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Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer

Pavel Exner Vladimir Lotoreichik

Spectral Theory and Operator Algebra mathscidoc:1910.43294

arXiv preprint arXiv:1805.12448, 2020.5
We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian \mathsf {H} on an unbounded, radially symmetric (generalized) parabolic layer \mathsf {H} . It was known before that \mathsf {H} has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for \mathsf {H} by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrdinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer \mathsf {H} at infinity.
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@inproceedings{pavel2020spectral,
  title={Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer},
  author={Pavel Exner, and Vladimir Lotoreichik},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020140134187835823},
  booktitle={arXiv preprint arXiv:1805.12448},
  year={2020},
}
Pavel Exner, and Vladimir Lotoreichik. Spectral asymptotics of the Dirichlet Laplacian on a generalized parabolic layer. 2020. In arXiv preprint arXiv:1805.12448. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020140134187835823.
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