Sticky eigenstates in systems with sharply-divided phase space

Abstract

We investigate mixed eigenstates in systems with sharply-divided phase space, using different piecewise-linear maps whose regular-chaotic boundaries are formed by marginally unstable periodic orbits (MUPOs) or by quasi-periodic orbits. With the overlap index and the entropy localization length, we classify mixed eigenstates and show that the contribution from dynamical tunneling scales as $\sim \hbar\, \exp(-b/\hbar)$, with $b>0$ associated with the relative size of the regular region. The dominant fraction of states that remain sticky to the boundaries, referred to as sticky eigenstates, scales as $\hbar^{1/2}$ in the MUPO case and oscillates around this algebraic behavior in the quasi-periodic case. This behavior generalizes established predictions for hierarchical states in KAM systems, which scale as $\hbar^{1 - 1/\gamma}$, with $\gamma$ set by the corresponding classical stickiness reflected in the algebraic decay of cumulative RTDs $t^{-\gamma}$. For the piecewise-linear maps studied here, $\gamma = 2$. These results reveal a clear quantum signature of classical stickiness in non-KAM systems.