On the spectrum of leaky surfaces with a potential bias

Pavel Exner

Spectral Theory and Operator Algebra mathscidoc:1910.43265

arXiv preprint arXiv:1701.06288, 2017.1
We discuss operators of the type H=-\Delta+ V (x)-lpha\delta (x-\Sigma) with an attractive interaction, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , in H=-\Delta+ V (x)-lpha\delta (x-\Sigma) , where H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is an infinite surface, asymptotically planar and smooth outside a compact, dividing the space into two regions, of which one is supposed to be convex, and H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is a potential bias being a positive constant H=-\Delta+ V (x)-lpha\delta (x-\Sigma) in one of the regions and zero in the other. We find the essential spectrum and ask about the existence of the discrete one with a particular attention to the critical case, H=-\Delta+ V (x)-lpha\delta (x-\Sigma) . We show that H=-\Delta+ V (x)-lpha\delta (x-\Sigma) is then empty if the bias is supported in theexterior'region, while in the opposite case isolated eigenvalues may exist.
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  title={On the spectrum of leaky surfaces with a potential bias},
  author={Pavel Exner},
  booktitle={arXiv preprint arXiv:1701.06288},
Pavel Exner. On the spectrum of leaky surfaces with a potential bias. 2017. In arXiv preprint arXiv:1701.06288. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020135230663304794.
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