c. at a and l.c.c. at $t=+\infty$.

We showed that the induced self-adjoint restrictions $\{H_{1,r}\} _{r=1}^{\infty}$ is spectrally exact for the given self-adjoint subspace extension $H_{1}$ in Cases 1 and 3 in [22]. Now, we shall further study how the spectrum $\sigma(H_{1})$ of $H_{1}$ is approximated by the eigenvalues of $H_{1,r}$. In addition, we also give their error estimates.

Lemma 3.1

Each self-adjoint subspace extension of $H_{0}$ has a pure discrete spectrum in Cases 1 and 3.

Proof

According to Theorems 6.7 and 6.10 in [24] and Lemma 2.1, it suffices to prove that $(zI-H_{1})^{-1}$ is a Hilbert-Schmidt operator for any $z\in\rho(H_{1})$.

We only prove that $(zI-H_{1})^{-1}$ is a Hilbert-Schmidt operator for any $z\in\rho(H_{1})$ in Case 1 with $I=[a,+\infty)$. For the other cases, it can be proved similarly.

By Proposition 3.1 in [22], for any $z\in\rho(H_{1})$ and any $f\in l_{w}^{2}(I)$,

$$(z I-H_{1})^{-1}(f) (t)=\sum^{+\infty}_{j=a}G(t,j,z)w(j)f(j), \quad t\in I, $$

where

$$G(t,j,z)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \sum_{k,l=1}^{2} m_{kl}\phi_{k}(t)\phi_{l}(j), & a\leq j\leq t< +\infty,\\ \sum_{k,l=1}^{2} n_{kl}\phi_{k}(t)\phi_{l}(j), & a\leq t< j< +\infty, \end{array}\displaystyle \right . $$

while $m_{kl}$, $n_{kl}$ ($1\leq k,l\leq2$) are constants, and $\phi_{1}$, $\phi_{2}$ are two linearly independent solutions of (1.1 _z ) satisfying certain initial conditions. It is evident that $\phi_{1}, \phi_{2} \in l_{w}^{2}(I)$. Denote

$$m_{0}:=\max_{k,l=1,2}\bigl\{ |m_{kl}|\bigr\} , \qquad \alpha_{0}:=\max_{k=1,2}\bigl\{ \|\phi_{k}\|\bigr\} . $$

Define

$$\begin{aligned}& \mathcal{F}_{1}(f) (t):=\sum^{+\infty }_{j=a}F_{1}(t,j,z)w(j)f(j), \\& \mathcal{F}_{2}(f) (t):=\sum^{+\infty }_{j=a}F_{2}(t,j,z)w(j)f(j), \quad t\in I, \end{aligned}$$

where

$$\begin{aligned}& F_{1}(t,j,z)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \sum_{k,l=1}^{2} m_{kl}\phi_{k}(t)\phi_{l}(j), & a\leq j\leq t< +\infty,\\ 0, & a\leq t< j< +\infty, \end{array}\displaystyle \right . \\& F_{2}(t,j,z)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0, & a\leq j\leq t< +\infty,\\ \sum_{k,l=1}^{2} n_{kl}\phi_{k}(t)\phi_{l}(j), & a\leq t< j< +\infty. \end{array}\displaystyle \right . \end{aligned}$$

Obviously, $(z I-H_{1})^{-1}=\mathcal{F}_{1}+\mathcal{F}_{2}$. Thus, it is sufficient to prove that $\mathcal{F}_{1}$ and $\mathcal {F}_{2}$ are both Hilbert-Schmidt operators.

We first prove that $\mathcal{F}_{1}$ is a Hilbert-Schmidt operator. Let $\{e_{n}: n\in\mathbf{N}\}$ be an orthonormal basis of $l_{w}^{2}(I)$. Then

$$\begin{aligned} \sum^{\infty}_{n=1}\bigl\| \mathcal{F}_{1}(e_{n}) \bigr\| ^{2} =&\sum^{\infty}_{n=1}\Biggl\| \sum ^{t}_{j=a}\sum_{k,l=1}^{2} m_{kl}\phi_{k}(t)\phi_{l}(j)w(j)e_{n}(j) \Biggr\| ^{2} \\ \leq& 8m_{0}^{2}\alpha_{0}^{2}\sum ^{2}_{l=1}\sum^{\infty }_{n=1}\bigl| \langle\phi_{l},e_{n}\rangle\bigr|^{2} \\ \leq& 16m_{0}^{2}\alpha_{0}^{4}< \infty, \end{aligned}$$

in which Parseval’s identity have been used. Therefore, $\mathcal{F}_{1}$ is a Hilbert-Schmidt operator. Similarly, one can show that $\mathcal{F}_{2}$ is a Hilbert-Schmidt operator and thus $(z I-H_{1})^{-1}$ is a Hilbert-Schmidt operator. The proof is complete. □

Remark 3.1

• (1) In Lemma 2.22 in [25], Teschl showed that each self-adjoint operator extension $H_{1}$ with separated boundary conditions has a pure discrete spectrum, and its resolvent is a Hilbert-Schmidt operator in Case 3.

• (2) By applying the Green functions of resolvents of $H_{1,r}$ given in Propositions 3.2 and 4.2 in [22], which still hold for $z\in\rho(H_{1,r})$, it can easily be verified that the resolvents of $H_{1,r}$ are Hilbert-Schmidt operators in Cases 1 and 3. In addition, by (2.25)-(2.26), it is evident that the resolvent of $\tilde{H}_{1,r}$ is also a Hilbert-Schmidt operator in Cases 1 and 3. Moreover, we point out that the results given in Propositions 3.1 and 4.1 in [22] still hold for $z\in\rho(H_{1})$.

The following useful lemma can be directly derived from (i)-(ii) of Theorem 3.6 in [21].

Lemma 3.2

Assume that $X_{1}$ is a proper closed subspace in X, $P: X\to X_{1}$ the orthogonal projection, and T a self-adjoint operator on $X_{1}$. Then

• (i) $T'=TP$ is a self-adjoint operator on X with $D(T')=D(T)\oplus X_{1}^{\perp}$;

• (ii) $\sigma(T')=\sigma(T)\cup\{0\}$.

By Lemma 3.1, $H_{1}$ has a discrete spectrum. Via translating it if necessary, we may suppose that 0 is not an eigenvalue of $H_{1}$. The eigenvalues of $H_{1}$ may be ordered as (multiplicity included):

$$ \cdots\leq\lambda_{-3}\leq\lambda_{-2} \leq \lambda_{-1}< 0 < \lambda_{1} \leq\lambda_{2}\leq \cdots\leq\lambda_{n}\leq\cdots. $$

(3.1)

For convenience, we briefly denote it by $\sigma(H_{1})=\{\lambda_{n}: n\in\Lambda\subset\mathbf{Z}\backslash\{0\}\}$, where Z denotes the set of all integer numbers. Recall that $\{H_{1,r}\}$ is spectrally exact for $H_{1}$ if $0\notin \sigma(H_{1})$ (see Theorems 3.2 and 4.2 in [22]). Since $0\notin\sigma(H_{1})$, there exists $r_{0}$ such that $0\notin \sigma(H_{1,r})$ for all $r\geq r_{0}$. Therefore, for $r\geq r_{0}$, the eigenvalues of $H_{1,r}$ may be ordered as (multiplicity included):

$$ \lambda_{-m(r)}^{(r)} \leq\cdots\leq \lambda_{-2}^{(r)} \leq\lambda _{-1}^{(r)}< 0< \lambda_{1}^{(r)} \leq\lambda_{2}^{(r)} \leq\cdots\leq\lambda_{n(r)}^{(r)}, $$

(3.2)

where $m(r)$ and $n(r)$ are the numbers of negative and positive eigenvalues of $H_{1,r}$, respectively. For convenience, we briefly denote it by $\sigma(H_{1,r})=\{\lambda _{n}^{(r)}: n\in\Lambda_{r}\subset\mathbf{Z}\backslash\{0\}\}$. Let

$$S=(-H_{1})^{-1},\qquad S_{r}=(- \tilde{H}_{1,r})^{-1},\quad r\geq r_{0}. $$

Then, according to (2) of Remark 3.1 and Lemma 3.2, it follows that $S_{r}P_{r}$ and S are both self-adjoint and Hilbert-Schmidt operators. Note that the results of Theorems 3.2 and 4.2 in [22] still hold for every $z\in\rho(H_{1})\cap\rho(\tilde{H}_{1,r})$. By Lemma 2.5, $\sigma(H_{1,r})=\sigma(\tilde{H}_{1,r})$, which implies that $0\in\rho(\tilde{H}_{1,r})$ as $r\geq r_{0}$, and thus $0\in\rho(H_{1})\cap\rho(\tilde{H}_{1,r})$ as $r\geq r_{0}$. Therefore, $S_{r}P_{r} {\to} S$ in norm as $r \to\infty$ by Theorems 3.2 and 4.2 in [22].

Theorem 3.1

In Cases 1 and 3, for each $n\in\Lambda$, there exists an $r_{n}\geq r_{0}$ such that for $r\geq r_{n}$, $n\in\Lambda_{r}$ and $\lambda_{n}^{(r)}\to\lambda _{n}$ as $r \to\infty$.

