New class of eigenstates in generic hamiltonian systems

Area of Science:

• Physics
• Nonlinear Dynamics
• Quantum Chaos

Background:

• Mixed dynamical systems exhibit both regular and chaotic behaviors.
• Chaotic regions often harbor complex structures, including hierarchical states near regular islands.

Purpose of the Study:

• To investigate the scaling properties of hierarchical states in mixed systems.
• To establish a relationship between these states and classical probability decay.
• To numerically verify these findings in a specific model.

Main Methods:

• Analysis of hierarchical states in the vicinity of regular islands.
• Derivation of scaling laws involving Planck's constant.
• Numerical simulations using the kicked rotor model.
• Examination of eigenfunction and level statistics.

Main Results:

• The fraction of hierarchical states scales as Planck's constant over 2pi raised to the power of alpha.
• The exponent alpha is related to the decay of classical staying probability P(t) by alpha = 1 - 1/gamma.
• Numerical confirmation of the scaling and its relation to probability decay was achieved for the kicked rotor.

Conclusions:

• Hierarchical states play a significant role in the dynamics of mixed systems.
• The scaling behavior provides a quantitative link between quantum and classical properties.
• The kicked rotor serves as a valid model for demonstrating these complex phenomena.