  Linear programming (polynomial) techniques are used to obtain lower and upper
bounds for the potential energy of spherical designs. This approach gives
unified bounds that are valid for a large class of potential functions. Our
lower bounds are optimal for absolutely monotone potentials in the sense that
for the linear programming technique they cannot be improved by using
polynomials of the same or lower degree. When additional information about the
structure (upper and lower bounds for the inner products) of the designs is
known, improvements on the bounds are obtained. Furthermore, we provide `test
functions' for determining when the linear programming lower bounds for energy
can be improved utilizing higher degree polynomials. We also provide some
asymptotic results for these energy bounds.
