<?xml version="1.0" encoding="utf-8"?>
<TEI xmlns="http://www.tei-c.org/ns/1.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:hal="http://hal.archives-ouvertes.fr/" xmlns:gml="http://www.opengis.net/gml/3.3/" xmlns:gmlce="http://www.opengis.net/gml/3.3/ce" version="1.1" xsi:schemaLocation="http://www.tei-c.org/ns/1.0 http://api.archives-ouvertes.fr/documents/aofr-sword.xsd">
  <teiHeader>
    <fileDesc>
      <titleStmt>
        <title>HAL TEI export of hal-00350061</title>
      </titleStmt>
      <publicationStmt>
        <distributor>CCSD</distributor>
        <availability status="restricted">
          <licence target="https://creativecommons.org/publicdomain/zero/1.0/">CC0 1.0 - Universal</licence>
        </availability>
        <date when="2026-05-17T15:18:41+02:00"/>
      </publicationStmt>
      <sourceDesc>
        <p part="N">HAL API Platform</p>
      </sourceDesc>
    </fileDesc>
  </teiHeader>
  <text>
    <body>
      <listBibl>
        <biblFull>
          <titleStmt>
            <title xml:lang="en">Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area</title>
            <author role="aut">
              <persName>
                <forename type="first">Francoise</forename>
                <surname>Truc</surname>
              </persName>
              <email type="md5">65c53aa1a7c717f75f22f566def75203</email>
              <email type="domain">univ-alpes-grenoble.fr</email>
              <idno type="idhal" notation="string">francoise-truc</idno>
              <idno type="idhal" notation="numeric">785</idno>
              <idno type="halauthorid" notation="string">2393-785</idno>
              <idno type="IDREF">https://www.idref.fr/034007113</idno>
              <affiliation ref="#struct-21"/>
            </author>
            <editor role="depositor">
              <persName>
                <forename>Francoise</forename>
                <surname>Truc</surname>
              </persName>
              <email type="md5">a53d2972e9d156efc844cf3cb48efe8c</email>
              <email type="domain">univ-grenoble-alpes.fr</email>
            </editor>
          </titleStmt>
          <editionStmt>
            <edition n="v1" type="current">
              <date type="whenSubmitted">2009-01-05 17:43:18</date>
              <date type="whenWritten">2007</date>
              <date type="whenModified">2025-10-16 09:21:28</date>
              <date type="whenReleased">2009-01-05 20:51:23</date>
              <date type="whenProduced">2008</date>
              <date type="whenEndEmbargoed">2009-01-05</date>
              <ref type="file" target="https://hal.science/hal-00350061v1/document">
                <date notBefore="2009-01-05"/>
              </ref>
              <ref type="file" subtype="author" n="1" target="https://hal.science/hal-00350061v1/file/suede.pdf" id="file-350061-357581">
                <date notBefore="2009-01-05"/>
              </ref>
              <ref type="externalLink" target="http://arxiv.org/pdf/0901.0539"/>
            </edition>
            <respStmt>
              <resp>contributor</resp>
              <name key="110987">
                <persName>
                  <forename>Francoise</forename>
                  <surname>Truc</surname>
                </persName>
                <email type="md5">a53d2972e9d156efc844cf3cb48efe8c</email>
                <email type="domain">univ-grenoble-alpes.fr</email>
              </name>
            </respStmt>
          </editionStmt>
          <publicationStmt>
            <distributor>CCSD</distributor>
            <idno type="halId">hal-00350061</idno>
            <idno type="halUri">https://hal.science/hal-00350061</idno>
            <idno type="halBibtex">truc:hal-00350061</idno>
            <idno type="halRefHtml">&lt;i&gt;Operator Theory: Advances and Applications&lt;/i&gt;, 2008, 186, pp.403-413</idno>
            <idno type="halRef">Operator Theory: Advances and Applications, 2008, 186, pp.403-413</idno>
            <availability status="restricted">
              <licence target="https://about.hal.science/hal-authorisation-v1/">HAL Authorization<ref corresp="#file-350061-357581"/></licence>
            </availability>
          </publicationStmt>
          <seriesStmt>
            <idno type="stamp" n="UGA">HAL Grenoble Alpes</idno>
            <idno type="stamp" n="CNRS">CNRS - Centre national de la recherche scientifique</idno>
            <idno type="stamp" n="FOURIER">Institut Fourier</idno>
            <idno type="stamp" n="TDS-MACS">Réseau de recherche en Théorie des Systèmes Distribués, Modélisation, Analyse et Contrôle des Systèmes</idno>
            <idno type="stamp" n="TEST-UGA">TEST-UGA</idno>
          </seriesStmt>
          <notesStmt>
            <note type="audience" n="2">International</note>
            <note type="popular" n="0">No</note>
            <note type="peer" n="1">Yes</note>
          </notesStmt>
          <sourceDesc>
            <biblStruct>
              <analytic>
                <title xml:lang="en">Born-Oppenheimer-type Approximations for Degenerate Potentials: Recent Results and a Survey on the area</title>
                <author role="aut">
                  <persName>
                    <forename type="first">Francoise</forename>
                    <surname>Truc</surname>
                  </persName>
                  <email type="md5">65c53aa1a7c717f75f22f566def75203</email>
                  <email type="domain">univ-alpes-grenoble.fr</email>
                  <idno type="idhal" notation="string">francoise-truc</idno>
