Abstract
As was noted in Section 1 of Chapter 1, a Sturm-Liouville problem is said to be singular if either the interval [a, b] is infinite or if the function q(x) on [a, b] is not summable (or both). Here, we obtain the expansion theorem for the singular problem, considering it as the limit of regular ones.
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Notes
See A. I. Markushevich (1950, p. 294).
The convergence of RZ,bfn to RZfn follows from the first part of Lemma 2.2.2. Note that, from now onwards, we consider those p(λ) which are associated with the sequences of bk and βk such that limk→∞l(λ, bk,βk) =m(λ) (see Section 2).
For real λ, F(λ) is also real-valued, since q(x) is real-valued in the equation (6.1).
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© 1991 Springer Science+Business Media Dordrecht
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Levitan, B.M., Sargsjan, I.S. (1991). Spectral Theory in the Singular Case. In: Sturm—Liouville and Dirac Operators. Mathematics and its Applications (Soviet Series), vol 59. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3748-5_2
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DOI: https://doi.org/10.1007/978-94-011-3748-5_2
Publisher Name: Springer, Dordrecht
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