Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition. (8th March 2016)

• Record Type: Journal Article
• Authors: Barseghyan, Diana; Exner, Pavel; Khrabustovskyi, Andrii; Tater, Miloš
• Is Part Of: Journal of physics. Volume 49:Number 16(2016)
• Publication Date: 2016-03-08
• Subjects: Schrödinger operator -- eigenvalue estimates -- spectral transition; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/49/16/165302
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: We analyze two-dimensional Schrödinger operators with the potential where and 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.