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Sturm–Liouville Problems with Finite Spectrum

✍ Scribed by Q. Kong; H. Wu; A. Zettl

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
122 KB
Volume
263
Category
Article
ISSN
0022-247X

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✦ Synopsis

For every positive integer n, we construct a class of regular self-adjoint and nonself-adjoint Sturm-Liouville problems with exactly n eigenvalues. These n eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case. The latter complements the well-known general result for right-definite Sturm-Liouville problems with an infinite number of eigenvalues, which must go to infinity asymptotically like n 2 does. With an appropriate and natural interpretation of a "zero," the eigenfunctions have the usual zero properties.

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✍ Michel Duhoux 📂 Article 📅 2001 🏛 John Wiley and Sons 🌐 English ⚖ 238 KB

We consider a Sturm -Liouville operator Lu = -(r(t)u ) +p(t)u, where r is a (strictly) positive continuous function on ]a, b[ and p is locally integrable on ]a, b[ . Let r 1 (t) = t a (1/r) ds and choose any c ∈ ]a, b[ . We are interested in the eigenvalue problem Lu = λm(t)u, u(a) = u(b) = 0, and t