Resonances for Open Quantum Maps and a Fractal Uncertainty Principle

Abstract

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, 1 2 − 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. 10 {0, 2, . . . 8} 12

Mathematical study of scattering resonances

“…Some lower bounds in strips have been obtained by Jakobson and Naud [146] but they are far from the upper bounds. Dyatlov-Jin [74] proved an analogue of Theorem 14 for open quantum maps of the form shown in Fig. 23.…”

“…In the example presented there the numerically computed best exponent in FUP (3.43) is sharp. Some other examples in [74] show however that it is not always the case. We will return to open quantum maps in Sect.…”

“…The fractal uncertainty principle (FUP) states that a wave function cannot be strongly localized in both position and frequency near a fractal set. To describe FUP rigorously we will use a simpler model studied recently by Dyatlov and Jin [74]: open quantum maps with products of exact Cantor sets as trapped sets.…”

“…The general principle of [75] and [74] for quantization of κ M,A goes as follows. In the notation of (3.38) and (3.39) we say that …”

“…10 and 26). Fractal Weyl laws have been considered (and numerically checked) for various open chaotic quantum maps-see [74,199] and references given there. The have also been proposed in other types of chaotic systems, ranging from dielectric cavities to communication and social networks-see recent review articles by CaoWiersig [42] and Ermann-Frahm-Shepelyansky [83] respectively.…”