Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions |
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Authors: | Guixia Wang Zhong Wang |
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Affiliation: | a Department of Mathematics, Inner Mongolia University, Hohhot 010021, China b Department of Mathematics, Inner Mongolia Normal University, Hohhot 010022, China c Department of Mathematics, ZhaoQing University, GuangDong 526061, China d Department of Mathematics, Northern Illinois University, DeKalb, IL 60115, USA |
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Abstract: | For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given. |
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Keywords: | Sturm-Liouville problems Indefinite leading coefficient functions Eigenvalues Number of zeros of eigenfunctions Interlacing relations Asymptotic formulas |
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