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

Abstract

We analyze two-dimensional Schrödinger 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.

20 pages, 4 figures

References in corpus (2)

• A regular version of Smilansky model
• Higher dimensional non standard eigenvalue asymptotics

Cited by in corpus (3)

• A magnetic version of the Smilansky-Solomyak model
• Spectral and resonance properties of Smilansky Hamiltonian
• A regular analogue of the Smilansky model: spectral properties