Spectral estimates for a class of Schrödinger operators with infinite phase space and potential unbounded from below

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 $\lambda \ge 0$. We show that there is a critical value of $\lambda$ such that the spectrum for $\lambda <{\lambda }_{\mathrm{crit}}$ is below bounded and purely discrete, while for $\lambda >{\lambda }_{\mathrm{crit}}$ it is unbounded from below. In the subcritical case we prove upper and lower bounds for the eigenvalue sums.

LaTeX, 16 pages with on ps figure

References in corpus (1)

• Sharp spectral estimates in domains of infinite volume

Cited by in corpus (5)

• A regular version of Smilansky model
• A magnetic version of the Smilansky-Solomyak model
• Higher dimensional non standard eigenvalue asymptotics
• Spectral analysis of a class of Schroedinger operators exhibiting a parameter-dependent spectral transition
• Bound states for logarithmic Schrodinger equations with potentials unbounded below