Fractal Weyl Laws for Chaotic Open Systems

Abstract

We present a conjecture relating the density of quantum resonances for an open chaotic system to the fractal dimension of the associated classical repeller. Mathematical arguments justifying this conjecture are discussed. Numerical evidence based on computation of resonances of systems of n disks on a plane are presented supporting this conjecture. The result generalizes the Weyl law for the density of states of a closed system to chaotic open systems.PACS numbers: 05.45. Mt, 03.65.Sq, 05.45.Ac, 31.15.Gy, 95.10.Fh The celebrated Weyl law concerning the density of eigenvalues of bound states is a central result in the spectroscopy of quantum systems [1]. The Weyl formula states that the asymptotic level number N (k), defined as the number of levels with k n < k (where k → ∞) is given after smoothing by.., for a quantum system bounded in a region R of D-dimensional space whose volume is V . For closed systems with smooth boundaries, the Weyl formula is well-established, and although primarily valid in the semi-classical limit, nevertheless can be applied with astonishing accuracy to very low energies extending almost down to the ground state of integrable and chaotic closed systems. Generalizations of the Weyl law to other situations have long been sought. A notable example is the conjecture by Berry [2] for the density of states of closed systems with fractal boundaries, i.e. "fractal drums".Open systems are characterized by resonances defined by complex wavevectork n = Re(k n ) + iIm(k n ), corresponding to states with finite life times arising from escape to infinity. Open chaotic systems, which occur in a variety of physical situations, are generically characterized by a classical phase space repeller that is fractal. In this letter we present a conjecture relating the density of resonances for an open chaotic system to the fractal dimension of the associated classical repeller. It can be stated as:where d H is the partial Hausdorff dimension of the repeller [3, §4.4]. This relation generalizes the Weyl law for the density of states of a closed system to chaotic open systems.In this letter, we will provide a heuristic argument for the validity of this conjecture, and present new computations that confirm its validity.The repeller in a scattering problem is defined as the set of points in phase space which do not escape to infinity at both positive or negative times. The Hausdorff dimension of the repeller is given by D H = 2d H +2, where we did not restrict ourselves to an energy surface. For closed two dimensional systems, such as compact surfaces of constant negative curvature, we have real zeros only and N (k) = #{k n : k n ≤ k} ∼ k 2 , which is consistent with (1) as d H = 1 then everything is trapped.Our motivation comes from rigorous work on quantum resonances and in particular from the work of Sjöstrand Here we consider a different but related problem. Suppose that Z(k) is the semi-classical Selberg-Ruelle zeta function, with k the wave number. In some situations the zeros of its meromorphic continua...

Fractal Weyl law for chaotic microcavities: Fresnel’s laws imply multifractal scattering

“…The modes (15) can be numerically computed with the boundary element method (BEM) using a root search algorithm in the complex frequency plane [25]. Even though the BEM is very efficient, it does not allow for obtaining a sufficient amount of resonances for a statistical analysis.…”

“…Quantum eigenenergies of open systems (resonance frequencies in the case of microcavities) are complex valued with the imaginary part being related to the lifetime of the state. Of particular interest is the fractal Weyl law for open chaotic systems [11,12,13,14,15,16,17]. This is an extension of the well-known Weyl's formula for bounded systems which states that the number of levels with wave number k n ≤ k is asymptotically N (k) ∼ k 2 for the particular case of a two-dimensional system which scales with the energy such as quantum billiards.…”

Fractal Weyl law for quantum fractal eigenstates

“…[9,10,11]. Thus the understanding of their properties in the semiclassical limit represents an important challenge.According to the fractal Weyl law [4,5] the number of Gamow eigenstates N γ , which have escape rates γ in a finite band width 0 ≤ γ ≤ γ b , scales aswhere d is a fractal dimension of a classical strange repeller formed by classical orbits nonescaping in future (or past) times. By numerical simulations it has been shown that the law (1) In view of the recent results described above I study numerically a simple model of the quantum Chirikov standard map (kicked rotator) with absorption introduced in [18] which allows to vary continuously the fractal dimension of the classical strange repeller.…”

Mathematical Physics of Quantum Mechanics

“…Several numerical studies have attempted to confirm the above estimate for a variety of scattering Hamiltonians [10,12,13,14], but with rather inconclusive results. Indeed, it is numerically demanding to compute resonances.…”

“…One method is to "complex rotate" the original Hamiltonian into a nonHermitian operator, the eigenvalues of which are the resonances. Another method uses the (approximate) relationship between, on one side, the resonance spectrum of H , one the other side, the set of zeros of some semiclassical zeta function, which is computed from the knowledge of classical periodic orbits [5,14]. In the case of the geodesic flow on a convex co-compact quotient of the Poincaré disk (which has a fractal trapped set), the resonances of the Laplace operator are exactly given by the zeros of Selberg's zeta function.…”