Spectral analysis of a class of Schrdinger operators exhibiting a parameter-dependent spectral transition

Diana Barseghyan Pavel Exner Andrii Khrabustovskyi Milo Tater

Spectral Theory and Operator Algebra mathscidoc:1910.43216

Journal of Physics A: Mathematical and Theoretical, 49, (16), 165302, 2016.3
We analyze two-dimensional Schrdinger operators with the potential |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} where |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} and |{xy}{|}^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+ 2)} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for below the critical value the spectrum is purely discrete and we establish a LiebThirring-type bound on its moments. In the critical case where the essential spectrum covers the positive halfline while the negative spectrum can only be discrete, we demonstrate numerically the existence of a ground-state eigenvalue.
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@inproceedings{diana2016spectral,
  title={Spectral analysis of a class of Schrdinger operators exhibiting a parameter-dependent spectral transition},
  author={Diana Barseghyan, Pavel Exner, Andrii Khrabustovskyi, and Milo Tater},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020133614379016745},
  booktitle={Journal of Physics A: Mathematical and Theoretical},
  volume={49},
  number={16},
  pages={165302},
  year={2016},
}
Diana Barseghyan, Pavel Exner, Andrii Khrabustovskyi, and Milo Tater. Spectral analysis of a class of Schrdinger operators exhibiting a parameter-dependent spectral transition. 2016. Vol. 49. In Journal of Physics A: Mathematical and Theoretical. pp.165302. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020133614379016745.
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