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Regular approximations of isolated eigenvalues of singular secondorder symmetric linear difference equations
Advances in Difference Equations volume 2016, Article number: 128 (2016)
Abstract
This paper is concerned with regular approximations of isolated eigenvalues of singular secondorder symmetric linear difference equations. It is shown that the kth eigenvalue of any given selfadjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular selfadjoint 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 selfadjoint subspace extension are exactly the limits of eigenvalues of suitably chosen induced regular selfadjoint subspace extensions in the case that at least one endpoint is in the limit point case.
Introduction
Consider the following secondorder symmetric linear difference equation:
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., \(\triangle x(t)=x(t+1)x(t)\), \(\nabla x(t)=x(t)x(t1)\); \(p(t)\) and \(q(t)\) are all realvalued with \(p(t)\neq0\) for \(t\in I\), \(p(a1)\neq0\) 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 wellbehaved 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 fourthorder 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 secondorder 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 multivalued (cf. [10–12]). So it cannot be treated by the methods described in [3, 5–9], which are based on selfadjoint extensions of densely defined Hermitian operators.
This major difficulty can be overcome by using the theory of selfadjoint 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 GlazmanKreinNaimark (briefly, GKN) theory to Hermitian subspaces [11], and based on this, she with her coauthor Sun presented complete characterizations of selfadjoint extensions for secondorder symmetric linear difference equation in both regular and singular cases [12]. Later, she studied some spectral properties of selfadjoint 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 selfadjoint subspaces [21].
Applying the results given in [12, 21], we studied regular approximations of spectra of singular secondorder symmetric linear difference equations [22]. We constructed suitable induced regular selfadjoint subspace extensions and proved that the sequence of induced regular selfadjoint subspace extensions is both spectrally inclusive and exact for a given selfadjoint 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 secondorder 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 selfadjoint 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 selfadjoint extensions have to be characterized by the theory of subspaces. These selfadjoint extensions are multivalued 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 secondorder symmetric linear difference equations are introduced. In addition, the induced regular selfadjoint subspace extensions for any given selfadjoint subspace extension are introduced. In particular, a sufficient condition is given for spectral exactness of a sequence of selfadjoint 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 selfadjoint subspace extension is exactly the limit of the kth eigenvalues of the induced regular selfadjoint subspace extensions. In addition, their error estimates are given. Spectral exactness in every gap of the essential spectrum of any selfadjoint subspace extension is obtained in the other three cases in Sections 46, separately.
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, preminimal, and minimal subspaces corresponding to equation (1.1) are introduced. In Section 2.3, some results about selfadjoint subspace extensions of the minimal subspace and their induced selfadjoint restrictions given in [22] are recalled.
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:
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
Further, denote
It is evident that \(T(0)=\{0\}\) if and only if T can uniquely determine a (singledvalued) 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
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
Consequently, if \(\rho(T)\ne\emptyset\), then
Let T and S be two subspaces in \(X^{2}\). If \(T\cap S=\{(0,0)\}\), denote
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
In addition, we introduce the following notation for convenience:
In 1961, Arens [23] introduced the following important decomposition for a closed subspace T in \(X^{2}\):
where
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 multivalued parts of T, respectively. In addition,
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
\(T_{s}\) is a closed Hermitian operator in \(T(0)^{\perp}\), \(T_{\infty}\) is a closed Hermitian subspace in \(T(0)^{2}\),
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 selfadjoint subspace in \(X^{2}\), then \(T_{\infty}\) and \(T_{s}\) are selfadjoint 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 selfadjoint subspace, which was introduced by Coddington and Dijksma in [15].
Let T be a selfadjoint subspace in \(X^{2}\). By Lemma 2.3, \(T_{s}\) is a selfadjoint operator in \(T(0)^{\perp}\). Then \(T_{s}\) has the following spectral resolution:
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
where O is the zero operator defined on \(T(0)\). It is obvious that for any \(t\in{\mathbf {R}}\) and any \(f\in X\),
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 selfadjoint subspace extension in Sections 46.
Lemma 2.4
Let \(\{T_{n}\}_{n=1}^{\infty}\) and T be selfadjoint 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
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 \(n\geq n_{0}\),
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 \(n\ge n_{0}\),
Note that
that is,
Since \(E(T_{n},(\alpha, \beta])\) and \(E(T,(\alpha, \beta])\) are orthogonal projections and
by (c) of Theorem 4.30 in [24] we have
It follows that
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. □
Maximal, preminimal and minimal subspaces
In this subsection, we first introduce the concepts of maximal, preminimal 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: \(a1\) means −∞ in the case of \(a=\infty\); \(b+1\) means +∞ in the case of \(b=+\infty\).
