Sticky eigenstates in systems with sharply-divided phase space

• URL: http://arxiv.org/abs/2512.05627v1
• Date: Fri, 05 Dec 2025 11:12:55 GMT
• Title: Sticky eigenstates in systems with sharply-divided phase space
• Abstract summary: We investigate mixed eigenstates in systems with sharply-divided phase space.<n>With the overlap index and the entropy localization length, we classify mixed eigenstates.<n>The dominant fraction of states that remain sticky to the boundaries, referred to as sticky eigenstates, scales as $hbar1/2$ in the MUPO case.
• Score: 4.589646880507029
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $\hbar^{1 - 1/γ}$, 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.

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