Counter-examples to the fractal Weyl law for semiclassical resonances

2025/12/03 by Jean‐François Bony, Bony, Jean-Francois, Setsuro Fujiié +5
Mathematics · Physics and Astronomy · #35B34 #35J10 #35P20 #37C29 #81Q20 #81U24 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Quantum chaos and dynamical systems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2512.03773

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

Under general assumptions, the numbers of semiclassical resonances is known to be bounded from above by a negative power of h which is given by the fractal dimension of the trapped set. In this paper we provide examples of operators with much less resonances, showing that these upper bounds are not always sharp.

Citations

• Resonances for homoclinic trapped sets
• Distribution of resonances for hyperbolic surfaces
• Fractal Weyl law for open quantum chaotic maps
• Non-Weyl Resonance Asymptotics for Quantum Graphs
• Microlocal kernel of pseudodifferential operators at an hyperbolic fixed point
• Fractal upper bounds on the density of semiclassical resonances

Related

• Fractal upper bounds on the density of semiclassical resonances
• Fractal Weyl law for open quantum chaotic maps
• Fractal Weyl Laws for Chaotic Open Systems
• Weyl law for semi-classical resonances with randomly perturbed potentials
• Semiclassical estimates of the cut-off resolvent for trapping perturbations