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arXiv · 1511.00097

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

Diana Barseghyan·Pavel Exner·Andrii Khrabustovskyi·Milos Tater
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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 $\lambda\ge 0$, which exhibit an abrupt change of its spectral properties at a critical value of the coupling constant $\lambda$. We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for $\lambda$ below the critical value the spectrum is purely discrete and we establish a Lieb-Thirring-type bound on its moments. In the critical case the essential spectrum covers the positive halfline while the negative spectrum can be only discrete, we demonstrate numerically the existence of a ground state eigenvalue.

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Diana Barseghyan, Pavel Exner, Andrii Khrabustovskyi, Milos Tater. 2015-10-31. Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition. https://doi.org/10.1088/1751-8113%2F49%2F16%2F165302

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