Spectral analysis of a class of Schrödinger operators exhibiting a parameterdependent spectral transition
Abstract
We analyze twodimensional Schrödinger operators with the potential  {xy}{ }^{p}λ {({x}^{2}+{y}^{2})}^{p/(p+2)} where p≥slant 1 and λ ≥slant 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 LiebThirringtype 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 groundstate eigenvalue.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 April 2016
 DOI:
 10.1088/17518113/49/16/165302
 arXiv:
 arXiv:1511.00097
 Bibcode:
 2016JPhA...49p5302B
 Keywords:

 Mathematical Physics;
 Mathematics  Spectral Theory;
 Quantum Physics;
 81Q10;
 35J10;
 35P15;
 35Q40
 EPrint:
 20 pages, 4 figures