                  <idno type="idhal" notation="numeric">785</idno>
                  <idno type="halauthorid" notation="string">2393-785</idno>
                  <idno type="IDREF">https://www.idref.fr/034007113</idno>
                  <affiliation ref="#struct-21"/>
                </author>
              </analytic>
              <monogr>
                <idno type="halJournalId" status="VALID">28837</idno>
                <idno type="issn">0255-0156</idno>
                <idno type="eissn">2296-4878</idno>
                <title level="j">Operator Theory: Advances and Applications</title>
                <imprint>
                  <publisher>Springer</publisher>
                  <biblScope unit="volume">186</biblScope>
                  <biblScope unit="pp">403-413</biblScope>
                  <date type="datePub">2008</date>
                </imprint>
              </monogr>
              <idno type="arxiv">0901.0539</idno>
            </biblStruct>
          </sourceDesc>
          <profileDesc>
            <langUsage>
              <language ident="en">English</language>
            </langUsage>
            <textClass>
              <keywords scheme="author">
                <term xml:lang="en">semi-classical asymptotics. minimax principle</term>
                <term xml:lang="en">Weyl formula</term>
                <term xml:lang="en">degenerate potentials</term>
                <term xml:lang="en">Born-Oppenheimer approximation</term>
                <term xml:lang="en">Eigenvalues</term>
              </keywords>
              <classCode scheme="classification">35P20</classCode>
              <classCode scheme="halDomain" n="math.math-mp">Mathematics [math]/Mathematical Physics [math-ph]</classCode>
              <classCode scheme="halTypology" n="ART">Journal articles</classCode>
              <classCode scheme="halOldTypology" n="ART">Journal articles</classCode>
              <classCode scheme="halTreeTypology" n="ART">Journal articles</classCode>
            </textClass>
            <abstract xml:lang="en">
              <p>This paper is devoted to the asymptotics of eigenvalues for a Schrö-dinger operator in the case when the potential V does not tend to infinity at infinity. Such a potential is called degenerate. The point is that the set in the phase space where the associated hamiltonian is smaller than a fixed energy E may have an infinite volume, so that the Weyl formula which gives the behaviour of the counting function has to be revisited. We recall various results in this area, in the classical context as well as in the semi-classical one and comment the different methods. In sections 3, 4 we present our joint works with A Morame , (Université de Nantes),concerning a degenerate potential V(x) =f(y) g(z) , where g is assumed to be a homogeneous positive function of m variables , and f is a smooth and strictly positive function of n variables, with a minimum in 0. In the case where f tends to infinity at infinity, we give the semi-classical asymptotic behaviour of the number of eigenvalues less than a fixed energy . Then we give a sharp estimate of the low eigenvalues, using a Born Oppenheimer approximation. With a refined approach we localize also higher energies . Finally we apply the previous methods to a class of potentials which vanish on a regular hypersurface.</p>
            </abstract>
          </profileDesc>
        </biblFull>
      </listBibl>
    </body>
    <back>
      <listOrg type="structures">
        <org type="laboratory" xml:id="struct-21" status="OLD">
          <idno type="IdRef">033629129</idno>
          <idno type="ISNI">0000000122896889</idno>
          <idno type="RNSR">199512018P</idno>
          <idno type="ROR">https://ror.org/05rwrfh97</idno>
          <orgName>Institut Fourier</orgName>
          <orgName type="acronym">IF</orgName>
          <date type="start">1973-01-01</date>
          <date type="end">2019-12-31</date>
          <desc>
            <address>
              <addrLine>100, rue des maths 38610 Gières</addrLine>
              <country key="FR"/>
            </address>
            <ref type="url">http://www-fourier.ujf-grenoble.fr/</ref>
          </desc>
          <listRelation>
            <relation name="UMR5582" active="#struct-441569" type="direct"/>
            <relation active="#struct-445543" type="direct"/>
          </listRelation>
        </org>
        <org type="regroupinstitution" xml:id="struct-441569" status="VALID">
          <idno type="IdRef">02636817X</idno>
          <idno type="ISNI">0000000122597504</idno>
          <idno type="ROR">https://ror.org/02feahw73</idno>
          <orgName>Centre National de la Recherche Scientifique</orgName>
          <orgName type="acronym">CNRS</orgName>
          <date type="start">1939-10-19</date>
          <desc>
            <address>
              <country key="FR"/>
            </address>
            <ref type="url">https://www.cnrs.fr/</ref>
          </desc>
        </org>
        <org type="institution" xml:id="struct-445543" status="OLD">
          <idno type="IdRef">188399275</idno>
          <idno type="ROR">https://ror.org/02rx3b187</idno>
          <orgName>Université Grenoble Alpes [2016-2019]</orgName>
          <orgName type="acronym">UGA [2016-2019]</orgName>
          <date type="start">2016-01-01</date>
          <date type="end">2019-12-31</date>
          <desc>
            <address>
              <addrLine>38058 Grenoble cedex</addrLine>
              <country key="FR"/>
            </address>
          </desc>
        </org>
      </listOrg>
    </back>
  </text>
</TEI>