Quantum systems exhibiting parameter-dependent spectral transitions

This talk is concerned with several examples of nonrelativistic quantum sys- tems with potentials which are below unbounded but their negative part is localized 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, corre- sponding to the particle escape to infinity. A prototype of such a behavior 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 xpyp potential is amended by a negative radially symmetric term; in the latter case the subcritical spectrum is purely discrete. The results come from a common work with Diana Barseghyan, Vladimir Lotoreichik and Milo?s Tater.

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