Denote
Then \(l_{w}^{2}(I)\) is a Hilbert space with the inner product
where \(x=y\) in \(l_{w}^{2}(I)\) if and only if \(\xy\=0\), 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
Now, we introduce the corresponding maximal, preminimal, and minimal subspaces corresponding to (1.1) in the interval I. Let
where H and \(H_{00}\) are called the maximal and preminimal 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 nondensely 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:
Selfadjoint subspace extensions and their induced selfadjoint restrictions
In this subsection, we recall the results about selfadjoint subspace extensions of \(H_{0}\) and their induced regular selfadjoint subspace extensions, i.e., induced selfadjoint 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 realvalued, 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 selfadjoint 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:
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 selfadjoint 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 selfadjoint subspace extension of \(H_{0}\). Then, by (3.5) in [22], we have
where
while
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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) can be given by
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 selfadjoint subspace extension of \(H_{0}\). Then, by (3.9) in [22], we have
where
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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) can be given by
where
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 selfadjoint subspace extension of \(H_{0}\). Then, by (4.4) in [22], we have
where
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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) can be given by
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 selfadjoint subspace extension of \(H_{0}\). According to the discussion for (4.8) in [22], we have
where
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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) can be given by
where
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 selfadjoint 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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) can be given by
where
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 selfadjoint subspace extension \(H_{1}\) of \(H_{0}\) is a selfadjoint operator in the case that \(I=(\infty, +\infty)\); that is, \(H_{1}\) can define a singlevalued selfadjoint operator in \(l_{w}^{2}(\infty,+\infty)\) whose graph is \(H_{1}\).
To end this section, we consider extensions of the induced selfadjoint restrictions from \(I_{r}\) to I.
Note that \(H_{1}\), \(H_{1,r}\) are selfadjoint 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 \(I=[a, +\infty)\),
In the case that \(I=(\infty, +\infty)\),
Let \(P_{r}\) be the orthogonal projection from \(l_{w}^{2}(I)\) onto \(\tilde {l}_{w}^{2}(I_{r})\). Define
Lemma 2.5
([22], Lemmas 3.1, 3.2, 3.3 and 4.1)
\(\tilde{H}_{1,r}\) and \(H_{1,r}'\) are selfadjoint 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
\(H_{1,r}'(0)=\tilde{H}_{1,r}(0)=\{\tilde{f}\in\tilde{l}_{w}^{2}(I_{r}) :\textit{ there exists }f\in H_{1,r}(0)\textit{ such that }\tilde{f}(t) =f(t)\mbox{ }\textit{for }t\in I_{r}\}\).
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 selfadjoint restrictions \(\{H_{1,r}\} _{r=1}^{\infty}\) is spectrally exact for the given selfadjoint 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 selfadjoint 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 \((zIH_{1})^{1}\) is a HilbertSchmidt operator for any \(z\in\rho(H_{1})\).
We only prove that \((zIH_{1})^{1}\) is a HilbertSchmidt 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 \(f\in l_{w}^{2}(I)\),
where
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
Define
where
Obviously, \((z IH_{1})^{1}=\mathcal{F}_{1}+\mathcal{F}_{2}\). Thus, it is sufficient to prove that \(\mathcal{F}_{1}\) and \(\mathcal {F}_{2}\) are both HilbertSchmidt operators.
We first prove that \(\mathcal{F}_{1}\) is a HilbertSchmidt operator. Let \(\{e_{n}: n\in\mathbf{N}\}\) be an orthonormal basis of \(l_{w}^{2}(I)\). Then
in which Parseval’s identity have been used. Therefore, \(\mathcal{F}_{1}\) is a HilbertSchmidt operator. Similarly, one can show that \(\mathcal{F}_{2}\) is a HilbertSchmidt operator and thus \((z IH_{1})^{1}\) is a HilbertSchmidt operator. The proof is complete. □
Remark 3.1

(1)
In Lemma 2.22 in [25], Teschl showed that each selfadjoint operator extension \(H_{1}\) with separated boundary conditions has a pure discrete spectrum, and its resolvent is a HilbertSchmidt 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 HilbertSchmidt 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 HilbertSchmidt 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 selfadjoint operator on \(X_{1}\). Then

(i)
\(T'=TP\) is a selfadjoint 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):
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):
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
Then, according to (2) of Remark 3.1 and Lemma 3.2, it follows that \(S_{r}P_{r}\) and S are both selfadjoint and HilbertSchmidt 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 selfadjoint and HilbertSchmidt operators for \(r\geq r_{0}\), and \(S_{r}P_{r} {\to} S\) in norm as \(r \to\infty\). Thus they are selfadjoint 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 selfadjoint 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 \(\alpha<\beta\),
which, together with Theorem 4.35 in [24], shows that
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 minmax principle, and the positive eigenvalues by a maxmin principle according to Section 12.1 in [27]; that is,
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,
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 ndimensional subspaces \(V_{n}\) and \(\tilde{V}_{n}\) of \(l^{2}_{w}(I)\) such that
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
Therefore, it follows that
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,
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
where
while \(\phi_{1}\) and \(\phi_{2}\) are two linearly independent solutions of (1.1_{ λ }) with \(\lambda=0\) satisfying the following initial conditions:
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
where
From (1.1_{ λ }) with \(\lambda=0\), we get
It follows that
where
Let \(x(a)\) and \(x(a1)\) 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
where
Denote
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 \(( x(a),\ x(a1))\) 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
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 \(r\geq r'_{n}\),
where \(e_{r}\) denotes the number on the righthand 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 \(n\in\Lambda\),
which shows that
Thus,
which implies that
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,
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
where
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
which shows that
where
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
where
Denote
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
In addition, since \(\phi_{1}\) and \(\phi_{2}\) satisfy (3.37)(3.38), one gets
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 \(r\geq r'_{n}\),
where \(\tilde{e}_{r}\) denotes the number on the righthand side in (3.30).
