Mathematics

This talk is concerned with several classes of Schrödinger operators with potentials that are unbounded below but their negative part is localised in narrow channels. They have the common property that they exhibit a parameter-dependent spectral transition: if the coupling constant exceeds a critical value, the spectrum will cover the whole real axis, corresponding to the particle escape to infinity. A prototype of such a behaviour can be found in the so-called Smilansky model. We review its properties and analyze a regular version of this model, as well as another system in which a x^p y^p potential is amended by a negative radially symmetric term; in the latter case the sub-critical spectrum is purely discrete. The results come from a common work with Diana Barseghyan, Vladimir Lotoreichik and Milos Tater.