A lattice potential investigation of quark mass and volume dependence of the $\Upsilon$ spectrum

We investigate bottomonia splittings by solving a Schrodinger-Pauli-type equation with parametrisations of QCD potentials around those that have been determined previously in lattice simulations. This is done both, in the continuum and on finite lattices with resolutions ranging from a=0.2 fm down to a=0.025 fm and extent of up to 12 fm or 144^3 lattice points. We find a strong dependence of some splittings, in particular the 2S-1S and 1P-1S splittings, on both the quark mass and the short range form of the static potential in the neighbourhood of the bottom quark mass, while splittings such as 3S-2S and 2P-2S show reduced dependence on the short distance potential. We conclude that the quenched quarkonium spectrum cannot be matched to experiment with a simple redefinition of the lattice spacing. We investigate the size of relativistic corrections as a function of the quark mass. Finite size effects are shown to die out rather rapidly as the volume is increased, and we demonstrate the restoration of rotational symmetry as the continuum limit is taken.