Resonances for Open Quantum Maps and a Fractal Uncertainty Principle

Abstract

© 2017, Springer-Verlag Berlin Heidelberg. We study eigenvalues of quantum open baker’s maps with trapped sets given by linear arithmetic Cantor sets of dimensions δ∈ (0 , 1). We show that the size of the spectral gap is strictly greater than the standard bound max(0,12-δ) for all values of δ, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.