  We develop an approach on how to define single-point interactions under the
application of external fields. The essential feature relies on an asymptotic
method based on the one-point approximation of multi-layered heterostructures
that are subject to bias potentials. In this approach , the zero-thickness
limit of the transmission matrices of specific structures is analyzed and shown
to result in matrices connecting the two-sided boundary conditions of the wave
function at the origin. The reflection and transmission amplitudes are computed
in terms of these matrix elements as well as biased data. Several one-point
interaction models of two- and three-terminal devices are elaborated. The
typical transistor in the semiconductor physics is modeled in the "squeezed
limit'' as a $\delta$- and a $\delta'$-potentialand referred to as a "point"
transistor. The basic property of these one-point interaction models is the
existence of several extremely sharp peaks as an applied voltage tunes, at
which the transmission amplitude is non-zero, while beyond these resonance
values, the heterostructure behaves as a fully reflecting wall. The location of
these peaks referred to as a "resonance set" is shown to depend on both system
parameters and applied voltages. An interesting effect of resonant transmission
through a $\delta$-like barrier under the presence of an adjacent well is
observed. This transmission occurs at a countable set of the well depth values.
