Abstract
In this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.
Highlights
We consider the following Sturm–Liouville differential equation: 1 ly := – py+ qy = λy, x ∈ (a, b), (1.1)w where λ is a complex parameter, –∞ ≤ a < b≤ +∞
We find that its spectrum can be approximated by the eigenvalues of a sequence of regular problems, based on the method of the strong graph limit, which is different from that of the previous papers such as [1, 22, 24]
(2) The endpoint b is of the limit point case if and only if [y, z](b) = 0 for any y, z ∈ Dmax
Summary
If the endpoint b is of the limit point case, [f , g](b) = 0 for any F, G ∈ D(A) by Lemma 1(2). If the endpoint b is of the limit point case, let the domain of Ar be
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