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Using Time Scales to Study Multi-Interval Sturm–Liouville Problems with Interface Conditions

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Abstract

We consider a Sturm–Liouville problem defined on multiple intervals with interface conditions. The existence of a sequence of eigenvalues is established and the zero counts of associated eigenfunctions are determined. Moreover, we reveal the continuous and discontinuous nature of the eigenvalues on the boundary condition. The approach in this paper is different from those in the literature: We transfer the Sturm–Liouville problem with interface conditions to a Sturm–Liouville problem on a time scale without interface conditions and then apply the Sturm–Liouville theory for equations on time scales. In this way, we are able to investigate the problem in a global view. Consequently, our results cover the cases when the potential function in the equation is not strictly greater than zero and when the domain consists of an infinite number of intervals.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL, 60115, USA

    Qingkai Kong

  2. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Qi-Ru Wang

Authors
  1. Qingkai Kong
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  2. Qi-Ru Wang
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Correspondence to Qingkai Kong.

Additional information

This paper is supported by the NNSF of China (No. 10971231).

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Kong, Q., Wang, QR. Using Time Scales to Study Multi-Interval Sturm–Liouville Problems with Interface Conditions. Results. Math. 63, 451–465 (2013). https://doi.org/10.1007/s00025-011-0208-8

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  • DOI: https://doi.org/10.1007/s00025-011-0208-8

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