Minimal spacing of eigenvalues on fractals

Abstract

On the unit interval (I) and the Sierpinski gasket (

), the spectral decimation function of the Laplacian has similar properties that result in a positive minimum spacing of eigenvalues. Other fractals, for example, the level-3

S G $SG$ upper S upper G

,

S G 3 $SG_3$ upper S upper G 3

, may not enjoy these properties. Our goal is to obtain a simple sufficient criterion for positive infimum spacing between eigenvalues in the spectrum, based on the properties of the corresponding spectral decimation function of the fractal. We also give a sufficient condition for zero infimum spacing between eigenvalues.