Numerical Study of Quantum Resonances in Chaotic Scattering
Kevin K. Lin
本文提出了数值证据,即对于具有混沌经典动力学的量子系统,在能量E尺度附近的散射共振数如ħ^-D(K_E)+1/2为ħ→0。 在这里,K_E表示经典能量表面{H=E}的子集,在Hamiltonian H和D(K_E)产生的流下,一直保持边界,表示其分形维度。 由于具有n自由度的量子系统中的绑定状态的数量,如ħ^-n,这表明数量D(K_E)+1/2代表了散射问题的自由度的有效数量。
This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like ħ^-D(K_E)+1/2 as ħ→0. Here, K_E denotes the subset of the classical energy surface {H=E} which stays bounded for all time under the flow generated by the Hamiltonian H and D(K_E) denotes its fractal dimension. Since the number of bound states in a quantum system with n degrees of freedom scales like ħ^-n, this suggests that the quantity...