{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,6]],"date-time":"2025-12-06T17:11:31Z","timestamp":1765041091004,"version":"3.46.0"},"reference-count":25,"publisher":"IOP Publishing","issue":"3","license":[{"start":{"date-parts":[[2020,10,19]],"date-time":"2020-10-19T00:00:00Z","timestamp":1603065600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/publishingsupport.iopscience.iop.org\/iop-standard\/v1"},{"start":{"date-parts":[[2020,10,19]],"date-time":"2020-10-19T00:00:00Z","timestamp":1603065600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/iopscience.iop.org\/info\/page\/text-and-data-mining"}],"funder":[{"name":"Centro de F\u00edsica das Universidades do Minho e do Porto","award":["UID\/FIS\/04650\/2019"],"award-info":[{"award-number":["UID\/FIS\/04650\/2019"]}]},{"name":"Propriedades \u00f3ticas n\u00e3o lineares de materiais em camadas","award":["POCI-01-0145-FEDER-028887"],"award-info":[{"award-number":["POCI-01-0145-FEDER-028887"]}]}],"content-domain":{"domain":["iopscience.iop.org"],"crossmark-restriction":false},"short-container-title":["J. Phys.: Condens. Matter"],"published-print":{"date-parts":[[2021,1,20]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>The Schr\u00f6dinger equation in a square or rectangle with hard walls is solved in every introductory quantum mechanics course. Solutions for other polygonal enclosures only exist in a very restricted class of polygons, and are all based on a result obtained by Lam\u00e9 in 1852. Any enclosure can, of course, be addressed by finite element methods for partial differential equations. In this paper, we present a variational method to approximate the low-energy spectrum and wave-functions for arbitrary convex polygonal enclosures, developed initially for the study of vibrational modes of plates. In view of the recent interest in the spectrum of quantum dots of two dimensional materials, described by effective models with massless electrons, we extend the method to the Dirac\u2013Weyl equation for a spin-1\/2 fermion confined in a quantum billiard of polygonal shape, with different types of boundary conditions. We illustrate the method\u2019s convergence in cases where the spectrum is known exactly, and apply it to cases where no exact solution exists.<\/jats:p>","DOI":"10.1088\/1361-648x\/abbe77","type":"journal-article","created":{"date-parts":[[2020,10,5]],"date-time":"2020-10-05T18:45:44Z","timestamp":1601923544000},"page":"035901","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":2,"title":["A polynomial approach to the spectrum of Dirac\u2013Weyl polygonal Billiards"],"prefix":"10.1088","volume":"33","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0961-9235","authenticated-orcid":false,"given":"M F C Martins","family":"Quintela","sequence":"first","affiliation":[]},{"given":"J M B","family":"Lopes dos Santos","sequence":"additional","affiliation":[]}],"member":"266","published-online":{"date-parts":[[2020,10,19]]},"reference":[{"year":"1852","author":"Lam\u00e9","key":"cmabbe77bib1","type":"book"},{"key":"cmabbe77bib2","doi-asserted-by":"publisher","first-page":"819","DOI":"10.1137\/0511073","type":"journal-article","article-title":"The eigenvalues of an equilateral triangle","volume":"11","author":"Pinsky","year":"1980","journal-title":"SIAM J. 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All rights, including for text and data mining, AI training, and similar technologies, are reserved.","name":"copyright_information","label":"Copyright Information"},{"value":"2020-08-08","name":"date_received","label":"Date Received","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-10-05","name":"date_accepted","label":"Date Accepted","group":{"name":"publication_dates","label":"Publication dates"}},{"value":"2020-10-19","name":"date_epub","label":"Online publication date","group":{"name":"publication_dates","label":"Publication dates"}}]}}