Proof

Based on the above discussion, S and $S_{r}P_{r}$ are self-adjoint and Hilbert-Schmidt operators for $r\geq r_{0}$, and $S_{r}P_{r} {\to} S$ in norm as $r \to\infty$. Thus they are self-adjoint and compact operators with eigenvalues $\mu _{n}=-1/{\lambda_{n}}$ for $n\in\Lambda$ and $\mu_{n}^{(r)}=-1/{\lambda_{n}^{(r)}}$ for $n\in\Lambda_{r}$, separately, by Lemma 2.1. ($S_{r}P_{r}$ also has 0 as an eigenvalue of infinite multiplicity. But it is not related to $H_{1,r}$ or $H_{1}$, and so can be ignored.) Furthermore, since $S_{r}P_{r} {\to} S$ in norm as $r \to\infty$, we can get $S_{r}P_{r} {\to} S$ in the norm resolvent sense as $r \to\infty$ according to the proof of Theorem 8.18 in [26] (for the concept of convergence of self-adjoint operators in the norm resolvent sense, please see [24, 26]). Let $E(S_{r}P_{r},\lambda)$ and $E(S,\lambda)$ be spectral families of $S_{r}P_{r}$ and S, respectively. Then, by (b) of Theorem 8.23 in [26], it follows that for any $\alpha,\beta\in\mathbf{R}\cap\rho(S)$ with $\alpha<\beta$,

$$\bigl\| \bigl\{ E(S_{r}P_{r},\beta)-E(S_{r}P_{r}, \alpha)\bigr\} -\bigl\{ E(S,\beta)-E(S,\alpha)\bigr\} \bigr\| \to 0 \quad\mbox{as } r \to \infty, $$

which, together with Theorem 4.35 in [24], shows that

$$\dim R\bigl\{ E(S_{r}P_{r},\beta)-E(S_{r}P_{r}, \alpha)\bigr\} =\dim R\bigl\{ E(S,\beta)-E(S,\alpha )\bigr\} $$

for all sufficiently large r. Hence, for each $n\in\Lambda$, there exists an $r_{n}\geq r_{0}$ such that for $r\geq r_{n}$, $\mu_{n}^{(r)}$ exists. This implies that $H_{1,r}$ has an eigenvalue $\lambda_{n}^{(r)}$ for all $r\geq r_{n}$; namely, $n\in\Lambda_{r}$ for all $r\geq r_{n}$.

Next, we show that $\lambda_{n}^{(r)}\to\lambda_{n}$ as $r \to\infty$. To do so, it suffices to prove that $\mu_{n}^{(r)}\to\mu_{n}$ as $r \to\infty$. The negative eigenvalues are described by a min-max principle, and the positive eigenvalues by a max-min principle according to Section 12.1 in [27]; that is,

$$\begin{aligned} \mu_{n}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \min_{V_{n}}\mathop{\max_{x\in{V_{n}},}}\limits_{\hphantom{aaaa}\|x\|=1}\langle Sx,x\rangle, & n\in\Lambda \mbox{ with } n>0,\\ \max_{V_{n}}\mathop{\min_{x\in{V_{n}},}}\limits_{\hphantom{aaaa}\|x\|=1}\langle Sx,x\rangle, & n\in\Lambda \mbox{ with } n< 0, \end{array}\displaystyle \right . \end{aligned}$$

(3.3)

where $V_{n}$ runs through all the $|n|$-dimensional subspaces of $l^{2}_{w}(I)$. For $r\geq r_{n}$, $\mu_{n}^{(r)}$ is similarly expressed in terms of $\langle S_{r}P_{r}x,x\rangle$; that is,

$$\begin{aligned} \mu_{n}^{(r)}=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \min_{V_{n}}\mathop{\max_{x\in{V_{n}},}}\limits_{\hphantom{aaaa}\|x\|=1}\langle S_{r}P_{r}x,x\rangle, & n\in\Lambda \mbox{ with } n>0,\\ \max_{V_{n}}\mathop{\min_{x\in{V_{n}},}}\limits_{\hphantom{aaaa}\|x\|=1}\langle S_{r}P_{r}x,x\rangle, & n\in\Lambda \mbox{ with } n< 0. \end{array}\displaystyle \right . \end{aligned}$$

(3.4)

We first consider the case that $n\in\Lambda$ with $n>0$. Let $r\geq r_{n}$. It follows from (3.3)-(3.4) that there exist two n-dimensional subspaces $V_{n}$ and $\tilde{V}_{n}$ of $l^{2}_{w}(I)$ such that

$$\begin{aligned} \mu_{n}=\mathop{\max_{x\in{V_{n}},}}_{\|x\|=1} \langle Sx,x\rangle,\qquad \mu_{n}^{(r)}=\mathop{\max _{x\in{\tilde{V}_{n}},}}_{\|x\|=1}\langle S_{r}P_{r}x,x \rangle. \end{aligned}$$

(3.5)

In addition, there exist $x_{1}\in\tilde{V}_{n}$ with $\|x_{1}\|=1$ and $x_{2}\in V_{n}$ with $\|x_{2}\|=1$ such that

$$\begin{aligned} \mathop{\max_{x\in{\tilde{V}_{n}},}}_{\|x\|=1}\langle Sx,x \rangle =\langle Sx_{1},x_{1}\rangle,\qquad \mathop{\max _{x\in{V}_{n},}}_{\|x\|=1}\langle S_{r}P_{r}x,x \rangle =\langle S_{r}P_{r}x_{2},x_{2} \rangle. \end{aligned}$$

(3.6)

From (3.3)-(3.6), we have

$$\begin{aligned}& \mu_{n}-\mu_{n}^{(r)} \leq\max _{{x\in{\tilde{V}_{n}},\atop \|x\|=1}}\langle Sx,x\rangle- \max_{{x\in{\tilde{V}_{n}},\atop \|x\|=1}}\langle S_{r}P_{r}x,x\rangle \leq\bigl\langle (S-S_{r}P_{r})x_{1},x_{1} \bigr\rangle , \\& \mu_{n}-\mu_{n}^{(r)}\geq \max _{{x\in{V_{n}},\atop \|x\|=1}}\langle Sx,x\rangle-\max_{{x\in{V_{n}},\atop \|x\|=1}}\langle S_{r}P_{r}x,x\rangle \geq\bigl\langle (S-S_{r}P_{r})x_{2},x_{2} \bigr\rangle . \end{aligned}$$

Therefore, it follows that

$$\begin{aligned} \bigl|\mu_{n}-\mu_{n}^{(r)}\bigr|&\leq\max \bigl\{ \bigl|\bigl\langle (S-S_{r}P_{r})x_{1},x_{1} \bigr\rangle \bigr|, \bigl|\bigl\langle (S-S_{r}P_{r})x_{2},x_{2} \bigr\rangle \bigr|\bigr\} \\ &\leq \|S-S_{r}P_{r}\| \to0 \quad\mbox{as } r \to\infty. \end{aligned}$$

(3.7)

Thus, $\mu_{n}^{(r)}\to\mu_{n}$ as $r \to\infty$ for $n\in\Lambda$ with $n>0$.

Similarly, we can get $\mu_{n}^{(r)}\to\mu_{n}$ as $r \to\infty$ for $n\in\Lambda$ with $n<0$. This completes the proof. □

At the end of this section, we shall try to give an error estimate for the approximation of $\lambda_{n}$ by $\lambda_{n}^{(r)}$ for each $n\in\Lambda$. Obviously, it is very important in numerical analysis and applications. In order to give error estimates of $\lambda_{n}^{(r)}$ to $\lambda_{n}$, in view of $\lambda_{n}=-1/{\mu_{n}}$ and $\lambda_{n}^{(r)}=-1/{\mu_{n}^{(r)}}$, we shall first investigate the error estimates of $\mu_{n}^{(r)}$ to $\mu_{n}$ for $n\in\Lambda$ instead.

In view of the arbitrariness of $\lambda\in\mathbf{R}$ in (2.8), we might as well take $\lambda=0$ in (2.8) in the following discussions.

Proposition 3.1

Assume that $\mathcal{L}$ is regular at $t=a$ and in l.c.c. at $t=+\infty$. Then, for each $n\in\Lambda$ and $r\geq r_{n}$, where $r_{n}$ is specified in Theorem 3.1,

$$\begin{aligned} \bigl|\mu_{n}^{(r)}-\mu_{n}\bigr|\leq4 \alpha_{0}\bigl(2m_{0}^{2}+5n_{0}^{2} \bigr)^{\frac {1}{2}}\biggl(2+\frac{1}{p^{2}(a-1)}\biggr)\varepsilon_{r}, \end{aligned}$$

(3.8)

where $\alpha_{0}$, $m_{0}$, and $n_{0}$ are constants and determined by (3.10)-(3.16), and $\varepsilon_{r}$ is completely determined by the coefficients of (1.1), more precisely, it is determined by (3.17)-(3.18), (3.21), (3.23)-(3.24). In addition, $\varepsilon_{r}\to0$ as $r\to\infty$.

