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arXiv · 1109.0168

Spectral estimates for a class of Schr\"odinger operators with infinite phase space and potential unbounded from below

Pavel Exner·Diana Barseghyan
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Abstract

We analyze two-dimensional Schr\"odinger operators with the potential $|xy|^p - \lambda (x^2+y^2)^{p/(p+2)}$ where $p\ge 1$ and $\lambda\ge 0$. 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.

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Pavel Exner, Diana Barseghyan. 2011-09-01. Spectral estimates for a class of Schr\"odinger operators with infinite phase space and potential unbounded from below. https://doi.org/10.1088/1751-8113/45/7/075204

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