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07 February 2025
 
  » arxiv » 1109.0168

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Spectral estimates for a class of Schr"odinger operators with infinite phase space and potential unbounded from below
Pavel Exner ; Diana Barseghyan ;
Date 1 Sep 2011
AbstractWe analyze two-dimensional Schr"odinger operators with the potential $|xy|^p - lambda (x^2+y^2)^{p/(p+2)}$ where $pge 1$ and $lambdage 0$. We show that there is a critical value of $lambda$ such that the spectrum for $lambda<lambda_mathrm{crit}$ is below bounded and purely discrete, while for $lambda>lambda_mathrm{crit}$ it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.
Source arXiv, 1109.0168
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