@article{1751-8121-49-16-165302,
author = {Diana Barseghyan and Pavel Exner and Andrii Khrabustovskyi and Miloš Tater},
title = {Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition},
journal = {Journal of Physics A: Mathematical and Theoretical},
volume = {49},
number = {16},
pages = {165302},
url = {http://stacks.iop.org/1751-8121/49/i=16/a=165302},
year = {2016},
abstract = {We analyze two-dimensional Schrödinger operators with the potential ##IMG## [http://ej.iop.org/images/1751-8121/49/16/165302/jpaaa1550ieqn1.gif] {$| {xy}{| }^{p}-\lambda {({x}^{2}+{y}^{2})}^{p/(p+2)}$} where ##IMG## [http://ej.iop.org/images/1751-8121/49/16/165302/jpaaa1550ieqn2.gif] {$p\geqslant 1$} and ##IMG## [http://ej.iop.org/images/1751-8121/49/16/165302/jpaaa1550ieqn3.gif] {$\lambda \geqslant 0$} which exhibit an abrupt change of spectral properties at a critical value of the coupling constant λ . We show that in the supercritical case the spectrum covers the whole real axis. In contrast, for λ below the critical value the spectrum is purely discrete and we establish a Lieb–Thirring-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.}
}