Numerical Study of Quantum Resonances in Chaotic Scattering

✍ Scribed by Kevin K. Lin

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 394 KB
• Volume: 176
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.2001.6986

✦ Synopsis

This paper presents numerical evidence that in quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy E scales like h --(D(K E )+1)/2 as h -→ 0. Here, K E denotes the subset of the energy surface {H = E} which stays bounded for all time under the flow generated by the classical 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 h --n , this suggests that the quantity (D(K E ) + 1)/2 represents the effective number of degrees of freedom in chaotic scattering problems. The calculations were performed using a recursive refinement technique for estimating the dimension of fractal repellors in classical Hamiltonian scattering, in conjunction with tools from modern quantum chemistry and numerical linear algebra.