- Research
- Open Access
Regular approximations of isolated eigenvalues of singular second-order symmetric linear difference equations
- Yan Liu^{1} and
- Yuming Shi^{1}Email author
https://doi.org/10.1186/s13662-016-0850-2
© Liu and Shi 2016
- Received: 15 March 2016
- Accepted: 28 April 2016
- Published: 9 May 2016
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.
Keywords
- symmetric linear difference equation
- self-adjoint subspace extension
- regular approximation
- spectral exactness
- error estimate
MSC
- 39A10
- 41A99
- 47A06
- 47A10
1 Introduction
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
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.
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)
- (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)
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)
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].
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
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 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 \(b=+\infty\).
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].
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.
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.
Case 3. Both endpoints are in l.c.c.
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 \(t=+\infty\).
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 \(H_{0}\).
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.
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.
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
- (i)
\(T'=TP\) is a self-adjoint operator on X with \(D(T')=D(T)\oplus X_{1}^{\perp}\);
- (ii)
\(\sigma(T')=\sigma(T)\cup\{0\}\).
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
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
Proof
Theorem 3.2
Proof
We now turn to error estimates of approximations of \(\lambda_{n}^{(r)}\) to \(\lambda_{n}\) in Case 3.
Proposition 3.2
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.
The proof of the following result is similar to that of Theorem 3.2 and so its details are omitted.
Theorem 3.3
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
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
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
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.
Theorem 5.1
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.
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.
Theorem 6.1
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 \(r\in{\mathbf {N}}\).
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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