{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:36:55Z","timestamp":1791322615489,"version":"4.3.1"},"reference-count":21,"publisher":"National Academy of Sciences","issue":"37","content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2012,9,11]]},"abstract":"<jats:p>Localization of stationary waves occurs in a large variety of vibrating systems, whether mechanical, acoustical, optical, or quantum. It is induced by the presence of an inhomogeneous medium, a complex geometry, or a quenched disorder. One of its most striking and famous manifestations is Anderson localization, responsible for instance for the metal-insulator transition in disordered alloys. Yet, despite an enormous body of related literature, a clear and unified picture of localization is still to be found, as well as the exact relationship between its many manifestations. In this paper, we demonstrate that both Anderson and weak localizations originate from the same universal mechanism, acting on any type of vibration, in any dimension, and for any domain shape. This mechanism partitions the system into weakly coupled subregions. The boundaries of these subregions correspond to the valleys of a hidden landscape that emerges from the interplay between the wave operator and the system geometry. The height of the landscape along its valleys determines the strength of the coupling between the subregions. The landscape and its impact on localization can be determined rigorously by solving one special boundary problem. This theory allows one to predict the localization properties, the confining regions, and to estimate the energy of the vibrational eigenmodes through the properties of one geometrical object. In particular, Anderson localization can be understood as a special case of weak localization in a very rough landscape.<\/jats:p>","DOI":"10.1073\/pnas.1120432109","type":"journal-article","created":{"date-parts":[[2012,8,27]],"date-time":"2012-08-27T23:18:09Z","timestamp":1346109489000},"page":"14761-14766","update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":180,"title":["Universal mechanism for Anderson and weak localization"],"prefix":"10.1073","volume":"109","author":[{"given":"Marcel","family":"Filoche","sequence":"first","affiliation":[{"name":"Physique de la Mati\u00e8re Condens\u00e9e, Ecole Polytechnique, Centre National de la Recherche Scientifique, 91128 Palaiseau, France;"},{"name":"Centre de Math\u00e9matiques et de Leurs Applications, Ecole Normale Sup\u00e9rieure de Cachan, Centre National de la Recherche Scientifique, UniverSud, 94230 Cachan, France; and"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Svitlana","family":"Mayboroda","sequence":"additional","affiliation":[{"name":"School of Mathematics, University of Minnesota, Minneapolis, 55455 MN"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2012,8,27]]},"reference":[{"key":"e_1_3_3_1_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRev.109.1492"},{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1016\/0370-1573(74)90029-5"},{"key":"e_1_3_3_3_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.42.673"},{"key":"e_1_3_3_4_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.57.287"},{"key":"e_1_3_3_5_2","doi-asserted-by":"publisher","DOI":"10.1103\/RevModPhys.80.1355"},{"key":"e_1_3_3_6_2","doi-asserted-by":"publisher","DOI":"10.1063\/1.3206091"},{"key":"e_1_3_3_7_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.99.253902"},{"key":"e_1_3_3_8_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1185080"},{"key":"e_1_3_3_9_2","doi-asserted-by":"publisher","DOI":"10.1364\/OL.36.000127"},{"key":"e_1_3_3_10_2","first-page":"624","article-title":"Localized eigenfunctions: Here you see them, there you don\u2019t","volume":"57","author":"Heilman S","year":"2010","unstructured":"S Heilman, R Strichartz, Localized eigenfunctions: Here you see them, there you don\u2019t. Notices Amer Math Soc 57, 624\u2013629 (2010).","journal-title":"Notices Amer Math Soc"},{"key":"e_1_3_3_11_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.70.3876"},{"key":"e_1_3_3_12_2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsv.2006.07.036"},{"key":"e_1_3_3_13_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.103.254301"},{"key":"e_1_3_3_14_2","doi-asserted-by":"publisher","DOI":"10.1142\/S0129055X94000419"},{"key":"e_1_3_3_15_2","doi-asserted-by":"publisher","DOI":"10.1080\/00018736400101061"},{"key":"e_1_3_3_16_2","doi-asserted-by":"publisher","DOI":"10.1088\/0022-3719\/11\/16\/012"},{"key":"e_1_3_3_17_2","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevE.60.4843"},{"key":"e_1_3_3_18_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1115660"},{"key":"e_1_3_3_19_2","doi-asserted-by":"publisher","DOI":"10.1126\/science.1183640"},{"key":"e_1_3_3_20_2","first-page":"110","article-title":"On the asymptotic distribution of eigenvalues","volume":"1911","author":"Weyl H","year":"1911","unstructured":"H Weyl, On the asymptotic distribution of eigenvalues. G\u00f6ttinger Nachr 1911, 110\u2013117, (in German). (1911).","journal-title":"G\u00f6ttinger Nachr"},{"key":"e_1_3_3_21_2","doi-asserted-by":"publisher","DOI":"10.1038\/nature09824"}],"container-title":["Proceedings of the National Academy of Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pnas.org\/doi\/pdf\/10.1073\/pnas.1120432109","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,15]],"date-time":"2022-04-15T07:16:18Z","timestamp":1650006978000},"score":1,"resource":{"primary":{"URL":"https:\/\/pnas.org\/doi\/full\/10.1073\/pnas.1120432109"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,8,27]]},"references-count":21,"journal-issue":{"issue":"37","published-print":{"date-parts":[[2012,9,11]]}},"alternative-id":["10.1073\/pnas.1120432109"],"URL":"https:\/\/doi.org\/10.1073\/pnas.1120432109","relation":{},"ISSN":["0027-8424","1091-6490"],"issn-type":[{"value":"0027-8424","type":"print"},{"value":"1091-6490","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,8,27]]},"assertion":[{"value":"2012-08-27","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}