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 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/γ}, with γ set by the corresponding classical stickiness reflected in the algebraic decay of cumulative RTDs 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.