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  • Cited by 5
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Sun, Huaqing Kong, Qingkai and Shi, Yuming 2016. Essential spectrum of singular discrete linear Hamiltonian systems. Mathematische Nachrichten, Vol. 289, Issue. 2-3, p. 343.


    Sun, Huaqing and Shi, Yuming 2015. On essential spectra of singular linear Hamiltonian systems. Linear Algebra and its Applications, Vol. 469, p. 204.


    Yao, Siqin Sun, Jiong and Zettl, Anton 2013. Self-adjoint domains, symplectic geometry, and limit-circle solutions. Journal of Mathematical Analysis and Applications, Vol. 397, Issue. 2, p. 644.


    Hao, Xiaoling Sun, Jiong and Zettl, Anton 2012. The spectrum of differential operators and square-integrable solutions. Journal of Functional Analysis, Vol. 262, Issue. 4, p. 1630.


    Zheng, Zhaowen and Zhang, Wenju 2012. Characterization of Eigenvalues in Spectral Gap for Singular Differential Operators. Abstract and Applied Analysis, Vol. 2012, p. 1.


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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 141, Issue 2
  • April 2011, pp. 417-430

On an open problem of Weidmann: essential spectra and square-integrable solutions

  • Jiangang Qi (a1) and Shaozhu Chen (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210509001681
  • Published online: 04 April 2011
Abstract

In 1987, Weidmann proved that, for a symmetric differential operator τ and a real λ, if there exist fewer square-integrable solutions of (τ−λ)y = 0 than needed and if there is a self-adjoint extension of τ such that λ is not its eigenvalue, then λ belongs to the essential spectrum of τ. However, he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed. In this paper, we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum. We then establish a formula of the numbers of linearly independent square-integrable solutions on the left and the right subintervals, and on the entire interval for nth-order differential operators. The latter formula ascertains the above conjecture. These results are crucial in criteria of essential spectra in terms of the numbers of square-integrable solutions for real values of the spectral parameter.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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