  The dynamics of a quantum particle bound by an accelerating delta-functional
potential is investigated. Three cases are considered, using the reference
frame moving along with the {\delta}-function, in which the acceleration is
converted into the additional linear potential. (i) A stationary regime, which
corresponds to a resonance state, with a minimum degree of delocalization,
supported by the accelerating potential trap. (ii) A pulling scenario: an
initially bound particle follows the accelerating delta-functional trap, within
a finite time. (iii) The pushing scenario: the particle, which was initially
localized to the right of the repulsive delta-function, is shoved to the right
by the accelerating potential. For the two latter scenarios, the life time of
the trapped particle, and the largest velocity to which it can be accelerated
while staying trapped, are found. Analytical approximations are developed for
the cases of small and large accelerations in the pulling regime, and also for
a small acceleration in the stationary situation, and in the regime of pushing.
The same regimes may be realized by Airy-like planar optical beams guided by a
narrow bending potential channel or crest. Physical estimates are given for an
atom steered by a stylus of a scanning tunneling microscope (STM), and for the
optical beam guided by a bending stripe.
