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Volume 40, Issue 3
Inversion of Trace Formulas for a Sturm-Liouville Operator

Xiang Xu & Jian Zhai

DOI: 10.4208/jcm.2010-m2019-0307

J. Comp. Math., 40 (2022), pp. 396-414.

Published online: 2022-02

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  • Abstract

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.

  • Keywords

Inverse spectral problem, Sturm-Liouville operator, Trace formulas.

  • AMS Subject Headings

65F18, 34B24

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

xxu@zju.edu.cn (Xiang Xu)

jian.zhai@outlook.com (Jian Zhai)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-396, author = {Xu , Xiang and Zhai , Jian}, title = {Inversion of Trace Formulas for a Sturm-Liouville Operator}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {3}, pages = {396--414}, abstract = {

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2010-m2019-0307}, url = {http://global-sci.org/intro/article_detail/jcm/20243.html} }
TY - JOUR T1 - Inversion of Trace Formulas for a Sturm-Liouville Operator AU - Xu , Xiang AU - Zhai , Jian JO - Journal of Computational Mathematics VL - 3 SP - 396 EP - 414 PY - 2022 DA - 2022/02 SN - 40 DO - http://doi.org/10.4208/jcm.2010-m2019-0307 UR - https://global-sci.org/intro/article_detail/jcm/20243.html KW - Inverse spectral problem, Sturm-Liouville operator, Trace formulas. AB -

This paper revisits the classical problem “Can we hear the density of a string?”, which can be formulated as an inverse spectral problem for a Sturm-Liouville operator. Instead of inverting the map from density to spectral data directly, we propose a novel method to reconstruct the density based on inverting a sequence of trace formulas which bridge the density and its spectral data clearly in terms of a series of nonlinear integral equations. Numerical experiments are presented to verify the validity and effectiveness of the proposed numerical algorithm. The impact of different parameters involved in the algorithm is also discussed.

Xu , Xiang and Zhai , Jian. (2022). Inversion of Trace Formulas for a Sturm-Liouville Operator. Journal of Computational Mathematics. 40 (3). 396-414. doi:10.4208/jcm.2010-m2019-0307
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