Proof

Note that the results of Propositions 3.1 and 3.2 and Theorem 3.2 in [22] still hold for every $z\in\rho(H_{1})\cap\rho(\tilde{H}_{1,r})$. By Lemma 2.5, $\sigma(H_{1,r})=\sigma(\tilde{H}_{1,r})$, which implies that $0\in\rho(\tilde{H}_{1,r})$ as $r\geq r_{0}$, and thus $0\in\rho(H_{1})\cap\rho(\tilde{H}_{1,r})$ as $r\geq r_{0}$. Consequently, by (3.43)-(3.44) in [22], one has

$$\begin{aligned} \|S-S_{r}P_{r}\|\leq4\alpha_{0} \bigl[3\alpha _{0}^{2}\bigl(m_{r}^{2}+n_{r}^{2} \bigr)+\bigl(2m_{0}^{2}+5n_{0}^{2}\bigr) \alpha_{r}^{2}\bigr]^{\frac{1}{2}},\quad r\geq r_{0}, \end{aligned}$$

(3.9)

where

$$\begin{aligned} &m_{0}:=\max_{k,l=1,2}\bigl\{ \bigl|m_{kl}^{0}\bigr|\bigr\} ,\qquad n_{0}:=\max _{k,l=1,2}\bigl\{ \bigl|n_{kl}^{0}\bigr|\bigr\} ,\qquad m_{r}:=\max_{k,l=1,2}\bigl\{ \bigl|m_{kl}^{0}-m_{kl}^{r}\bigr| \bigr\} , \\ &n_{r}:=\max_{k,l=1,2}\bigl\{ \bigl|n_{kl}^{0}-n_{kl}^{r}\bigr| \bigr\} , \qquad \alpha_{0}:=\max_{i=1,2}\bigl\{ \| \phi_{i}\|\bigr\} , \\ &\alpha_{r}:=\max_{i=1,2}\Biggl\{ \sum ^{\infty}_{t=b_{r}+1}\bigl|\phi _{i}(t)\bigr|^{2}w(t) \Biggr\} , \end{aligned}$$

(3.10)

while $\phi_{1}$ and $\phi_{2}$ are two linearly independent solutions of (1.1 _λ ) with $\lambda=0$ satisfying the following initial conditions:

$$\begin{aligned} \begin{aligned} &\phi_{1}(a-1)=0, \qquad\phi_{2}(a-1)=-1,\\ &p(a-1)\triangle\phi_{1}(a-1)=1, \qquad p(a-1)\triangle \phi_{2}(a-1)=0, \end{aligned} \end{aligned}$$

(3.11)

$$\begin{aligned} &M_{0}= \begin{pmatrix} -m_{12}^{0} & m_{11}^{0}\\ -m_{22}^{0} & m_{21}^{0} \end{pmatrix}=I+N_{0}, \end{aligned}$$

(3.12)

$$\begin{aligned} &N_{0}= \begin{pmatrix} -n_{12}^{0} & n_{11}^{0}\\ -n_{22}^{0} & n_{21}^{0} \end{pmatrix} =-\bigl( \hat{Y}_{0}^{*}(a-1)+K\bigr)^{-1}K, \qquad K=\lim _{t\rightarrow+\infty}\hat {Y}_{0}^{*}(t)J\Phi(t), \end{aligned}$$

(3.13)

$$\begin{aligned} &M_{r}= \begin{pmatrix} -m_{12}^{r} & m_{11}^{r}\\ -m_{22}^{r} & m_{21}^{r} \end{pmatrix}=I+N_{r}, \end{aligned}$$

(3.14)

$$\begin{aligned} &N_{r}= \begin{pmatrix} -n_{12}^{r} & n_{11}^{r}\\ -n_{22}^{r} & n_{21}^{r} \end{pmatrix} =-\bigl( \hat{Y}_{0}^{*}(a-1)+K_{r}\bigr)^{-1}K_{r},\qquad K_{r}=\hat{Y}_{0}^{*}(b_{r})J\Phi(b_{r}), \end{aligned}$$

(3.15)

$$\begin{aligned} &\Phi(t)= \begin{pmatrix} \phi_{1}(t) & \phi_{2}(t)\\ p(t)\triangle\phi_{1}(t) & p(t)\triangle\phi_{2}(t) \end{pmatrix},\qquad \hat{Y}_{0}(t)= \begin{pmatrix} \hat{y}_{1}(t) & \hat{y}_{2}(t)\\ p(t)\triangle\hat{y}_{1}(t) & p(t)\triangle\hat{y}_{2}(t) \end{pmatrix}, \end{aligned}$$

(3.16)

where $\hat{y}_{1}(t)$ and $\hat{y}_{2}(t)$ are defined by (2.10). Noting that $\hat{y}_{1}$, $\hat{y}_{2}$, $\phi_{1}$, and $\phi_{2}$ are both solutions of (1.1 _λ ) with $\lambda=0$ in $[c_{0},+\infty)$, where $c_{0}$ is the same as in (2.10). Since $b_{r}> c_{0}$, by Lemma 2.3 in [12], we get $K=K_{r}$, which shows that $M_{r}=M_{0}$, $N_{r}=N_{0}$, and thus $m_{r}=n_{r}=0$.

Now, it remains to estimate $\alpha_{r}$. Let

$$\begin{aligned} W(t):= \begin{pmatrix} w(t+1) & 0\\ 0 & w(t) \end{pmatrix},\qquad U(t):= \begin{pmatrix} c(t) & -d(t)\\ 1 & 0 \end{pmatrix}, \quad t\geq a, \end{aligned}$$

(3.17)

where

$$\begin{aligned} c(t):=1+\frac{q(t)}{p(t)}+\frac{p(t-1)}{p(t)}, \qquad d(t):= \frac {p(t-1)}{p(t)}, \quad t\geq a. \end{aligned}$$

(3.18)

From (1.1 _λ ) with $\lambda=0$, we get

$$x(t+1)=c(t)x(t)-d(t)x(t-1), \quad t\geq a. $$

It follows that

$$\begin{aligned} \begin{pmatrix} x(t+1) \\ x(t) \end{pmatrix} =U(t) \begin{pmatrix} x(t) \\ x(t-1) \end{pmatrix},\quad t\geq a. \end{aligned}$$

(3.19)

By (3.17) and (3.19), we get

$$\begin{aligned} &w(t+1)x^{2}(t+1)+w(t)x^{2}(t) \\ &\quad= \bigl(x(t+1),\ x(t)\bigr) W(t) \bigl(x(t+1),\ x(t)\bigr)^{\top} \\ &\quad=\bigl(x(t),\ x(t-1)\bigr) U^{\top}(t)W(t)U(t) \bigl( x(t),\ x(t-1)\bigr)^{\top} \\ &\quad=\bigl(x(a),\ x(a-1)\bigr) V^{\top}(t)W(t)V(t) \bigl(x(a),\ x(a-1) \bigr)^{\top}, \quad t\geq a, \end{aligned}$$

(3.20)

where

$$\begin{aligned} V(t):= U(t)U(t-1)\cdots U(a+1)U(a),\quad t\geq a. \end{aligned}$$

(3.21)

Let $x(a)$ and $x(a-1)$ be any real numbers. Since $\mathcal{L}$ is regular at $t=a$ and in l.c.c. at $t=+\infty$, it follows from (3.20) that

$$\begin{aligned} &\sum^{\infty}_{t=b_{r}+1}w(t)x^{2}(t) \\ &\quad=\sum^{\infty }_{i=0}\bigl[w(b_{r}+2i+2)x^{2}(b_{r}+2i+2)+w(b_{r}+2i+1)x^{2}(b_{r}+2i+1) \bigr] \\ &\quad=\bigl(x(a),\ x(a-1)\bigr) D_{r}\bigl(x(a),\ x(a-1) \bigr)^{\top}\to0 \quad\mbox{as } r\to\infty, \end{aligned}$$

(3.22)

where

$$\begin{aligned} D_{r}:=\sum^{\infty}_{i=0}V^{\top}(b_{r}+2i+1)W(b_{r}+2i+1)V(b_{r}+2i+1). \end{aligned}$$

(3.23)

Denote

$$\begin{aligned} \varepsilon_{r}:=\|D_{r}\|, \end{aligned}$$

(3.24)

where the norm of the matrix $D_{r}=(d^{r}_{ij})_{2\times2}$ is defined as $\|D_{r}\|=\sum^{2}_{i,j=1}|d^{r}_{ij}|$. Since $D_{r}$ is symmetric and $( x(a),\ x(a-1))$ is arbitrary, it follows from (3.22) that $\varepsilon_{r}\to0$ as $r\to\infty$. In addition, since $\phi_{1}$ and $\phi_{2}$ satisfy (3.22), it follows that

$$\begin{aligned} \alpha_{r}&=\max_{i=1,2} \bigl\{ \bigl( \phi_{i}(a),\ \phi_{i}(a-1)\bigr) D_{r} \bigl( \phi_{i}(a),\ \phi_{i}(a-1)\bigr)^{\top}\bigr\} \\ &\leq\varepsilon_{r} \max_{i=1,2}\bigl\{ \phi_{i}^{2}(a)+\phi_{i}^{2}(a-1)\bigr\} \\ &\leq\varepsilon_{r}\biggl(2+\frac{1}{p^{2}(a-1)}\biggr), \end{aligned}$$

(3.25)

in which (3.11) has been used. This, together with $m_{r}=n_{r}=0$, (3.9), and (3.7), shows that (3.8) holds. This completes the proof. □

Theorem 3.2

Assume that $\mathcal{L}$ is regular at $t=a$ and in l.c.c. at $t=+\infty$. Then, for each $n\in\Lambda$, there exists an $r'_{n}\geq r_{n}$ such that for all $r\geq r'_{n}$,

$$\begin{aligned} &\bigl|\lambda_{n}^{(r)}-\lambda_{n}\bigr| \leq\frac{|\lambda _{n}^{(r)}|^{2}e_{r}}{1-|\lambda_{n}^{(r)}|e_{r}}, \end{aligned}$$

(3.26)

$$\begin{aligned} &\bigl|\lambda_{n}^{(r)}-\lambda_{n}\bigr| \leq\frac{|\lambda _{n}|^{2}e_{r}}{1-|\lambda_{n}|e_{r}}, \end{aligned}$$

(3.27)

where $e_{r}$ denotes the number on the right-hand side in (3.8).

