  At least 25% of white dwarfs show atmospheric pollution by metals, sometimes
accompanied by detectable circumstellar dust/gas discs or (in the case of WD
1145+017) transiting disintegrating asteroids. Delivery of planetesimals to the
white dwarf by orbiting planets is a leading candidate to explain these
phenomena. Here, we study systems of planets and planetesimals undergoing
planet-planet scattering triggered by the star's post-main sequence mass loss,
and test whether this can maintain high rates of delivery over the several Gyr
that they are observed. We find that low-mass planets (Earth to Neptune mass)
are efficient deliverers of material and can maintain the delivery for Gyr.
Unstable low-mass planetary systems reproduce the observed delayed onset of
significant accretion, as well as the slow decay in accretion rates at late
times. Higher-mass planets are less efficient, and the delivery only lasts a
relatively brief time before the planetesimal populations are cleared. The
orbital inclinations of bodies as they cross the white dwarf's Roche limit are
roughly isotropic, implying that significant collisional interactions of
asteroids, debris streams and discs can be expected. If planet-planet
scattering is indeed responsible for the pollution of white dwarfs, many such
objects, and their main-sequence progenitors, can be expected to host
(currently undetectable) super-Earth planets on orbits of several au and
beyond.
