Abstract
Let L bea linear operator defined on a certain set of elements. An element y ≠ 0 is called an eigenelement of L if Ly = λy, and A is called the corresponding eigenvalue of L.
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Notes
Assuming for the present that neither h nor H are infinite. These cases will be considered in the sequel.
This least zero exists because a nontrivial solution y(x) of a linear differential equation has only isolated zeros. Indeed, assume that x0 ≠ ∞ is the limit point of the zeros x1, x2,…, xn,… of a solution of the differential equation. Then y(x0) = 0 by the continuity of y(x). Furthermore, (y(x0) — y(xn))/(x0-xn) = 0. Passing to the limit for n → ∞, we obtain y′(x0) = 0, i.e., y(x) = 0.
In the sequel, we will often write ϕ, ϕ′, ϑ, ϑ′ instead of ϕ(a, λ), ϕ′(a, λ), ϑ(a, λ) and ϑ′(a, λ), respectively.
See I. G. Petrovsky (1948, p. 68).
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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 Regular 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_1
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DOI: https://doi.org/10.1007/978-94-011-3748-5_1
Publisher Name: Springer, Dordrecht
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