Proof

For each $n\in\Lambda$, $\lambda_{n}$ and $\lambda_{n}^{(r)}$ have the same sign for sufficiently large r. In view of $\lambda_{k}=-1/{\mu_{k}}$ and $\lambda_{k}^{(r)}=-1/{\mu _{k}^{(r)}}$, it follows from (3.8) that for each $n\in\Lambda$,

$$\biggl|\frac{1}{\lambda_{n}^{(r)}}-\frac{1}{\lambda_{n}}\biggr|\leq e_{r}, \quad r\geq r_{n}, $$

which shows that

$$\begin{aligned} \bigl|\lambda_{n}^{(r)}-\lambda_{n}\bigr|\leq e_{r}\bigl|\lambda_{n}^{(r)}\bigr||\lambda _{n}|, \quad r\geq r_{n}. \end{aligned}$$

(3.28)

Thus,

$$|\lambda_{n}|=\bigl|\lambda_{n}+\lambda_{n}^{(r)}- \lambda_{n}^{(r)}\bigr|\leq \bigl|\lambda_{n}- \lambda_{n}^{(r)}\bigr|+\bigl|\lambda_{n}^{(r)}\bigr|\leq e_{r}\bigl|\lambda_{n}^{(r)}\bigr||\lambda_{n}|+\bigl| \lambda_{n}^{(r)}\bigr|, \quad r\geq r_{n}, $$

which implies that

$$\begin{aligned} |\lambda_{n}|\bigl(1-\bigl|\lambda_{n}^{(r)}\bigr|e_{r} \bigr)\leq\bigl|\lambda_{n}^{(r)}\bigr|. \end{aligned}$$

(3.29)

By Theorem 3.1 and Proposition 3.1, there exists an $r'_{n}\geq r_{n}$ such that $1-|\lambda_{n}^{(r)}|e_{r}>0$. Hence, it follows from (3.28) and (3.29) that (3.26) holds. With a similar argument, one can show that (3.27) holds. This completes the proof. □

We now turn to error estimates of approximations of $\lambda_{n}^{(r)}$ to $\lambda_{n}$ in Case 3.

Proposition 3.2

Assume that $\mathcal{L}$ is in l.c.c. at $t=\pm\infty$. Then, for each $n\in\Lambda$ and $r\geq r_{n}$, where $r_{n}$ is specified in Theorem  3.1,

$$\begin{aligned} \bigl|\mu_{n}^{(r)}-\mu_{n}\bigr|\leq8 \sqrt{2}\tilde{\alpha}_{0}\bigl(\tilde {m}_{0}^{2}+ \tilde{n}_{0}^{2}\bigr)^{\frac{1}{2}} \biggl(2+ \frac{1}{p^{2}(d_{0}-1)}\biggr)\tilde{\varepsilon}_{r}, \end{aligned}$$

(3.30)

where $\tilde{\alpha}_{0}$, $\tilde{m}_{0}$, and $\tilde{n}_{0}$ are constants and given by (3.32), $d_{0}$ is the same as in (2.8), and $\tilde{\varepsilon}_{r}$ is completely determined by the coefficients of (1.1), more precisely, it is determined by (3.17)-(3.18), (3.36), (3.39)-(3.41). In addition, $\tilde{\varepsilon}_{r}\to0$ as $r\to\infty$.

Proof

The main idea of the proof is similar to that of Proposition 3.1, where the interval $I=[a,+\infty)$ is replaced by $I=(-\infty,+\infty)$. For completeness, we now give its detailed proof.

Note that the results of Propositions 4.1 and 4.2 and Theorem 4.2 in [22] still hold for every $z\in\rho(H_{1})\cap\rho(\tilde{H}_{1,r})$. By Lemma 2.5, $\sigma(H_{1,r})=\sigma(\tilde{H}_{1,r})$, which implies that $0\in\rho(\tilde{H}_{1,r})$ as $r\geq r_{0}$, and thus $0\in\rho(H_{1})\cap\rho(\tilde{H}_{1,r})$ as $r\geq r_{0}$. Consequently, by Theorem 4.2 in [22] one has

$$\begin{aligned} \|S-S_{r}P_{r}\|\leq8\tilde{ \alpha}_{0} \bigl[\tilde{\alpha}_{0}^{2}\bigl( \tilde{m}_{r}^{2}+\tilde{n}_{r}^{2} \bigr)+2\bigl(\tilde{m}_{0}^{2}+\tilde {n}_{0}^{2} \bigr)\tilde{\alpha}_{r}^{2}\bigr]^{\frac{1}{2}},\quad r\geq r_{0}, \end{aligned}$$

(3.31)

where

$$\begin{aligned} &\tilde{ m}_{0}:=\max_{k,l=1,2}\bigl\{ \bigl|m_{kl}^{0}\bigr|\bigr\} , \qquad \tilde {n}_{0}:=\max _{k,l=1,2}\bigl\{ \bigl|n_{kl}^{0}\bigr|\bigr\} , \\ &\tilde{m}_{r}:=\max_{k,l=1,2}\bigl\{ \bigl|m_{kl}^{0}-m_{kl}^{r}\bigr| \bigr\} ,\qquad \tilde{n}_{r}:=\max_{k,l=1,2}\bigl\{ \bigl|n_{kl}^{0}-n_{kl}^{r}\bigr|\bigr\} , \\ &\tilde{\alpha}_{0}:=\max_{i=1,2}\bigl\{ \| \phi_{i}\|\bigr\} ,\qquad \tilde{\alpha}_{r}:=\max _{i=1,2} \Biggl\{ \sum^{a_{r}-1}_{t=-\infty}\bigl| \phi_{i}(t)\bigr|^{2}w(t)+\sum^{\infty }_{t=b_{r}+1}\bigl| \phi_{i}(t)\bigr|^{2}w(t)\Biggr\} , \end{aligned}$$

(3.32)

while $m_{kl}^{0}$, $n_{kl}^{0}$ and $m_{kl}^{r}$, $n_{kl}^{r}$ are given by (4.19) and (4.21) in [22], separately; $\phi_{1}$ and $\phi_{2}$ are two linearly independent solutions of (1.1 _λ ) with $\lambda=0$ satisfying the initial conditions (3.11), in which a is replaced by $d_{0}$ given in (2.8). With a similar discussion to that in the proof of Proposition 3.1, we can prove that $\tilde{m}_{r}=\tilde{n}_{r}=0$. Therefore, it remains to estimate $\tilde{\alpha}_{r}$.

Let $W(t)$, $U(t)$, $c(t)$, $d(t)$ be defined as these in (3.17)-(3.18) for every $t\in I$. It is evident that $U(t)$ is invertible for any $t\in I$. From (1.1 _λ ) with $\lambda=0$, we get

$$x(t+1)=c(t)x(t)-d(t)x(t-1), \quad t\in I, $$

which shows that

$$\begin{aligned} \begin{pmatrix} x(t+1) \\ x(t) \end{pmatrix} =U(t) \begin{pmatrix} x(t) \\ x(t-1) \end{pmatrix}, \quad t\in I. \end{aligned}$$

(3.33)

By (3.17) and (3.33), we get

$$\begin{aligned} &w(t+1)x^{2}(t+1)+w(t)x^{2}(t) \\ &\quad=\bigl(x(t+1),\ x(t)\bigr) W(t) \bigl(x(t+1),\ x(t)\bigr)^{\top} \\ &\quad=\bigl(x(t),\ x(t-1)\bigr) U^{\top}(t)W(t)U(t) \bigl( x(t),\ x(t-1) \bigr)^{\top} \\ &\quad=\bigl( x(d_{0}),\ x(d_{0}-1)\bigr) V_{+}^{\top}(t)W(t)V_{+}(t) \bigl( x(d_{0}),\ x(d_{0}-1) \bigr)^{\top},\quad t\geq d_{0}, \end{aligned}$$

(3.34)

$$\begin{aligned} &w(t)x^{2}(t)+w(t-1)x^{2}(t-1) \\ &\quad=\bigl( x(t),\ x(t-1)\bigr) W(t-1) \bigl( x(t),\ x(t-1) \bigr)^{\top} \\ &\quad=\bigl( x(t+1),\ x(t)\bigr) \bigl(U^{-1}\bigr)^{\top}(t)W(t-1)U^{-1}(t) \bigl( x(t+1),\ x(t)\bigr)^{\top} \\ &\quad=\bigl(x(d_{0}),\ x(d_{0}-1)\bigr) V_{-}^{\top}(t)W(t-1)V_{-}(t) \bigl( x(d_{0}),\ x(d_{0}-1) \bigr)^{\top},\quad t\leq d_{0}-1, \end{aligned}$$

(3.35)

where

$$ \begin{aligned} &V_{+}(t)=U(t)U(t-1)\cdots U(d_{0}+1)U(d_{0}), \quad t\geq d_{0}, \\ &V_{-}(t)=U^{-1}(t)U^{-1}(t+1)\cdots U^{-1}(d_{0}-2)U^{-1}(d_{0}-1), \quad t\leq d_{0}-1. \end{aligned} $$

(3.36)

Let $x(d_{0})$ and $x(d_{0}-1)$ be any real numbers. Recall that $a_{r}< d_{0}-1$, $b_{r}>d_{0}$. Since $\mathcal{L}$ is in l.c.c. at $t=\pm\infty$, it follows from (3.34)-(3.35) that

$$\begin{aligned} &\sum^{\infty}_{t=b_{r}+1}w(t)x^{2}(t) \\ &\quad=\sum^{\infty }_{i=0}\bigl[w(b_{r}+2i+2)x^{2}(b_{r}+2i+2)+w(b_{r}+2i+1)x^{2}(b_{r}+2i+1) \bigr] \\ &\quad=\bigl(x(d_{0}),\ x(d_{0}-1)\bigr) D_{r_{+}} \bigl(x(d_{0}),\ x(d_{0}-1)\bigr)^{\top}\to0 \quad \mbox{as } r\to\infty, \end{aligned}$$

(3.37)

$$\begin{aligned} &\sum^{a_{r}-1}_{t=-\infty}w(t)x^{2}(t) \\ &\quad=\sum^{\infty }_{i=0}\bigl[w(a_{r}-2i-1)x^{2}(a_{r}-2i-1)+w(a_{r}-2i-2)x^{2}(a_{r}-2i-2) \bigr] \\ &\quad=\bigl(x(d_{0}),\ x(d_{0}-1)\bigr) D_{r_{-}} \bigl(x(d_{0}),\ x(d_{0}-1)\bigr)^{\top}\to0 \quad \mbox{as } r\to\infty, \end{aligned}$$

