Abstract: In the model suggested by Smilansky one studies an operator describ- ing the interaction between a quantum graph and a nite system of one- dimensional oscillators. We discuss a modi cation of a Smilansky model in which a singular potential is replaced by a regular, below unbounded poten- tial. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and de- termine the critical value of this constant. This is a common work with Pavel Exner.