Hyperbolic Scar Patterns in Phase Space

Abstract

We develop a semiclassical approximation for the spectral Wigner and Husimi functions in the neighbourhood of a classically unstable periodic orbit of chaotic two dimensional maps. The prediction of hyperbolic fringes for the Wigner function, asymptotic to the stable and unstable manifolds, is verified computationally for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus. The characteristic fringe patterns can be distinguished even for quasi-energies where the fixed point is not Bohr-quantized. The corresponding Husimi function dampens these fringes with a Gaussian envelope centered on the periodic point. Even though the hyperbolic structure is then barely perceptible, more periodic points stand out due to the weakened interference.

12 pages, 10 figures, Submited to Phys. Rev. E

Cited by in corpus (5)

• Scarred eigenstates for quantum cat maps of minimal periods
• Wigner distributions for finite state systems without redundant phase point operators
• Wavefunction Statistics using Scar States
• Semi-classical Scar functions in phase space
• Semiclassical matrix elements for a chaotic propagator in the Scar functions basis