(3.38)

where

$$ \begin{aligned} &D_{r_{+}}:=\sum ^{\infty}_{i=0}V_{+}^{\top}(b_{r}+2i+1)W(b_{r}+2i+1)V_{+}(b_{r}+2i+1), \\ &D_{r_{-}}:=\sum^{\infty}_{i=0}V_{-}^{\top}(a_{r}-2i-1)W(a_{r}-2i-2)V_{-}(a_{r}-2i-1). \end{aligned} $$

(3.39)

Denote

$$\begin{aligned} \varepsilon_{r_{+}}:=\|D_{r_{+}}\|, \qquad \varepsilon_{r_{-}}:=\|D_{r_{-}}\|, \end{aligned}$$

(3.40)

where the norm of the matrix $D=(d_{ij})_{2\times2}$ is defined as $\| D\|=\sum^{2}_{i,j=1}|d_{ij}|$. Since $D_{r_{+}}$ and $D_{r_{+}}$ are both symmetric and $(x(d_{0}), \ x(d_{0}-1))$ is arbitrary, it follows from (3.37)-(3.38) that

$$\begin{aligned} \tilde{\varepsilon}_{r}:=\varepsilon_{r_{+}}+ \varepsilon_{r_{-}}\to0 \quad\mbox{as } r\to\infty. \end{aligned}$$

(3.41)

In addition, since $\phi_{1}$ and $\phi_{2}$ satisfy (3.37)-(3.38), one gets

$$\begin{aligned} \alpha_{r}&=\max_{i=1,2} \bigl\{ \bigl( \phi_{i}(d_{0}),\ \phi_{i}(d_{0}-1)\bigr) (D_{r_{+}}+D_{r_{-}}) \bigl(\phi_{i}(d_{0}),\ \phi_{i}(d_{0}-1)\bigr)^{\top}\bigr\} \\ &\leq\tilde{\varepsilon}_{r} \max_{i=1,2}\bigl\{ \phi_{i}^{2}(d_{0})+\phi _{i}^{2}(d_{0}-1) \bigr\} \\ &\leq\tilde{\varepsilon}_{r}\biggl(2+\frac{1}{p^{2}(d_{0}-1)}\biggr), \end{aligned}$$

(3.42)

in which (3.11) with $a=d_{0}$ has been used. This, together with $\tilde{m}_{r}=\tilde{n}_{r}=0$, (3.31), and (3.7), shows that (3.30) holds. This completes the proof. □

The proof of the following result is similar to that of Theorem 3.2 and so its details are omitted.

Theorem 3.3

Assume that $\mathcal{L}$ is in l.c.c. at $t=\pm\infty$. Then, for each $n\in\Lambda$, there exists an $r'_{n}\geq r_{n}$ such that for all $r\geq r'_{n}$,

$$\bigl|\lambda_{n}^{(r)}-\lambda_{n}\bigr|\leq \frac{|\lambda_{n}^{(r)}|^{2}\tilde {e}_{r}}{1-|\lambda_{n}^{(r)}|\tilde{e}_{r}},\qquad \bigl|\lambda_{n}^{(r)}- \lambda_{n}\bigr|\leq\frac{|\lambda_{n}|^{2}\tilde {e}_{r}}{1-|\lambda_{n}|\tilde{e}_{r}}, $$

where $\tilde{e}_{r}$ denotes the number on the right-hand side in (3.30).

Remark 3.2

The authors in [1, 4] and [3] gave similar results to Theorem 3.1 for singular second-order and fourth-order differential Sturm-Liouville problems, respectively, where the results in [1, 4] hold under the assumption that each endpoint is regular or in l.c.c. and non-oscillatory. However, they did not give any error estimate for the approximations of isolated eigenvalues. To the best of our knowledge, there have been no results about error estimates for approximations of isolated eigenvalues of singular differential and difference equations in the existing literature.

4 One endpoint is regular and the other in l.p.c

In this section, we shall study spectral exactness in an open interval laking essential spectral points in Case 2. Without loss of generality, we only consider the case that $\mathcal{L}$ is regular at a and in l.p.c. at $t=+\infty$.

In [22], we proved that the sequence of induced self-adjoint restrictions $\{H_{1,r}\}_{r=1}^{\infty}$ is spectrally inclusive for a given self-adjoint subspace extension $H_{1}$ in Case 2 and pointed out that it is not spectrally exact in general. In this section, we will choose a sequence of special induced self-adjoint restrictions, still denoted by $\{H_{1,r}\}_{r=1}^{\infty}$ without any confusion, such that it is spectrally exact for $H_{1}$ in an interval laking essential spectral points.

The following are some useful lemmas.

Lemma 4.1

([24], Exercise 7.37)

Let T be a self-adjoint operator with spectral family E, and S a subspace in $D(T)$ such that $\|(\lambda-T)f\|\leq c\|f\|$ for all $f\in S$. Then $\dim R\{E(\lambda+c)-E(\lambda-c-)\}\geq\dim S$.

Lemma 4.2

([28], Lemma 8.1.23)

If P and $P_{n}$ are orthogonal projections on X with $\dim R(P_{n})\leq\dim R(P)<\infty$ for $n\geq1$ and $P_{n}$ is strongly convergent to P as $n {\to} \infty$, denoted by $P_{n}\stackrel{s}{\to} P$, then $\dim R(P_{n})=\dim R(P)$ for sufficiently large n.

Lemma 4.3

Let $\mathcal{L}$ be regular or in l.c.c. at one endpoint and in l.p.c. at the other endpoint, i.e., in Case 2 or Case 4. If for some $\lambda\in\mathbf{R}$ the equation (1.1 _λ ) has no nontrivial square summable solutions, then λ belongs to the essential spectrum of every self-adjoint subspace extension $H_{1}$ of $H_{0}$.

Proof

By the assumption, λ is not an eigenvalue of $H_{1}$. Denote

$$M_{\lambda}:=\bigl\{ (x, \lambda x)\in H\bigr\} . $$

Then $M_{\lambda}=M_{\bar{\lambda}}=\{(0, 0)\}$. Hence, by Lemma 2.2 in [11] we have $R(H_{0}-\lambda I)^{\bot}=D(M_{\bar {\lambda}})=\{0\}$. In addition, since the deficiency indices of $H_{0}$ are (1,1), λ is not in the regularity domain of $H_{0}$ by Theorem 2.3 in [11], and therefore it is not in the resolvent set of $H_{1}$. Hence, λ is in the essential spectrum of $H_{1}$, i.e., $\lambda\in\sigma_{e}(H_{1})$. This completes the proof. □

Remark 4.1

Teschl in Lemma 2.2 in [25] showed the same statement as Lemma 4.3 when one endpoint is finite and the other endpoint is in l.p.c. and $H_{1}$ is an operator. The authors in Corollary 6.4 in [29] showed a similar result to Lemma 4.3 when it is regular at one endpoint and in l.p.c. at the other endpoint. Since our proof of Lemma 4.3 is more simple, we list it here.

Let $E_{s}(H_{1},\lambda)$, $E_{s}(H_{1,r},\lambda)$, $E_{s}(\tilde {H}_{1,r},\lambda)$, and $E_{s}(H'_{1,r},\lambda)$ be spectral families of $H_{1,s}$, $H_{1,r,s}$, $\tilde{H}_{1,r,s}$, and $H_{1,r,s}'$, respectively, which denote the operator parts of $H_{1}$, $H_{1,r}$, $\tilde{H}_{1,r}$, and $H_{1,r}'$, respectively.

Theorem 4.1

Assume that $\mathcal{L}$ is regular at $t=a$ and in l.p.c. at $t=+\infty$. Let $H_{1}$ be any fixed self-adjoint subspace extension of $H_{0}$ given by (2.12). Assume that $0\notin I_{0}\subset{\mathbf {R}}$ is an open interval with $I_{0} \cap\sigma_{e}(H_{1})=\emptyset$ and $I_{0} \cap\sigma_{d}(H_{1})\ne \emptyset$. Let v be a nontrivial real square summable solution of (1.1 _γ ) with any fixed $\gamma\in I_{0}$, $H_{1,r}$ the induced self-adjoint restriction of $H_{1}$ on $I_{r}$ defined by

$$\begin{aligned} H_{1,r}=\bigl\{ (x,f)\in H^{r}: (x,\hat{y}) (a-1)=0, (x,v) (b_{r})=0\bigr\} , \end{aligned}$$

(4.1)

where ŷ is defined by (2.13) and $\{b_{r}\}_{r=1}^{\infty}$ specified in Section 2.3 satisfies $v(b_{r})\neq0$ for $r\in{\mathbf {N}}$. Then $\{H_{1,r}\}_{r=1}^{\infty}$ is spectrally exact for $H_{1}$ in $I_{0}$.

By Lemma 4.3, there exists at least one nontrivial square summable solution v of (1.1 _γ ) for any $\gamma\in I_{0}$, where $I_{0}$ is specified in Theorem 4.1. Consequently, there are infinite $t\in I=\{t\}_{t=a}^{+\infty}$ such that $v(t)\neq0$, and so we can choose $\{b_{r}\}_{r=1}^{\infty}$ specified in Section 2.3 such that $v(b_{r})\neq0$ for $r\in{\mathbf {N}}$ in (4.1). Hence, $H_{1,r}$ given by (4.1) is well defined.

