Localization in fractal spaces: Exact results on the Sierpinski gasket

Localization due to space structure, rather than due to randomness, is investigated by studying the usual tight-binding model on the Sierpinski gasket. Some exact results are obtained from the decimation--renormalization-group method. It is surprising that there exist an infinite number of extended states on the Sierpinski gasket. This set of extended states forms a Cantor set. The rest of the states are exponentially localized except for two states that are localized in a power-law fashion. It can be shown that exponential localization of lengths of states on the Sierpinski gasket reveal a self-similar pattern.