{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:32:25Z","timestamp":1777447945384,"version":"3.51.4"},"reference-count":21,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2018,12,10]],"date-time":"2018-12-10T00:00:00Z","timestamp":1544400000000},"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":[[2019,1]]},"abstract":"<jats:p>\n                    In this paper, we describe the asymptotic behavior of the scattering matrix\n                    <jats:italic toggle=\"yes\">S<\/jats:italic>\n                    associated with the\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mn>2<\/mml:mn>\n                        <mml:mo>\u00d7<\/mml:mo>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    matrix Schr\u00f6dinger operator\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"italic\">P<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mo>\u2212<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">h<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mstyle displaystyle=\"false\">\n                          <mml:mfrac>\n                            <mml:mrow>\n                              <mml:msup>\n                                <mml:mrow>\n                                  <mml:mi mathvariant=\"italic\">d<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mrow>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mi mathvariant=\"italic\">d<\/mml:mi>\n                              <mml:msup>\n                                <mml:mrow>\n                                  <mml:mi mathvariant=\"italic\">x<\/mml:mi>\n                                <\/mml:mrow>\n                                <mml:mrow>\n                                  <mml:mn>2<\/mml:mn>\n                                <\/mml:mrow>\n                              <\/mml:msup>\n                            <\/mml:mrow>\n                          <\/mml:mfrac>\n                        <\/mml:mstyle>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">I<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">V<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">x<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                        <mml:mo>+<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">h<\/mml:mi>\n                        <mml:mi mathvariant=\"italic\">R<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">x<\/mml:mi>\n                        <mml:mo mathvariant=\"normal\">,<\/mml:mo>\n                        <mml:mi mathvariant=\"italic\">h<\/mml:mi>\n                        <mml:msub>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">D<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">x<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:msub>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    on\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">L<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                        <mml:mo>\u2295<\/mml:mo>\n                        <mml:msup>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"italic\">L<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:msup>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">(<\/mml:mo>\n                        <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                        <mml:mo fence=\"true\" mathvariant=\"normal\" stretchy=\"false\">)<\/mml:mo>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . We study the situation where the diagonal terms of\n                    <jats:italic toggle=\"yes\">V<\/jats:italic>\n                    cross on the real axis. In particular it is proven that, due to the real crossing, the off-diagonal terms of\n                    <jats:italic toggle=\"yes\">S<\/jats:italic>\n                    are no longer exponentially small with respect to the semi-classical parameter\n                    <jats:italic toggle=\"yes\">h<\/jats:italic>\n                    .\n                  <\/jats:p>","DOI":"10.3233\/asy-181485","type":"journal-article","created":{"date-parts":[[2018,12,11]],"date-time":"2018-12-11T11:55:36Z","timestamp":1544529336000},"page":"15-42","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":1,"title":["Interacting double-state scattering system"],"prefix":"10.1177","volume":"111","author":[{"given":"Boulbeba","family":"Abdelmoumen","sequence":"first","affiliation":[{"name":"Sfax University","place":["Tunisia"]}]},{"given":"Hamadi","family":"Baklouti","sequence":"additional","affiliation":[{"name":"Sfax University","place":["Tunisia"]}]},{"given":"Slaheddine Ben","family":"Abdeljeilil","sequence":"additional","affiliation":[{"name":"ElManar University","place":["Tunisia"]}]}],"member":"179","published-online":{"date-parts":[[2018,12,10]]},"reference":[{"issue":"68","key":"e_1_3_3_2_2","first-page":"179","article-title":"Asymptotique des largeurs des r\u00e9sonnances pour un mod\u00e8le d\u2019effet tunnel microlocal","volume":"2","author":"Baklouti H.","year":"1998","unstructured":"H.\u00a0Baklouti, Asymptotique des largeurs des r\u00e9sonnances pour un mod\u00e8le d\u2019effet tunnel microlocal, Ann. 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