**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 modication 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.