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 $\delta \in (0,1)$. We show that the size of the spectral gap is strictly greater than the standard bound $\max (0,\frac{1}{2}-\delta )$ for all values of $\delta$, 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.

53 pages, 10 figures, 2 tables. Simplified the proof of Lemma 2.2 and revised according to referee's comments. To appear in Communications in Mathematical Physics

References in corpus (9)

• Fractal Weyl laws for chaotic open systems
• Spectral gaps without the pressure condition
• Semiclassical structure of chaotic resonance eigenfunctions
• Resonances in open quantum maps
• Weyl asymptotics: From closed to open systems
• On the resonance eigenstates of an open quantum baker map
• Short periodic orbits theory for partially open quantum maps
• Decoupling and near-optimal restriction estimates for Cantor sets
• Resonant eigenstates in quantum chaotic scattering

Cited by in corpus (15)

• Spectral gaps without the pressure condition
• Fourier dimension and spectral gaps for hyperbolic surfaces
• An introduction to fractal uncertainty principle
• A higher dimensional Bourgain-Dyatlov fractal uncertainty principle
• Dolgopyat's method and the fractal uncertainty principle
• The role of short periodic orbits in quantum maps with continuous openings
• Resonance states of the three-disk scattering system
• Spectral gap for obstacle scattering in dimension 2
• Fractal Uncertainty Principle with Explicit Exponent
• Mathematical Study of Scattering Resonances
• Fractal Weyl laws and wave decay for general trapping
• Fractal uncertainty for discrete 2D Cantor sets
• The fractal uncertainty principle via Dolgopyat's method in higher dimensions
• Fractal uncertainty principle for discrete Cantor sets with random alphabets
• Eigenstates and spectral projection for quantized baker's map