Abstract 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 ) that 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 give 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 the target encounter: the minimal MFPT is lower than the Kramer 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.