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A bound for the eigenvalue counting function for higher-order Krein Laplacians on open sets

Fritz Gesztesy Pavel Exner S Sukhtaiev A Laptev

Spectral Theory and Operator Algebra mathscidoc:1910.43219

3-29, 2015
For an arbitrary nonempty, open set <sup><i>n</i></sup>, <i>n</i> of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian (-\Delta)^m|_{{C^\infty_0}(\Omega)}, <i>m</i> , and its Kreinvon Neumann extension <i>A</i><sub><i>K</i>,,<i>m</i></sub> in <i>L</i><sup>2</sup>(). with <i>N</i>(, <i>A</i><sub><i>K</i>,,<i>m</i>)</sub>, (-\Delta)^m|_{{C^\infty_0}(\Omega)}, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of <i>A</i><sub><i>K</i>,,<i>m</i></sub>, we derive the bound <i>N</i>(, <i>A</i><sub><i>K</i>,,<i>m</i></sub>) (2<i></i>)<sup>-<i>n</i></sup><i></i><sub>n</sub>||{1 + [2<i>m</i>/(2<i>m</i> + <i>n</i>)]}<sup><i>n</i>/(2<i>m</i>)</sup><sup><i>n</i>/(2<i>m</i>)</sup>, &gt; 0, where <i><sub>n</sub></i> <i></i><sup><i>n</i>/2</sup> /((<i>n</i> + 2)/2) denotes the (Euclidean) volume of the unit ball in <sup><i>n</i></sup>.
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@inproceedings{fritz2015a,
  title={A bound for the eigenvalue counting function for higher-order Krein Laplacians on open sets},
  author={Fritz Gesztesy, Pavel Exner, S Sukhtaiev, and A Laptev},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020133705131784748},
  pages={3-29},
  year={2015},
}
Fritz Gesztesy, Pavel Exner, S Sukhtaiev, and A Laptev. A bound for the eigenvalue counting function for higher-order Krein Laplacians on open sets. 2015. pp.3-29. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020133705131784748.
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