Proof of Theorem 4.1

By (2.2) and Lemmas 2.4 and 2.5, it suffices to show that

$$\begin{aligned} H_{1,r}'(0)=H_{1}(0), \end{aligned}$$

(4.2)

and for any given $\alpha,\beta\in I_{0}\cap\rho(H_{1})$ with $\alpha <\gamma\leq\beta$, there exists an integer $\tilde{r}\geq1$ such that for all $r\geq \tilde{r}$,

$$\begin{aligned} \dim R\bigl\{ E_{s}\bigl(H'_{1,r}, \beta\bigr)-E_{s}\bigl(H'_{1,r},\alpha\bigr)\bigr\} =\dim R\bigl\{ E_{s}(H_{1},\beta )-E_{s}(H_{1}, \alpha)\bigr\} . \end{aligned}$$

(4.3)

We first prove (4.2). By (2.7) and (2.12), it can be deduced that

$$\begin{aligned} H_{1}(0)=\bigl\{ f\in l_{w}^{2}(I): \hat{y}(a)f(a)=0, f(t)=0, t\geq a+1\bigr\} , \end{aligned}$$

(4.4)

By (4.1) and $v(b_{r})\neq0$ we get

$$H_{1,r}(0)=\bigl\{ f\in l_{w}^{2}(I_{r}): \hat{y}(a)f(a)=0, f(t)=0, a+1\leq t\leq b_{r}\bigr\} , $$

which, together with Lemma 2.6, shows that

$$H_{1,r}'(0)=\tilde{H}_{1,r}(0)=\bigl\{ f\in l_{w}^{2}(I): \hat{y}(a)f(a)=0, f(t)=0, t\geq a+1\bigr\} . $$

Therefore, (4.2) holds.

Next, we show that (4.3) holds. Fix any $\alpha,\beta\in I_{0}\cap\rho(H_{1})$ with $\alpha<\gamma\leq \beta$. For any fixed r, let $\lambda_{1},\ldots,\lambda_{k_{r}}$ be all $k_{r}$ (counting multiplicity) eigenvalues of $H_{1,r}$ in $(\alpha,\beta]$, and $\eta_{1},\ldots,\eta_{k_{r}}$ the corresponding orthonormal eigenfunctions. By Lemma 2.5 and the assumption that $0\notin I_{0}$ we get

$$\sigma\bigl(H'_{1,r}\bigr)\cap(\alpha, \beta]=\sigma( \tilde{H}_{1,r})\cap(\alpha, \beta]=\sigma(H_{1,r})\cap(\alpha, \beta] =\{\lambda_{1},\ldots,\lambda_{k_{r}}\}. $$

By (i) of Theorem 3.9 in [20] we have

$$\begin{aligned} &\dim R\bigl\{ E_{s}\bigl(H'_{1,r}, \beta\bigr)-E_{s}\bigl(H'_{1,r},\alpha\bigr)\bigr\} \\ &\quad=\dim R\bigl\{ E_{s}(\tilde{H}_{1,r}, \beta)-E_{s}(\tilde{H}_{1,r},\alpha)\bigr\} \\ &\quad=\dim R\bigl\{ E_{s}(H_{1,r},\beta)-E_{s}(H_{1,r}, \alpha)\bigr\} =k_{r}. \end{aligned}$$

(4.5)

In addition, it follows from Lemma 2.2 that $\lambda_{1},\ldots,\lambda _{k_{r}}$ are all $k_{r}$ (counting multiplicity) eigenvalues of $H_{1,r,s}$ in $(\alpha,\beta]$, and $\eta_{1},\ldots,\eta _{k_{r}}$ are the corresponding orthonormal eigenfunctions. Since $\eta_{j}\in D(H_{1,r})$, by (4.1) we have $\eta_{j}(b_{r}+1)=\frac{v(b_{r}+1)}{v(b_{r})}\eta_{j}(b_{r})$, $1\leq j\leq k_{r}$. Hence, there exist constants $c_{j}$, $1\leq j\leq k_{r}$, such that

$$\begin{aligned} \begin{pmatrix} \eta_{j}(b_{r}) \\ \eta_{j}(b_{r}+1) \end{pmatrix} =c_{j} \begin{pmatrix} v(b_{r}) \\ v(b_{r}+1) \end{pmatrix}. \end{aligned}$$

(4.6)

For every $j\in\{1,\ldots,k_{r}\}$, let

$$\psi_{j}(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \eta_{j}(t), & a-1\leq t\leq b_{r}, \\ c_{j}v(t), & t\geq b_{r}+1. \end{array}\displaystyle \right . $$

Then $\psi_{j}\in D(H)$. This, together with (4.1) and $\eta_{j}\in D(H_{1,r})$, shows that $\psi _{j}\in D(H_{1})=D(H_{1,s})$, where (2.1) has been used. Let

$$S:=L\{\psi_{1},\ldots,\psi_{k_{r}}\}. $$

Then S obviously is $k_{r}$-dimensional, and every $\psi\in S$ is of the form

$$\begin{aligned} \psi(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} \sum^{k_{r}}_{j=1}d_{j}\eta_{j}(t), & a-1\leq t\leq b_{r}, \\ cv(t), & t\geq b_{r}+1, \end{array}\displaystyle \right . \end{aligned}$$

(4.7)

where $d_{j}$, $j=1,\ldots,k_{r}$, are constants, and $c=\sum^{k_{r}}_{j=1}d_{j}c_{j}$. It follows from (4.7) that

$$\|\psi\|^{2}=\sum^{+\infty}_{t=a}w(t)\bigl| \psi(t)\bigr|^{2} =\sum^{k_{r}}_{j=1}|d_{j}|^{2}+|c|^{2} \sum^{+\infty }_{t=b_{r}+1}w(t)\bigl|v(t)\bigr|^{2}. $$

Therefore,

$$\begin{aligned} &\biggl\| \biggl(H_{1,s}-\frac{\alpha+\beta}{2}\biggr)\psi\biggr\| ^{2} \\ &\quad=\sum^{+\infty }_{t=a}w(t)\biggl| \biggl(H_{1,s}-\frac{\alpha+\beta}{2}\biggr)\psi(t)\biggr|^{2} \\ &\quad=\sum^{k_{r}}_{j=1}|d_{j}|^{2} \sum^{b_{r}}_{t=a}w(t)\biggl|\biggl(\lambda _{j}-\frac{\alpha+\beta}{2}\biggr)\eta_{j}(t)\biggr|^{2} +|c|^{2}\sum^{+\infty}_{t=b_{r}+1}w(t)\biggl| \biggl(\gamma-\frac{\alpha+\beta }{2}\biggr)v(t)\biggr|^{2} \\ &\quad=\sum^{k_{r}}_{j=1}|d_{j}|^{2}\biggl| \lambda_{j}-\frac{\alpha+\beta}{2}\biggr|^{2} +|c|^{2}\biggl| \gamma-\frac{\alpha+\beta}{2}\biggr|^{2}\sum^{+\infty }_{t=b_{r}+1}w(t)\bigl|v(t)\bigr|^{2} \\ &\quad\leq\biggl(\frac{\beta-\alpha}{2}\biggr)^{2}\|\psi\|^{2}. \end{aligned}$$

This shows that

$$\biggl\| \biggl(H_{1,s}-\frac{\alpha+\beta}{2}\biggr)\psi\biggr\| \leq\frac{\beta-\alpha}{2} \| \psi\|. $$

Consequently, by Lemma 4.1 one has

$$\begin{aligned} k_{r}=\dim S\leq\dim R\bigl\{ E_{s}(H_{1}, \beta)-E_{s}(H_{1},\alpha-)\bigr\} < \infty. \end{aligned}$$

(4.8)

Further, note that $\alpha\in\rho(H_{1})=\rho(H_{1,s})$ by Lemma 2.2. Hence, by Theorem 7.22 in [24] we have

$$\dim R\bigl\{ E_{s}(H_{1},\beta)-E_{s}(H_{1}, \alpha-)\bigr\} =\dim R\bigl\{ E_{s}(H_{1},\beta )-E_{s}(H_{1},\alpha)\bigr\} , $$

which, together with (4.5) and (4.8), shows that

$$\begin{aligned} \dim R\bigl\{ E_{s}\bigl(H'_{1,r}, \beta\bigr)-E_{s}\bigl(H'_{1,r},\alpha\bigr)\bigr\} \leq\dim R\bigl\{ E_{s}(H_{1},\beta)-E_{s}(H_{1}, \alpha)\bigr\} . \end{aligned}$$

(4.9)

On the other hand, one can show that $\{H_{1,r}'\}$ converges to $H_{1}$ in the strong resolvent sense with a completely similar argument to that in the proof of Theorem 3.3 in [22] (for the concept of convergence of self-adjoint subspaces in the strong resolvent sense, please see Definition 4.1 in [21]). From (4.2), it follows that $\{{H_{1,r,s}'}\}$ converges to $H_{1,s}$ in the strong resolvent sense by Theorem 4.2 in [21]. Therefore, by Theorem 9.19 in [24] we get

$$\begin{aligned} E_{s}\bigl(H_{1,r}',\lambda \bigr) \stackrel{s}{\to} E_{s}(H_{1},\lambda) \quad \mbox{for } \lambda\in{\mathbf {R}}\setminus\sigma_{p}(H_{1}), \end{aligned}$$

(4.10)

where $\sigma_{p}(H_{1})=\sigma_{p}(H_{1,s})$ has been used. By (4.9)-(4.10), (4.3) follows from Lemma 4.2. This completes the proof. □

The following result is a direct consequence of Theorem 4.1.

Corollary 4.1

Assume that $\mathcal{L}$ is regular at $t=a$ and in l.p.c. at $t=+\infty$. Let $H_{1}$ be any fixed self-adjoint subspace extension of $H_{0}$ given by (2.12). If $H_{1}$ has a pure discrete spectrum, then the sequence $\{H_{1,r}\}_{r=1}^{\infty}$ defined by (4.1) is spectrally exact for $H_{1}$ if $0\notin\sigma(H_{1})$.

