Fractal Weyl law for open quantum chaotic maps. (English) Zbl 1293.81022

The purpose of the article under review is the study of the semiclassical quantizations of open hyperbolic maps arising in chaotic scattering problems. More precisely, the general aim of the article is to study the spectral properties of \(\hbar\)-Fourier integral operators quantizing open hyperbolic (symplectic) maps. Such operators (or their projection on a finite dimensional space) are called by the authors hyperbolic open quantum maps, and holomorphic families of such maps are called hyperbolic quantum monodromy operators. The main result of the article (Theorem 4) gives a fractal Weyl upper bound on the number of “resonances” for hyperbolic quantum monodromy operators in terms of the Minkowski dimension of the trapped set.
As an application of their general study of these hyperbolic open quantum maps, the authors derive a fractal Weyl upper bound in the case of scattering by several convex obstacles. Precisely, consider \(\mathcal{O}=\bigcup_{j=1}^J\mathcal{O}_j\) the union of open, bounded strictly convex subsets of \(\mathbb{R}^n\) with smooth boundaries and satisfying Ikawa condition \[ \overline{\mathcal{O}}_k\cap\text{convex hull}(\overline{\mathcal{O}}_l\cup \overline{\mathcal{O}}_j)=\emptyset,\;\;j\neq k\neq l. \] The classical Hamiltonian flow associated to this geometric situation is defined by free motion away from obstacles and normal reflections on the obstacles. The information that will be relevant in the study of scattering resonances will be contained in the geometry of the trapped set \(K\) which consists of points that do not escape at infinity when \(t\rightarrow\pm\infty\). In this geometric context, the open quantum maps will be associated to the billiard map on \(B^*\partial\mathcal{O}\) constructed from the classical Hamiltonian flow.
One is then interested in counting the number of scattering resonances, i.e., counting with their multiplicity the poles of the meromorphic continuation of \[ R(\lambda):=(-\Delta-\lambda^2)^{-1}:L^2_{\mathrm{comp}}(\mathbb{R}^n\backslash\mathcal{O})\longrightarrow L^2_{\mathrm{loc}}(\mathbb{R}^n\backslash\mathcal{O}). \] Under the above geometric assumptions, the authors deduce from their main result the following fractal Weyl upper bound for any fixed \(\alpha>0\): \[ \sum_{-\alpha<\text{Im} \lambda, r\leq|\lambda|\leq r+1}m_R(\lambda)=\mathcal{O}(r^{\mu+0}),\;\text{as}\;r\rightarrow+\infty, \] where \(2\mu+1\) is the box dimension of the trapped set, and \(m_R(\lambda)\) is the multiplicity of a (nonzero) resonance, i.e., \[ m_R(\lambda)=\text{rank}\oint_{\gamma}R(\zeta)d\zeta,\;\gamma:t\mapsto \lambda+\epsilon e^{2i\pi t},\;0<\epsilon\ll 1. \] This kind of upper bound was predicted by the second author in [Duke Math. J. 60, No. 1, 1–57 (1990; Zbl 0702.35188)] where similar bounds were obtained for families of semiclassical Schrödinger operators satisfying some analyticity assumptions.


81Q50 Quantum chaos
35P25 Scattering theory for PDEs
35S30 Fourier integral operators applied to PDEs
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
81S22 Open systems, reduced dynamics, master equations, decoherence
81U05 \(2\)-body potential quantum scattering theory
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
35B34 Resonance in context of PDEs
58J40 Pseudodifferential and Fourier integral operators on manifolds


Zbl 0702.35188
Full Text: DOI arXiv


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