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Inverse problems for radial Schrödinger operators with the missing part of eigenvalues

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Abstract

We study inverse spectral problems for radial Schrödinger operators in L2(0, 1). It is well known that for a radial Schrödinger operator, two spectra for the different boundary conditions can uniquely determine the potential. However, if the spectra corresponding to the radial Schrödinger operators with the two potential functions miss a finite number of eigenvalues, what is the relationship between the two potential functions? Inspired by Hochstadt (1973)’s work, which handled the Sturm-Liouville operator with the potential q ∈ L1(0, 1), we give a corresponding result for radial Schrödinger operators with a larger class of potentials than L1(0, 1). When q ∈ L1(0, 1), we also consider the case where the spectra corresponding to the radial Schrödinger operators with the two potential functions miss an infinite number of eigenvalues and the eigenvalues are close in some sense.

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Acknowledgements

Xin-Jian Xu and Chuan-Fu Yang were supported by National Natural Science Foundation of China (Grant No. 11871031) and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK 20201303). The authors thank the referees for their insightful comments and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, China

    Xin-Jian Xu & Chuan-Fu Yang

  2. Department of Mathematics, Saratov State University, Saratov, 410012, Russia

    Vjacheslav A. Yurko

  3. Department of Mathematics, School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Ran Zhang

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  1. Xin-Jian Xu
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  2. Chuan-Fu Yang
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  3. Vjacheslav A. Yurko
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  4. Ran Zhang
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Correspondence to Chuan-Fu Yang.

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Xu, XJ., Yang, CF., Yurko, V.A. et al. Inverse problems for radial Schrödinger operators with the missing part of eigenvalues. Sci. China Math. 66, 1831–1848 (2023). https://doi.org/10.1007/s11425-022-2024-8

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  • DOI: https://doi.org/10.1007/s11425-022-2024-8

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