{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:49:17Z","timestamp":1777448957230,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"3-4","license":[{"start":{"date-parts":[[2012,9,1]],"date-time":"2012-09-01T00:00:00Z","timestamp":1346457600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2012,9]]},"abstract":"<jats:p>\n                    We consider the unperturbed operator H\n                    <jats:sub>0<\/jats:sub>\n                    :=(\u2212i\u2207\u2212A)\n                    <jats:sup>2<\/jats:sup>\n                    +W, self-adjoint in L\n                    <jats:sup>2<\/jats:sup>\n                    (R\n                    <jats:sup>2<\/jats:sup>\n                    ). Here A is a magnetic potential which generates a constant magnetic field b&gt;0, and the edge potential W=W\u00af is a \ud835\udcaf-periodic non-constant bounded function depending only on the first coordinate x\u2208R of (x,y)\u2208R\n                    <jats:sup>2<\/jats:sup>\n                    . Then the spectrum \u03c3(H\n                    <jats:sub>0<\/jats:sub>\n                    ) of H\n                    <jats:sub>0<\/jats:sub>\n                    has a band structure, the band functions are b\ud835\udcaf-periodic, and generically there are infinitely many open gaps in \u03c3(H\n                    <jats:sub>0<\/jats:sub>\n                    ). We establish explicit sufficient conditions which guarantee that a given band of \u03c3(H\n                    <jats:sub>0<\/jats:sub>\n                    ) has a positive length, and all the extremal points of the corresponding band function are non-degenerate. Under these assumptions we consider the perturbed operators H\n                    <jats:sub>\u00b1<\/jats:sub>\n                    =H\n                    <jats:sub>0<\/jats:sub>\n                    \u00b1V where the electric potential V\u2208L\n                    <jats:sup>\u221e<\/jats:sup>\n                    (R\n                    <jats:sup>2<\/jats:sup>\n                    ) is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of H\n                    <jats:sub>\u00b1<\/jats:sub>\n                    in the spectral gaps of H\n                    <jats:sub>0<\/jats:sub>\n                    . We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian could be interpreted as a 1D Schr\u00f6dinger operator with infinite-matrix-valued potential. Further, we restrict our attention on perturbations V of compact support. We find that there are infinitely many discrete eigenvalues in any open gap in the spectrum \u03c3(H\n                    <jats:sub>0<\/jats:sub>\n                    ), and the convergence of these eigenvalues to the corresponding spectral edge is asymptotically Gaussian.\n                  <\/jats:p>","DOI":"10.3233\/asy-2012-1103","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:23:12Z","timestamp":1575055392000},"page":"325-345","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Discrete spectrum of quantum Hall effect Hamiltonians II: Periodic edge potentials"],"prefix":"10.1177","volume":"79","author":[{"given":"Pablo","family":"Miranda","sequence":"first","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Ciencias, Universidad de Chile, Santiago de Chile, Chile. E-mail: pmirandar@ug.uchile.cl"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georgi","family":"Raikov","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Facultad de Matem\u00e1ticas, Pontificia Universidad Cat\u00f3lica de Chile, Santiago de Chile, Chile. E-mail: graikov@mat.puc.cl"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2012,9,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2012-1103","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2012-1103","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:39:35Z","timestamp":1777379975000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2012-1103"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,9]]},"references-count":0,"journal-issue":{"issue":"3-4","published-print":{"date-parts":[[2012,9]]}},"alternative-id":["10.3233\/ASY-2012-1103"],"URL":"https:\/\/doi.org\/10.3233\/asy-2012-1103","relation":{"is-cited-by":[{"id-type":"doi","id":"10.1088\/1751-8113\/49\/36\/365205","asserted-by":"object"}]},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,9]]}}}