A regular analogue of the Smilansky model: spectral properties

Diana Barseghyan Pavel Exner

Spectral Theory and Operator Algebra mathscidoc:1910.43229

Reports on Mathematical Physics, 80, (2), 177-192, 2017.10
We analyze spectral properties of the operator <i>H</i> = <i></i><sup>2</sup>/<i>x</i><sup>2</sup> <i></i><sup>2</sup>/<i>y</i><sup>2</sup> + <i></i><sup>2</sup><i>y</i><sup>2</sup> <i>y</i><sup>2</sup><i>V</i>(<i>xy</i>) in <i>L</i><sup>2</sup>(<sup>2</sup>), where <i></i> 0 and <i>V </i> 0 is a compactly supported and sufficiently regular potential. It is known that the spectrum of <i>H</i> depends on the one-dimensional Schrdinger operator <i>L</i> = <i>d</i><sup>2</sup>/<i>dx</i><sup>2</sup> + <i></i><sup>2</sup> <i>V</i>(<i>x</i>) and it changes substantially as inf(<i>L</i>) switches sign. We prove that in the critical case, inf(<i>L</i>) = 0, the spectrum of <i>H</i> is purely essential and covers the interval [0, ). In the subcritical case, inf <i></i>(<i>L</i>) &gt; 0, the essential spectrum starts from <i></i> and there is a nonvoid discrete spectrum in the interval [0, <i></i>). We also derive a bound on the corresponding eigenvalue moments.
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  title={A regular analogue of the Smilansky model: spectral properties},
  author={Diana Barseghyan, and Pavel Exner},
  booktitle={Reports on Mathematical Physics},
Diana Barseghyan, and Pavel Exner. A regular analogue of the Smilansky model: spectral properties. 2017. Vol. 80. In Reports on Mathematical Physics. pp.177-192. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020134008357758758.
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