{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:44:33Z","timestamp":1777448673531,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2008,5,1]],"date-time":"2008-05-01T00:00:00Z","timestamp":1209600000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2008,5]]},"abstract":"<jats:p>We are interested in the asymptotic behavior of the solution to the Cauchy problem for the linear evolution equation<\/jats:p>\n                  <jats:p>\n                    i\u03b5\u2009\u2202\n                    <jats:sub>t<\/jats:sub>\n                    \u03c8=A(t)\u03c8,\u2003A(t)=A\n                    <jats:sub>0<\/jats:sub>\n                    +V(t), \u03c8(0)=\u03c8\n                    <jats:sub>0<\/jats:sub>\n                    ,\n                  <\/jats:p>\n                  <jats:p>\n                    in the limit \u03b5\u21920. A case of special interest is when the operator A(t) has continuous spectrum and the initial data \u03c8\n                    <jats:sub>0<\/jats:sub>\n                    is, in particular, an improper eigenfunction of the continuous spectrum of A(0). Under suitable assumptions on A(t), we derive a formal asymptotic solution of the problem whose leading order has an explicit representation.\n                  <\/jats:p>\n                  <jats:p>\n                    A key ingredient is a reduction of the original Cauchy problem to the study of the semiclassical pseudo-differential operator \u2133=M(t, i\u03b5\u2009\u2202\n                    <jats:sub>t<\/jats:sub>\n                    ) with compact operator-valued symbol M(t, E)=V\n                    <jats:sub>1<\/jats:sub>\n                    (t)(A\n                    <jats:sub>0<\/jats:sub>\n                    \u2212EI)\n                    <jats:sup>\u22121<\/jats:sup>\n                    V\n                    <jats:sub>2<\/jats:sub>\n                    (t), V(t)=V\n                    <jats:sub>2<\/jats:sub>\n                    (t)V\n                    <jats:sub>1<\/jats:sub>\n                    (t), and an asymptotic analysis of its spectral properties. We illustrate our approach with a detailed presentation of the example of the Schr\u00f6dinger equation on the axis with the \u03b4-function potential: A(t)=\u2212\u2202\n                    <jats:sub>xx<\/jats:sub>\n                    +\u03b1(t)\u03b4(x).\n                  <\/jats:p>","DOI":"10.3233\/asy-2008-0874","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:01:13Z","timestamp":1575054073000},"page":"17-45","source":"Crossref","is-referenced-by-count":2,"title":["Linear adiabatic dynamics generated by operators with continuous spectrum. I"],"prefix":"10.1177","volume":"58","author":[{"given":"Vladimir S.","family":"Buslaev","sequence":"first","affiliation":[{"name":"Institute for Physics, St. Petersburg State University, St. Petersburg\u2013Petrodvorets 198904, Russia. E-mail: buslaev@mph.phys.spbu.ru"}],"role":[{"role":"author","vocab":"crossref"}]},{"given":"Catherine","family":"Sulem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Toronto, Toronto, M5S 2E4 Canada. E-mail: sulem@math.toronto.edu"}],"role":[{"role":"author","vocab":"crossref"}]}],"member":"179","published-online":{"date-parts":[[2008,5,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2008-0874","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-2008-0874","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:38:35Z","timestamp":1777379915000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-2008-0874"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,5]]},"references-count":0,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2008,5]]}},"alternative-id":["10.3233\/ASY-2008-0874"],"URL":"https:\/\/doi.org\/10.3233\/asy-2008-0874","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,5]]}}}