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

The e ffects of the local accumulation of charges in resonant tunnelling heterostructures have been described using 1D Shrödinger–Poisson Hamiltonians in the asymptotic regime of quantum wells. Taking into account the features of the underling physical system, the corresponding linearized model is naturally related to the adiabatic evolution of shape resonances on a time scale which is exponentially large w.r.t. the asymptotic parameter $h$. A possible strategy to investigate this problem consists of using a complex dilation to identify the resonances with the eigenvalues of a deformed operator. Then, the adiabatic evolution problem for a sheet-density of charges can be reformulated using the deformed dynamical system which, under suitable initial conditions, is expected to evolve following the instantaneous resonant states.
After recalling the main technical di culties related to this approach, we introduce a modi ed model where $h$-dependent arti cial interface conditions, occurring at the boundary of the interaction region, allow one to obtain adiabatic approximations for the relevant resonant states, while producing a small perturbation of the dynamics on the scale $h^{N_0}$. According to these results, we nally suggest an alternative formulation of the adiabatic problem. An a posteriori justi cation of our method is obtained by considering an explicitly-solvable case.