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.
Highlights
Consider the following second-order symmetric linear difference equation:–∇ p(t) x(t) + q(t)x(t) = λw(t)x(t), t ∈ I,( . λ) where I is the integer set {t}bt=a, a is a finite integer or –∞ and b is a finite integer or +∞; and ∇ are the forward and backward difference operators, respectively, i.e., x(t) =x(t + ) – x(t), ∇x(t) = x(t) – x(t – ); p(t) and q(t) are all real-valued with p(t) = for t ∈ I, p(a – ) = if a is finite and p(b + ) = if b is finite; w(t) > for t ∈ I; and λ is a complex spectral parameter.Spectral problems can be divided into two classifications
The second author of the present paper extended the classical Glazman-Krein-Naimark theory to Hermitian subspaces [ ], and based on this, she with her coauthor Sun presented complete characterizations of selfadjoint extensions for second-order symmetric linear difference equation in both regular and singular cases [ ]
Applying the results given in [, ], we studied regular approximations of spectra of singular second-order symmetric linear difference equations [ ]
Summary
Consider the following second-order symmetric linear difference equation:. ( . λ) where I is the integer set {t}bt=a, a is a finite integer or –∞ and b is a finite integer or +∞; and ∇ are the forward and backward difference operators, respectively, i.e., x(t) =. The second author of the present paper extended the classical Glazman-Krein-Naimark (briefly, GKN) theory to Hermitian subspaces [ ], and based on this, she with her coauthor Sun presented complete characterizations of selfadjoint extensions for second-order symmetric linear difference equation in both regular and singular cases [ ] Later, she studied some spectral properties of self-adjoint subspaces together with her coauthors Shao and Ren [ ]. 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 , regular approximations of isolated eigenvalues of equation
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