Sticky eigenstates in systems with sharply divided phase space

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 quasiperiodic orbits. With the overlap index and the entropy localization length, we classify mixed eigenstates and show that the contribution from dynamical tunneling scales as ∼ℏexp(-b/ℏ), 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 approximately as ℏ^{1/2} in the MUPO case and oscillates around this algebraic behavior in the quasiperiodic case. This behavior generalizes established predictions for hierarchical states in Kolmogorov-Arnold-Moser (KAM) systems, which scale as ℏ^{1-1/γ}, with γ set by the corresponding classical stickiness reflected in the algebraic decay of cumulative recurrence-time distributions t^{-γ}. For the piecewise-linear maps studied here, γ=2. These results reveal a clear quantum signature of classical stickiness in non-KAM systems.