New Class of Eigenstates in Generic Hamiltonian Systems

In mixed systems, besides regular and chaotic states, there are states supported by the chaotic region mainly living in the vicinity of the hierarchy of regular islands. We show that the fraction of these hierarchical states scales as ${ħ}^{\alpha }$ and we relate the exponent $\alpha =1-1\mathrm{}/\gamma$ to the decay of the classical staying probability $P\left(t\right)\sim t^{-\gamma }$. This is numerically confirmed for the kicked rotor by studying the influence of hierarchical states on eigenfunction and level statistics.