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

Abstract

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 Schrödinger 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-selfadjont perturbations. In order to obtain uniform-in-time estimates of the perturbed semigroup, our strategy consists in constructing stationary waves operators allowing to intertwine the modified non-selfadjoint Schrödinger 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 L 2 -operator norm.

Non-autonomous quantum systems with scale-dependent interface conditions

“…in [2], [7], [3]). Using specific spectral assumptions, a small-θ expansion for the modified dynamical system e −itH h θ , similar to the one given in [9], has been obtained in this case (see Theorem 4.4 in [10]); nevertheless, when both h and θ are small, the resulting error term is small uniformly in time only for initial states belonging to an appropriate subspace with prescribed energy conditions. This prevents to extend the result to the most relevant case of time dependent potentials.…”

“…In the case h = 1, it has been shown that the interface conditions (1.1) do not modify the spectrum provided that θ is small (see [9,Proposition 2.6]). In the present case, the dilation: y = (x − (b + a) /2) /h transforms the boundary conditions (1.1) into…”

“…A detailed analysis of their properties have been given in [13] for generic L 1 -potentials, while the particular case of a potential barrier is explicitly considered in [9] for h = 1. The h-dependent case can be considered as a rescaled problem and the result presented in [9] rephrase as follows…”

A linearized model of quantum transport in the asymptotic regime of quantum wells

“…In this connection, H h > 0, it has been shown that H h θ has a purely continuous spectrum coinciding with R + , provided that θ is small (depending on h; see [14]). A detailed analysis concerned with operators of this class has been developed (in a slightly more general framework) in [8]- [15].…”

“…The generalized eigenfunctions of H h θ , next denoted with ψ h θ (·, k, V h ), solve of the boundary value problem (we refer to [14]- [15])…”