Remark 3.2
The authors in [1, 4] and [3] gave similar results to Theorem 3.1 for singular secondorder and fourthorder differential SturmLiouville problems, respectively, where the results in [1, 4] hold under the assumption that each endpoint is regular or in l.c.c. and nonoscillatory. 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.
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 selfadjoint restrictions \(\{H_{1,r}\}_{r=1}^{\infty}\) is spectrally inclusive for a given selfadjoint 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 selfadjoint 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 selfadjoint operator with spectral family E, and S a subspace in \(D(T)\) such that \(\(\lambdaT)f\\leq c\f\\) for all \(f\in S\). Then \(\dim R\{E(\lambda+c)E(\lambdac)\}\geq\dim S\).
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 selfadjoint subspace extension \(H_{1}\) of \(H_{0}\).
Proof
By the assumption, λ is not an eigenvalue of \(H_{1}\). Denote
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 selfadjoint 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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) defined by
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
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 \(r\geq \tilde{r}\),
We first prove (4.2). By (2.7) and (2.12), it can be deduced that
By (4.1) and \(v(b_{r})\neq0\) we get
which, together with Lemma 2.6, shows that
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
By (i) of Theorem 3.9 in [20] we have
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
For every \(j\in\{1,\ldots,k_{r}\}\), let
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
Then S obviously is \(k_{r}\)dimensional, and every \(\psi\in S\) is of the form
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
Therefore,
This shows that
Consequently, by Lemma 4.1 one has
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
which, together with (4.5) and (4.8), shows that
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 selfadjoint 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
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 selfadjoint 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 \(0\notin\sigma(H_{1})\).
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 selfadjoint restrictions \(\{H_{1,r}\}\) is spectrally inclusive for any given selfadjoint 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 selfadjoint operator extension of \(H_{0}\). Now, we shall try to choose a sequence of induced regular selfadjoint 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 selfadjoint 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
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
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
Therefore, \(H_{1}\) determined by (2.19) can be expressed as
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 selfadjoint 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 selfadjoint restriction of \(H_{1}\) on \(I_{r}\) defined by
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 selfadjoint operator extension of \(H_{0}^{r}\). This, together with Lemma 2.6, shows that \(\tilde{H}_{1,r}\) and \(H'_{1,r}\) are selfadjoint 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,
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
On the other hand, it is obvious that \(\eta_{j}\in D(H_{1,r})\) and thus by (5.3) we have
Hence, there exist constants \(c_{j}\) and \(d_{j}\), \(1\leq j\leq k_{r}\), such that
For every \(j\in\{1,\ldots,k_{r}\}\), let
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
Then S obviously is \(k_{r}\)dimensional, and every \(\psi\in S\) is of the form
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
Therefore,
Thus,
Consequently, by Lemma 4.1 one has
Since \(\alpha\in\rho(H_{1})\), by Theorem 7.22 in [24] we have
which, together with (5.5) and (5.9), shows that
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
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 selfadjoint 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 \(0\notin\sigma(H_{1})\).
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 selfadjoint operator. In [22], it was shown that the sequence of induced selfadjoint 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 selfadjoint 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 selfadjoint 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
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 selfadjoint restriction of \(H_{0}\) on \(I_{r}\) defined by
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 \(r\in{\mathbf {N}}\).
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Acknowledgements
The authors are grateful to the editor and referees for their valuable suggestions. This research was supported by the NNSF of China (Grant 11571202) and the China Scholarship Council (Grant 201406220019).
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YL carried out the results of this article and drafted the manuscript. YS proposed this study and inspected the manuscript. All authors read and approved the final manuscript.
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Liu, Y., Shi, Y. Regular approximations of isolated eigenvalues of singular secondorder symmetric linear difference equations. Adv Differ Equ 2016, 128 (2016). https://doi.org/10.1186/s1366201608502
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MSC
 39A10
 41A99
 47A06
 47A10
Keywords
 symmetric linear difference equation
 selfadjoint subspace extension
 regular approximation
 spectral exactness
 error estimate