Bandgap Extremization: Some Exact Results

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.