  This paper is concerned with a linearized version of the transport problem
where the Schr\"{o}dinger-Poisson operator is replaced by a non-autonomous
Hamiltonian, slowly varying in time. We consider an explicitly solvable model
where a semiclassical island is described by a flat potential barrier, while a
time dependent 'delta' interaction is used as a model for a single quantum
well. Introducing, in addition to the complex deformation, a further
modification formed by artificial interface conditions, we give a reduced
equation for the adiabatic evolution of the sheet density of charges
accumulating around the interaction point.
