Smilansky model, Schrödinger operators. spectral transition

We discuss a model which modifies the system studied first by Smilansky. In our case a singular potential ?channel? is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential ?channels?.