{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,1]],"date-time":"2026-10-01T19:10:37Z","timestamp":1790881837079,"version":"4.1.0"},"reference-count":68,"publisher":"American Physical Society (APS)","issue":"3","license":[{"start":{"date-parts":[[2026,9,25]],"date-time":"2026-09-25T00:00:00Z","timestamp":1790294400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004329","name":"The Slovenian Research and Innovation Agency","doi-asserted-by":"publisher","award":["J1-4387"],"award-info":[{"award-number":["J1-4387"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100004329","name":"The Slovenian Research and Innovation Agency","doi-asserted-by":"publisher","award":["P1-0306"],"award-info":[{"award-number":["P1-0306"]}],"id":[{"id":"10.13039\/501100004329","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["journals.aps.org"],"crossmark-restriction":true},"short-container-title":["Phys. Rev. E"],"accepted":{"date-parts":[[2026,9,14]]},"abstract":"<jats:p>\n                    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\n                    <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <a:mrow>\n                        <a:mo>\u223c<\/a:mo>\n                        <a:mi>\u210f<\/a:mi>\n                        <a:mspace width=\"0.16em\"\/>\n                        <a:mo form=\"prefix\">exp<\/a:mo>\n                        <a:mo>(<\/a:mo>\n                        <a:mo>\u2212<\/a:mo>\n                        <a:mi>b<\/a:mi>\n                        <a:mo>\/<\/a:mo>\n                        <a:mi>\u210f<\/a:mi>\n                        <a:mo>)<\/a:mo>\n                      <\/a:mrow>\n                    <\/a:math>\n                    , with\n                    <d:math xmlns:d=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <d:mrow>\n                        <d:mi>b<\/d:mi>\n                        <d:mo>&gt;<\/d:mo>\n                        <d:mn>0<\/d:mn>\n                      <\/d:mrow>\n                    <\/d:math>\n                    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\n                    <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <e:msup>\n                        <e:mi>\u210f<\/e:mi>\n                        <e:mrow>\n                          <e:mn>1<\/e:mn>\n                          <e:mo>\/<\/e:mo>\n                          <e:mn>2<\/e:mn>\n                        <\/e:mrow>\n                      <\/e:msup>\n                    <\/e:math>\n                    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\n                    <f:math xmlns:f=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <f:msup>\n                        <f:mi>\u210f<\/f:mi>\n                        <f:mrow>\n                          <f:mn>1<\/f:mn>\n                          <f:mo>\u2212<\/f:mo>\n                          <f:mn>1<\/f:mn>\n                          <f:mo>\/<\/f:mo>\n                          <f:mi>\u03b3<\/f:mi>\n                        <\/f:mrow>\n                      <\/f:msup>\n                    <\/f:math>\n                    , with\n                    <g:math xmlns:g=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <g:mi>\u03b3<\/g:mi>\n                    <\/g:math>\n                    set by the corresponding classical stickiness reflected in the algebraic decay of cumulative recurrence-time distributions\n                    <h:math xmlns:h=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <h:msup>\n                        <h:mi>t<\/h:mi>\n                        <h:mrow>\n                          <h:mo>\u2212<\/h:mo>\n                          <h:mi>\u03b3<\/h:mi>\n                        <\/h:mrow>\n                      <\/h:msup>\n                    <\/h:math>\n                    . For the piecewise-linear maps studied here,\n                    <i:math xmlns:i=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <i:mrow>\n                        <i:mi>\u03b3<\/i:mi>\n                        <i:mo>=<\/i:mo>\n                        <i:mn>2<\/i:mn>\n                      <\/i:mrow>\n                    <\/i:math>\n                    . These results reveal a clear quantum signature of classical stickiness in non-KAM systems.\n                  <\/jats:p>","DOI":"10.1103\/dwhh-42vk","type":"journal-article","created":{"date-parts":[[2026,9,14]],"date-time":"2026-09-14T18:58:17Z","timestamp":1789412297000},"update-policy":"https:\/\/doi.org\/10.1103\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Sticky eigenstates in systems with sharply divided phase space"],"prefix":"10.1103","volume":"114","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3199-4648","authenticated-orcid":true,"given":"Hua","family":"Yan","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/01d5jce07","id-type":"ROR","asserted-by":"publisher"}],"name":"University of Maribor"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"16","published-online":{"date-parts":[[2026,9,25]]},"reference":[{"key":"dwhh-42vkCc1R1","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3700\/6\/9\/002","type":"journal-article"},{"key":"dwhh-42vkCc2R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/10\/12\/016","type":"journal-article"},{"key":"dwhh-42vkCc3R1","first-page":"1","type":"journal-article","volume":"1","author":"M. Robnik","year":"1998","journal-title":"Nonlinear Phenom. Complex Syst."},{"key":"dwhh-42vkCc4R1","volume-title":"Chaos in Classical and Quantum Mechanics","author":"M.\u00a0C. Gutzwiller","year":"2013","type":"book"},{"key":"dwhh-42vkCc5R1","volume-title":"Quantum Coherence in Mesoscopic Systems","author":"F. Haake","year":"1991","type":"conference-proceedings"},{"key":"dwhh-42vkCc6R1","volume-title":"Quantum Chaos An Introduction","author":"H.-J. St\u00f6ckmann","year":"2007","type":"book"},{"key":"dwhh-42vkCc7R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/17\/12\/013","type":"journal-article"},{"key":"dwhh-42vkCc8R1","doi-asserted-by":"publisher","DOI":"10.1088\/0305-4470\/27\/24\/017","type":"journal-article"},{"key":"dwhh-42vkCc9R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.111.054213","type":"journal-article"},{"key":"dwhh-42vkCc10R1","doi-asserted-by":"publisher","DOI":"10.1016\/0370-1573(93)90109-Q","type":"journal-article"},{"key":"dwhh-42vkCc11R1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.50.145","type":"journal-article"},{"key":"dwhh-42vkCc12R1","doi-asserted-by":"crossref","DOI":"10.1201\/b10712","type":"book","volume-title":"Dynamical Tunneling: Theory and Experiment","author":"S. 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All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright_statement","label":"Copyright statement","group":{"name":"copyright","label":"copyright"}},{"value":"2026","name":"copyright_year","label":"Copyright year","group":{"name":"copyright","label":"copyright"}},{"value":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/","name":"creative_commons_license","label":"Creative Commons license","group":{"name":"licenses","label":"Licenses"}},{"value":"\u00a92026 The authors","name":"copyright_holder","label":"Copyright holder","group":{"name":"copyright","label":"copyright"}}],"article-number":"034222"}}