Wave operators, similarity and dynamics for a class of Schr?dinger operators with generic non-mixed interface conditions in 1D

摘要

We consider a simple modification of the 1D-Laplacian where non-mixed interface conditions occur at the boundaries of a finite interval. It has recently been shown that Schr?dinger 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 non-selfadjoint perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary wave operators allowing to intertwine the modified non-selfadjoint Schr?dinger operator with a "physical" Hamiltonian. For small values of a deformation parameter "θ," this yields a dynamical comparison between the two models showing that the distance between the corresponding semigroups is dominated by |θ| uniformly in time in the L2-operator norm.