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Numerical algorithms for inverse Sturm-Liouville problems

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Abstract

In this paper, two classical inverse spectral problems are investigated, namely, the inverse second-order Sturm-Liouville problem and the inverse fourth-order Sturm-Liouville problem. Based on Lidskii’s theorem, we derive trace formulas showing relations between the unknown coefficients and eigenvalues explicitly for both problems. According to those trace formulas, two efficient algorithms are proposed to recover the symmetric potential from one spectrum for second-order Sturm-Liouville problem and two coefficients simultaneously from three spectra for fourth-order Sturm-Liouville problem, respectively. Numerical results are presented to illustrate the effectiveness of the proposed reconstruction algorithms.

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Funding

This work was partly supported by NSFC 11621101, 12071430 and the Fundamental Research Funds for the Central Universities.

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  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, China

    Xiaoying Jiang, Xiaowen Li & Xiang Xu

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  1. Xiaoying Jiang
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  2. Xiaowen Li
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  3. Xiang Xu
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Correspondence to Xiang Xu.

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Jiang, X., Li, X. & Xu, X. Numerical algorithms for inverse Sturm-Liouville problems. Numer Algor 89, 1287–1309 (2022). https://doi.org/10.1007/s11075-021-01153-2

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