{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:53:25Z","timestamp":1777449205409,"version":"3.51.4"},"reference-count":0,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"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":[[2015,1]]},"abstract":"<jats:p>Motivated by the study of resonances for molecular systems in the Born\u2013Oppenheimer approximation, we consider a semiclassical 2\u00d72 matrix Schr\u00f6dinger operator of the form<\/jats:p>\n                  <jats:p>\n                    P=\u2212h\n                    <jats:sup>2<\/jats:sup>\n                    \u0394I\n                    <jats:sub>2<\/jats:sub>\n                    +diag\u2009(x\n                    <jats:sub>n<\/jats:sub>\n                    \u2212\u03bc,\u03c4V\n                    <jats:sub>2<\/jats:sub>\n                    (x))+hR(x,hD\n                    <jats:sub>x<\/jats:sub>\n                    ),\n                  <\/jats:p>\n                  <jats:p>\n                    where \u03bc and \u03c4 are two small positive constants, V\n                    <jats:sub>2<\/jats:sub>\n                    is real-analytic and admits a nondegenerate minimum at 0, and R=(r\n                    <jats:sub>j,k<\/jats:sub>\n                    (x,hD\n                    <jats:sub>x<\/jats:sub>\n                    ))\n                    <jats:sub>1\u2264j,k\u22642<\/jats:sub>\n                    is a symmetric off-diagonal 2\u00d72 matrix of first-order differential operators with analytic coefficients. Then, denoting by e\n                    <jats:sub>1<\/jats:sub>\n                    the first eigenvalue of \u2212\u0394+\u3008\u03c4V\n                    <jats:sub>2<\/jats:sub>\n                    \u2033(0)x,x\u3009\/2, and under some ellipticity condition on r\n                    <jats:sub>1,2<\/jats:sub>\n                    =r\n                    <jats:sub>2,1<\/jats:sub>\n                    <jats:sup>*<\/jats:sup>\n                    , we show that, for any \u03bc sufficiently small, and for 0&lt;\u03c4\u2264\u03c4(\u03bc) with some \u03c4(\u03bc)&gt;0, the unique resonance \u03c1 of P such that \u03c1=\u03c4V\n                    <jats:sub>2<\/jats:sub>\n                    (0)+(e\n                    <jats:sub>1<\/jats:sub>\n                    +r\n                    <jats:sub>2,2<\/jats:sub>\n                    (0,0))h+\ud835\udcaa(h\n                    <jats:sup>2<\/jats:sup>\n                    ) (as h\u21920\n                    <jats:sub>+<\/jats:sub>\n                    ) satisfies\n                  <\/jats:p>\n                  <jats:p>\n                    Im \u03c1=\u2212h\n                    <jats:sup>3\/2<\/jats:sup>\n                    f(h,ln\u20091\/h)e\n                    <jats:sup>\u22122S\/h<\/jats:sup>\n                    ,\n                  <\/jats:p>\n                  <jats:p>\n                    where f(h,ln\u20091\/h)~\u03a3\n                    <jats:sub>0\u2264m\u2264\u2113<\/jats:sub>\n                    f\n                    <jats:sub>\u2113,m<\/jats:sub>\n                    h\n                    <jats:sup>\u2113<\/jats:sup>\n                    (ln\u20091\/h)\n                    <jats:sup>m<\/jats:sup>\n                    is a symbol with f\n                    <jats:sub>0,0<\/jats:sub>\n                    &gt;0, and S is the imaginary part of the complex action along some convenient closed path containing (0,0) and consisting of a union of complex nul-bicharacteristics of p\n                    <jats:sub>1<\/jats:sub>\n                    :=\u03be\n                    <jats:sup>2<\/jats:sup>\n                    \u2212x\n                    <jats:sub>n<\/jats:sub>\n                    \u2212\u03bc and p\n                    <jats:sub>2<\/jats:sub>\n                    :=\u03be\n                    <jats:sup>2<\/jats:sup>\n                    +\u03c4V\n                    <jats:sub>2<\/jats:sub>\n                    (x) (broken instanton). This broken instanton is described in terms of the outgoing and incoming complex Lagrangian manifolds associated with p\n                    <jats:sub>2<\/jats:sub>\n                    at the point (0,0), and their intersections with the characteristic set p\n                    <jats:sub>1<\/jats:sub>\n                    <jats:sup>\u22121<\/jats:sup>\n                    (0) of p\n                    <jats:sub>1<\/jats:sub>\n                    .\n                  <\/jats:p>","DOI":"10.3233\/asy-141256","type":"journal-article","created":{"date-parts":[[2019,11,29]],"date-time":"2019-11-29T19:37:31Z","timestamp":1575056251000},"page":"33-90","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":5,"title":["Resonance widths in a case of multidimensional phase space tunneling"],"prefix":"10.1177","volume":"91","author":[{"given":"Alain","family":"Grigis","sequence":"first","affiliation":[{"name":"LAGA UMR CNRS 7539, D\u00e9partement de Math\u00e9matiques, Universit\u00e9 Paris 13, Av. J.-B. Cl\u00e9ment, 93430 Villetaneuse, France. E-mail: grigis@math.univ-paris13.fr"}]},{"given":"Andr\u00e9","family":"Martinez","sequence":"additional","affiliation":[{"name":"Dipartimento di Matematica, Universit\u00e0 di Bologna, Piazza di Porta San Donato, 40127 Bologna, Italy. E-mail: martinez@dm.unibo.it"}]}],"member":"179","published-online":{"date-parts":[[2015,1,1]]},"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-141256","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-141256","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:40:20Z","timestamp":1777380020000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-141256"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":0,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.3233\/ASY-141256"],"URL":"https:\/\/doi.org\/10.3233\/asy-141256","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}