  During a random search, resetting the searcher's position from time to time
to the starting point often reduces the mean completion time of the process.
Although many different resetting models have been studied over the past ten
years, only a few can be physically implemented. Here we study theoretically a
protocol that can be realised experimentally and which exhibits unusual
optimization properties. A Brownian particle is subject to an arbitrary
confining potential $v(x)$ which is switched on and off intermittently at fixed
rates. Motion is constrained between an absorbing wall located at the origin
and a reflective wall. When the walls are sufficiently far apart, the interplay
between free diffusion during the "off" phases and attraction toward the
potential minimum during the "on" phases gives rise to rich behaviours, not
observed in ideal resetting models. For potentials of the form
$v(x)=k|x-x_0|^n/n$, with $n>0$, the switch-on and switch-off rates that
minimise the mean first passage time (MFPT) to the origin undergo a continuous
phase transition as the potential stiffness $k$ is varied. When $k$ is above a
critical value $k_c$, potential intermittency enhances target encounter: the
minimal MFPT is lower than the Kramer's time and is attained for a
non-vanishing pair of switching rates. We focus on the harmonic case $n=2$,
extending previous results for the piecewise linear potential ($n=1$) in
unbounded domains. We also study the non-equilibrium stationary states emerging
in this process.
