1210.6475 (Andrea Mantile)

Wave operators, similarity and dynamics for a class of Schroedinger
operators with generic non-mixed interface conditions in 1D    [PDF]

Andrea Mantile

We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occurs at the boundaries of a finite interval. It has recently been shown that Schr\"odinger operators having this form allow a new approach to the transverse quantum transport through resonant heterostructures. In this perspective, it is important to control the deformations effects introduced on the spectrum and on the time propagator by this class of perturbations. 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. Our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schroedinger operator with a corresponding 'physical' Hamiltonian. For small values of a deformation parameter '\theta', this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |\theta| uniformly in time in the L^2-operator norm. When a suitable energy constraint condition is assumed for the initial states, the above analysis adapts to the modelling of 1D quantum systems in the regime of quantum wells in a semiclassical island. In this framework, we show that artificial interface conditions introduce perturbations on the dynamics which are controlled by the ratio |\theta|/h^2, being h>0 a parameter fixing the 'quantum scale' of the system.

View original: http://arxiv.org/abs/1210.6475