5 One endpoint is in l.c.c. and the other in l.p.c

In this section, we shall study spectral exactness in an open interval laking essential spectral points in Case 4. Without loss of generality, we only consider the case that $\mathcal{L}$ is in l.c.c. at $t=-\infty$ and in l.p.c. at $t=+\infty$.

In this case, it was shown that the sequence of induced self-adjoint restrictions $\{H_{1,r}\}$ is spectrally inclusive for any given self-adjoint subspace extension $H_{1}$ but not spectrally exact for it in general in [22]. By Remark 2.1, every $H_{1}$ given by (2.19) is a self-adjoint operator extension of $H_{0}$. Now, we shall try to choose a sequence of induced regular self-adjoint operator extensions, still denoted by $\{H_{1,r}\}$ without any confusion, such that it is spectrally exact for $H_{1}$ in an open interval laking essential spectral points.

Let $H_{1}$ be any fixed self-adjoint operator extension of $H_{0}$ given by (2.19). Denote $I_{1}:=\{t\}_{t=-\infty}^{d_{0}-1}$, where $d_{0}$ is the same as in (2.8). Let $H_{a}$ and $H_{a,0}$ be the left maximal and minimal subspaces corresponding to (1.1) or $\mathcal{L}$ on $I_{1}$, respectively. Let

$$\hat{H}_{a,0}=\bigl\{ (x,f)\in H_{a}: x(d_{0}-1)=x(d_{0})=0 \mbox{ and } (x,y) (-\infty)=0 \mbox{ for all } y\in D\bigl(H_{a,0}^{*} \bigr)\bigr\} . $$

Assume that $0\notin I_{0}\subset{\mathbf {R}}$ is an open interval and $I_{0} \cap\sigma_{e}(H_{1})=\emptyset$, $I_{0} \cap\sigma_{d}(H_{1})\ne \emptyset$. Then for ŷ defined by (2.20) and any fixed $\delta\in I_{0}$, by virtue of Theorem 3.8 in [12], there exist uniquely $y_{0}\in D(\hat{H}_{a,0})$ and one solution h in $l_{w}^{2}(I_{1})$ of (1.1 _δ ) such that

$$\begin{aligned} \hat{y}(t)=y_{0}(t)+h(t), \quad t\leq d_{0}-2. \end{aligned}$$

(5.1)

We assert that h is nontrivial. In fact, if the assertion would not hold, then $\hat{y}(t)=y_{0}(t)$ for $t\leq d_{0}-2$. Hence, for any $(x,f)\in H$, one has $(x,\hat{y})(-\infty )=(x,y_{0})(-\infty)=0$, where the definition of $\hat{H}_{a,0}$ and $H_{a}\subset H_{a,0}^{*}$ have been used. So, it follows from (2.19) that $H_{1}=H$. This leads to a contradiction. Thus, this assertion holds. For any $(x,f)\in H$, it follows from (5.1) that

$$\begin{aligned} (x,\hat{y}) (-\infty)=(x,h) (-\infty). \end{aligned}$$

(5.2)

Therefore, $H_{1}$ determined by (2.19) can be expressed as

$$H_{1}=\bigl\{ (x,f)\in H: (x,h) (-\infty)=0\bigr\} . $$

In addition, by Lemma 4.3, there exists at least one nontrivial square summable solution v of (1.1 _γ ) for any $\gamma\in I_{0}$. Consequently, we can choose $\{a_{r}\}_{r=1}^{\infty}$ and $\{b_{r}\} _{r=1}^{\infty}$ specified in Section 2.3 such that $h(a_{r})\neq0$ and $v(b_{r})\neq0$ for $r\in{\mathbf {N}}$.

Theorem 5.1

Assume that $\mathcal{L}$ is in l.c.c. at $t=-\infty$ and in l.p.c. at $t=+\infty$. Let $H_{1}$ be any fixed self-adjoint operator extension of $H_{0}$ given by (2.19). Assume that $0\notin I_{0}\subset{\mathbf {R}}$ is an open interval with $I_{0} \cap\sigma_{e}(H_{1})=\emptyset$ and $I_{0} \cap\sigma_{d}(H_{1})\ne \emptyset$. Let v be a nontrivial real square summable solution of (1.1 _γ ) with any fixed $\gamma\in I_{0}$, $H_{1,r}$ the induced self-adjoint restriction of $H_{1}$ on $I_{r}$ defined by

$$\begin{aligned} H_{1,r}=\bigl\{ (x,f)\in H^{r}: (x,h) (a_{r}-1)=0, (x,v) (b_{r})=0\bigr\} , \end{aligned}$$

(5.3)

where h is determined by (5.1) with $\delta=\gamma$ and $\{a_{r}\}_{r=1}^{\infty}$ and $\{b_{r}\}_{r=1}^{\infty}$ specified in Section 2.3 satisfy $h(a_{r})\neq0$ and $v(b_{r})\neq0$ for $r\in{\mathbf {N}}$, respectively. Then $\{H_{1,r}\}_{r=1}^{\infty}$ is spectrally exact for $H_{1}$ in $I_{0}$.

Proof

The main idea of the proof is similar to that of the proof of Theorem 4.1. For completeness, we shall give its details.

Since $v(b_{r})\neq0$ and $h(a_{r})\neq0$, we have $H_{1,r}(0)=\{0\}$. Therefore, $H_{1,r}$ given by (5.3) is a self-adjoint operator extension of $H_{0}^{r}$. This, together with Lemma 2.6, shows that $\tilde{H}_{1,r}$ and $H'_{1,r}$ are self-adjoint operators in $\tilde {l}_{w}^{2}(I_{r})$ and $l_{w}^{2}(I)$, respectively.

By Lemmas 2.4 and 2.5, in order to prove $\{H_{1,r}\}_{r=1}^{\infty}$ is spectrally exact for $H_{1}$ in $I_{0}$, it suffices to show that for any given $\alpha,\beta\in I_{0}\cap\rho (H_{1})$ with $\alpha<\gamma\leq\beta$ and sufficiently large r,

$$\begin{aligned} \dim R\bigl\{ E\bigl(H'_{1,r},\beta\bigr)-E \bigl(H'_{1,r},\alpha\bigr)\bigr\} =\dim R\bigl\{ E(H_{1},\beta )-E(H_{1},\alpha)\bigr\} . \end{aligned}$$

(5.4)

Fix any $\alpha,\beta\in I_{0}\cap\rho(H_{1})$ with $\alpha<\gamma\leq \beta$. For any fixed r, let $\lambda_{1},\ldots,\lambda_{k_{r}}$ be all $k_{r}$ (counting multiplicity) eigenvalues of $H_{1,r}$ in $(\alpha,\beta]$, and $\eta_{1},\ldots,\eta_{k_{r}}$ the corresponding orthonormal eigenfunctions. Then, by Lemma 2.5, the assumption that $0\notin I_{0}$, and the first proposition in [24], p.204, we get

$$\begin{aligned} &\dim R\bigl\{ E\bigl(H'_{1,r},\beta\bigr)-E \bigl(H'_{1,r},\alpha\bigr)\bigr\} \\ &\quad=\dim R\bigl\{ E(\tilde {H}_{1,r},\beta)-E(\tilde{H}_{1,r}, \alpha)\bigr\} \\ &\quad=\dim R\bigl\{ E(H_{1,r},\beta)-E(H_{1,r},\alpha)\bigr\} =k_{r}. \end{aligned}$$

(5.5)

On the other hand, it is obvious that $\eta_{j}\in D(H_{1,r})$ and thus by (5.3) we have

$$\eta_{j}(a_{r}-1)=\frac{h(a_{r}-1)}{h(a_{r})}\eta_{j}(a_{r}), \qquad \eta_{j}(b_{r}+1)=\frac{v(b_{r}+1)}{v(b_{r})} \eta_{j}(b_{r}),\quad 1\leq j\leq k_{r}. $$

Hence, there exist constants $c_{j}$ and $d_{j}$, $1\leq j\leq k_{r}$, such that

$$\begin{aligned} \begin{pmatrix} \eta_{j}(a_{r}-1) \\ \eta_{j}(a_{r}) \end{pmatrix} =c_{j} \begin{pmatrix} h(a_{r}-1) \\ h(a_{r}) \end{pmatrix},\qquad \begin{pmatrix} \eta_{j}(b_{r}) \\ \eta_{j}(b_{r}+1) \end{pmatrix} =d_{j} \begin{pmatrix} v(b_{r}) \\ v(b_{r}+1) \end{pmatrix}. \end{aligned}$$

(5.6)

For every $j\in\{1,\ldots,k_{r}\}$, let

$$\begin{aligned} \psi_{j}(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} c_{j}h(t), & t\leq a_{r}-1 , \\ \eta_{j}(t), & a_{r}\leq t\leq b_{r}, \\ d_{j}v(t), & t\geq b_{r}+1. \end{array}\displaystyle \right . \end{aligned}$$

(5.7)

Then $\psi_{j}\in D(H)$. Since $\hat{y}\in D(H_{1})$, it follows from (5.2) that $(h,\hat {y})(-\infty)=(\hat{y},\hat{y})(-\infty)=0$ and thus $\psi_{j}\in D(H_{1})$ by (2.19). Let

$$S:=L\{\psi_{1},\ldots,\psi_{k_{r}}\}. $$

Then S obviously is $k_{r}$-dimensional, and every $\psi\in S$ is of the form

$$\begin{aligned} \psi(t)=\left \{ \textstyle\begin{array}{@{}l@{\quad}l} ch(t), & t\leq a_{r}-1,\\ \sum^{k_{r}}_{j=1}l_{j}\eta_{j}(t), & a_{r}\leq t\leq b_{r}, \\ dv(t), & t\geq b_{r}+1, \end{array}\displaystyle \right . \end{aligned}$$

