A quantum evolution problem in the regime of quantum wells in a semiclassical island with artificial interface conditions

We introduce a modified Schr\"odinger operator where the semiclassical Laplacian is perturbed by artificial interface conditions occurring at the boundaries of the potential's support. The corresponding dynamics is analysed in the regime of quantum wells in a semiclassical island. Under a suitable energy constraint for the initial states, we show that the time propagator is stable w.r.t. the non-selfadjont perturbation, provided that this is parametrized through infinitesimal functions of the semiclassical parameter 'h'. It has been recently shown that h-dependent artificial interface conditions allow a new approach to the adiabatic evolution problem for the shape resonances in models of resonant heterostructures. Our aim is to provide with a rigorous justification of this method.