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Stability of the inverse transmission eigenvalue problem for the Schrödinger operator with a radial potential

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Abstract

In the work, we consider the stability of the inverse transmission eigenvalue problem for the Schrödinger operator with a radial potential \(q\in W_2^1[0,1]\). Under the assumption that the mean value of the potential is zero and \(q(1)\ne 0\), the local solvability and stability are obtained by Buterin et al. (Inverse Probl 36:105002, 2020), whereas they need the condition that the difference of two spectral sequences of two problems, in the sense of \(l^2\)-norm, is small enough. In this paper, we shall extend their stability result by using a different method, without this condition. By using the theory of transformation operators and the properties of Riesz basis, we give the estimates for the difference of two potentials in the sense of the weak form and \(W_2^1\)-norm, in terms of the difference of two corresponding spectral data. As a corollary, we obtain that if only finite spectral data are close enough, then the corresponding potentials are also close in the sense of the weak form.

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Acknowledgements

The authors would like to thank the referees for the insightful comments and helpful suggestions. The author Xu and Guo was supported by the National Natural Science Foundation of China (11901304). The author Yang was supported in part by the National Natural Science Foundation of China (11871031) and by the Natural Science Foundation of the Jiangsu Province of China (BK 20201303).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, Jiangsu, People’s Republic of China

    Xiao-Chuan Xu & Yan Guo

  2. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, People’s Republic of China

    Chuan-Fu Yang

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  1. Xiao-Chuan Xu
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  2. Yan Guo
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Correspondence to Xiao-Chuan Xu.

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Xu, XC., Guo, Y. & Yang, CF. Stability of the inverse transmission eigenvalue problem for the Schrödinger operator with a radial potential. Lett Math Phys 112, 82 (2022). https://doi.org/10.1007/s11005-022-01577-4

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