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Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations

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Abstract

This paper is concerned with approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations with at least one endpoint in the limit point case. A sufficient condition is firstly given for that the k-th eigenvalue of a self-adjoint subspace (relation) below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of self-adjoint subspaces. Then, by applying it to singular second-order symmetric linear difference equations, the approximation of eigenvalues below the essential spectra is obtained, i.e., for any given self-adjoint subspace extension of the corresponding minimal subspace, its k-th eigenvalue below its essential spectrum is exactly the limit of the k-th eigenvalues of a sequence of constructed induced regular self-adjoint subspace extensions.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11571202) and the China Scholarship Council (Grant No. 201406220019).

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  1. Department of Mathematics, Shandong University, Jinan, 250100, China

    Yan Liu & YuMing Shi

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  1. Yan Liu
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  2. YuMing Shi
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Correspondence to YuMing Shi.

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Liu, Y., Shi, Y. Approximation of eigenvalues below the essential spectra of singular second-order symmetric linear difference equations. Sci. China Math. 60, 1661–1678 (2017). https://doi.org/10.1007/s11425-016-0223-9

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