  We present here a variational method for maximizing the bandgap in a
one-dimensional system where the potential is subject to given constraints. Two
specific examples are studied in detail. In the first, we show that if the
potential is constrained to lie between two values, the largest bandgap is
obtained by a mixture of the highest and lowest potential - an exact result
valid in any dimension. The second example fixes the first and second moments
of the potential and seeks to extremize the bandgap. An exact result is
obtained. Finally, we indicate how our techniques may be applied to photonic
bandgaps.
