Abstract:
In a recent collaboration with F.Nier and A.Faraj, it has been shown that
a simple modifcation of the Laplacian through artificial interface
conditions allows an alternative approach to the adiabatic evolution
of quantum resonances.
The use of this modifed framework, may hopefully provide with
effective equations for the non-linear dynamics of Schrödinger-Poisson
systems in the regime of quantum wells in a semiclassical island.
In this perspective, it is important to control the deformations
effects introduced on the spectrum and on the time propagator by such
interface conditions.
In particular we are interested in uniform-in-time estimates of the perturbed
semigroup. The main difficulty is due to the non-selfadjont character of our
class of operators, involving a lack of accretivity for the
corresponding generator of the quantum dynamics. In this framework, a
standard approach would only provide with finite-time estimates for
the dynamical system. An alternative startegy consists in constructing
intertwining operators leading to a dynamical comparison between the
modified non-selfadjoint model and the corresponding `physical'
Hamiltonian. |