(5.8)

where $l_{j}$, $1\leq j\leq k_{r}$, are constants, $c=\sum^{k_{r}}_{j=1}l_{j}c_{j}$, and $d=\sum^{k_{r}}_{j=1}l_{j}d_{j}$. It follows from (5.8) that

$$\begin{aligned} \|\psi\|^{2}&=\sum^{+\infty}_{t=-\infty}w(t)\bigl| \psi(t)\bigr|^{2} \\ &=|c|^{2}\sum^{a_{r}-1}_{t=-\infty}w(t)\bigl|h(t)\bigr|^{2}+ \sum^{k_{r}}_{j=1}|l_{j}|^{2}+ |d|^{2}\sum^{+\infty}_{t=b_{r}+1}w(t)\bigl|v(t)\bigr|^{2}. \end{aligned}$$

Therefore,

$$\begin{aligned} &\biggl\| \biggl(H_{1}-\frac{\alpha+\beta}{2}\biggr)\psi\biggr\| ^{2} \\ &\quad=\sum^{+\infty }_{t=-\infty}w(t)\biggl| \biggl(H_{1}-\frac{\alpha+\beta}{2}\biggr)\psi(t)\biggr|^{2} \\ &\quad=\sum^{a_{r}-1}_{t=-\infty}w(t)\biggl|\biggl(\gamma- \frac{\alpha+\beta}{2}\biggr)ch(t)\biggr|^{2} +\sum^{b_{r}}_{t=a_{r}}w(t)\Biggl| \biggl(\lambda_{j}-\frac{\alpha+\beta}{2}\biggr)\sum ^{k_{r}}_{j=1}l_{j}\eta_{j}(t)\Biggr|^{2} \\ &\qquad{}+\sum^{+\infty}_{t=b_{r}+1}w(t)\biggl|\biggl( \gamma-\frac{\alpha+\beta }{2}\biggr)\,dv(t)\biggr|^{2} \\ &\quad=|c|^{2}\biggl|\gamma-\frac{\alpha+\beta}{2}\biggr|^{2}\sum ^{a_{r}-1}_{t=-\infty }w(t)\bigl|h(t)\bigr|^{2} +\sum ^{k_{r}}_{j=1}|l_{j}|^{2}\biggl| \lambda_{j}-\frac{\alpha+\beta}{2}\biggr|^{2} \\ &\qquad{}+|d|^{2}\biggl|\gamma-\frac{\alpha+\beta}{2}\biggr|^{2}\sum ^{+\infty }_{t=b_{r}+1}w(t)\bigl|v(t)\bigr|^{2} \\ &\quad\leq\biggl(\frac{\beta-\alpha}{2}\biggr)^{2}\|\psi\|^{2}. \end{aligned}$$

Thus,

$$\biggl\| \biggl(H_{1}-\frac{\alpha+\beta}{2}\biggr)\psi\biggr\| \leq\frac{\beta-\alpha}{2} \|\psi \|. $$

Consequently, by Lemma 4.1 one has

$$\begin{aligned} k_{r}=\dim S\leq\dim R\bigl\{ E(H_{1}, \beta)-E(H_{1},\alpha-)\bigr\} < \infty. \end{aligned}$$

(5.9)

Since $\alpha\in\rho(H_{1})$, by Theorem 7.22 in [24] we have

$$\dim R\bigl\{ E(H_{1},\beta)-E(H_{1},\alpha-)\bigr\} =\dim R \bigl\{ E(H_{1},\beta)-E(H_{1},\alpha )\bigr\} , $$

which, together with (5.5) and (5.9), shows that

$$\begin{aligned} \dim R\bigl\{ E\bigl(H'_{1,r},\beta\bigr)-E \bigl(H'_{1,r},\alpha\bigr)\bigr\} \leq\dim R\bigl\{ E(H_{1},\beta )-E(H_{1},\alpha)\bigr\} . \end{aligned}$$

(5.10)

On the other hand, one can show that $\{H_{1,r}'\}$ converges to $H_{1}$ in the strong resolvent sense with a completely similar argument to that in the proof of Theorem 4.3 in [22] and so we omit the details. Therefore, by Theorem 9.19 in [24], it follows that

$$E\bigl(H_{1,r}',\lambda\bigr) \stackrel{s}{\to} E(H_{1},\lambda) \quad\mbox{for } \lambda\in{\mathbf {R}}\setminus \sigma_{p}(H_{1}). $$

Together with (5.10), (5.4) follows from Lemma 4.2. This completes the proof. □

The following result is a direct consequence of Theorem 5.1.

Corollary 5.1

Assume that $\mathcal{L}$ is in l.c.c. at $t=-\infty$ and in l.p.c. at $t=+\infty$. Let $H_{1}$ be any fixed self-adjoint subspace extension of $H_{0}$ given by (2.19). If $H_{1}$ has a pure discrete spectrum, then the sequence $\{H_{1,r}\}_{r=1}^{\infty}$ defined by (5.3) is spectrally exact for $H_{1}$ if $0\notin\sigma(H_{1})$.

6 Both endpoints are in l.p.c

In this section, we shall study spectral exactness in an open interval laking essential spectral points in Case 5. In this case, $H_{1}=H_{0}=H$ is a self-adjoint operator. In [22], it was shown that the sequence of induced self-adjoint restrictions $\{H_{1,r}\}$ is spectrally inclusive for $H_{0}$ but not spectrally exact for $H_{0}$ in general. Now, we shall try to choose a sequence of induced regular self-adjoint operator extensions, denoted by $\{H_{0,r}\}$, which is spectrally exact for $H_{0}$ in an open interval laking essential spectral points.

Denote $I_{2}:=\{t\}_{t=d_{0}}^{+\infty}$, where $d_{0}$ is the same as in (2.8). Let $H_{b}$ and $H_{b,0}$ be the right maximal and minimal subspaces corresponding to (1.1) or $\mathcal{L}$ on $I_{2}$, respectively. $I_{1}$, $H_{a}$, and $H_{a,0}$ are specified in Section 5. Let $H_{a,1}$ and $H_{b,1}$ be any self-adjoint subspace extensions of $H_{a,0}$ and $H_{b,0}$, separately. Then, by Theorem 3.3 in [29] and Corollary 2.1 in [30], one has

$$\begin{aligned} \sigma_{e}(H_{1})=\sigma_{e}(H_{0})= \sigma_{e}(H_{a,1})\cup\sigma_{e}(H_{b,1}) \end{aligned}$$

(6.1)

in Case 5. Assume that $0\notin I_{0}\subset{\mathbf {R}}$ is an open interval with $I_{0} \cap\sigma_{e}(H_{0})=\emptyset$ and $I_{0} \cap\sigma_{d}(H_{0})\ne \emptyset$. Then, by Lemma 4.3 and (6.1), there exist two nontrivial real solutions $v_{1}$ in $l_{w}^{2}(I_{1})$ and $v_{2}$ in $l_{w}^{2}(I_{2})$ of (1.1 _γ ) with any $\gamma\in I_{0}$. Consequently, we can choose $\{a_{r}\}_{r=1}^{\infty}$ and $\{b_{r}\} _{r=1}^{\infty}$ specified in Section 2.3 such that $v_{1}(a_{r})\neq0$ and $v_{2}(b_{r})\neq0$ for $r\in{\mathbf {N}}$.

Theorem 6.1

Assume that $\mathcal{L}$ is in l.p.c. at $t=\pm\infty$, $0\notin I_{0}\subset{\mathbf {R}}$ is an open interval with $I_{0} \cap\sigma_{e}(H_{0})=\emptyset$ and $I_{0} \cap\sigma_{d}(H_{0})\ne \emptyset$. Let $v_{1}$ and $v_{2}$ be two nontrivial real solutions of (1.1 _γ ) with any fixed $\gamma\in I_{0}$, which are square summable near ∓∞, respectively, $H_{0,r}$ the induced self-adjoint restriction of $H_{0}$ on $I_{r}$ defined by

$$\begin{aligned} H_{0,r}=\bigl\{ (x,f)\in H^{r}: (x,v_{1}) (a_{r}-1)=0, (x,v_{2}) (b_{r})=0\bigr\} , \end{aligned}$$

(6.2)

where $\{a_{r}\}_{r=1}^{\infty}$ and $\{b_{r}\}_{r=1}^{\infty}$ specified in Section 2.3 satisfy $v_{1}(a_{r})\neq0$ and $v_{2}(b_{r})\neq0$ for $r\in{\mathbf {N}}$, respectively. Then $\{H_{0,r}\}_{r=1}^{\infty}$ is spectrally exact for $H_{0}$ in $I_{0}$.

Proof

The main idea of the proof is similar to that of the proof of Theorem 5.1. So we omit its details. This completes the proof. □

Corollary 6.1

Assume that $\mathcal{L}$ is in l.p.c. at $t=\pm\infty$, and $H_{0}$ has a pure discrete spectrum. Then the sequence $\{H_{0,r}\}_{r=1}^{\infty}$ defined by (6.2) is spectrally exact for $H_{0}$ if $0\notin\sigma(H_{0})$.

Remark 6.1

$H_{1,r}$ defined by (4.1), (5.3), and (6.2) can be viewed as special cases of those defined by (2.14), (2.18), and (2.21), respectively. For example, consider $H_{1,r}$ defined by (4.1). It can be obtained by taking $\lambda=\gamma$ in (2.8) and $u=v$ in (2.14) and choosing $\{b_{r}\}_{t=1}^{\infty}$ specified in Section 2.3 such that $v(b_{r})\neq0$ for $r\in{\mathbf {N}}$.