Upper bound on the energy gap in terms of the localization in tight-binding systems

We present an upper bound on the energy gap above the ground energy in terms of the variance of the particle position in one-particle tight-binding systems. Our inequality gives a complementary relationship between the energy gap and the localization of the particle; that is, the energy gap narrows as the ground state spreads spatially, while the energy gap can broaden as the ground state becomes localized. We first consider the ground state in one-dimensional tight-binding systems and relate the energy gap to the variance of the particle position in the ground state. We then extend the result to higher-dimensional systems. We compare our upper bound with the exact energy gap as calculated numerically in one and two dimensions; this comparison confirms that our upper bound places a tight restriction on the energy gap.