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1、DynamicalResonancesandLifetimesinRiggedHilbertSpace(II)MeishanZhaoTheJamesFranckInstitute,TheUniversityofChicago,Chicago,Illinois60637,USADecember9,1997AbstractAsimpleschemeforlocatingthresholdresonancecenterinriggedHilbertspaceusingvirialtheoremisdiscussed.Asasimpletest,
2、ap-plicationsareperformedonmodelpotentialsresemblingH+Hand2O+Dreactions.21Therearemanydierenttheoreticalapproachesinlocatingdynamicalresonances.Amongstalltheseapproaches,quantumscatteringtheory[1-4]isthemostpopularlyusedone,becausearesonanceisdenedrigorouslyasapoleofthe
3、scatteringmatrix.However,adirectscatteringcalculationcansometimesbeverytediousanditmayleadtoveryheavynumericalcomputation.Inrecentyears,thequasi-bound-satetheoryinriggedHilbertspacegen-eratedbycomplexscalingofthecoordinatesthroughadilatationtransfor-mationhasgainedwidespr
4、eadacceptance[5-9],becauseofitseectivewayoflocatingresonances,itsconvenienceinborrowingthecomputationalmeth-odsusedinboundstatecalculationsanditsnovelthinkinginnon-Hermitiandomain.BasedontheriggedHilbertspaceanalysis,inourpreviousstudies[10,11],wehaveshownthatonecandene
5、aresonancewidthoperatorwhoseexpectationvalueistheimaginarypartoftheenergyforaresonancestate.Inthisletterweshowthatinvokingtheboundstatenatureoftheresonancetherealpartoftheenergyofathresholdresonancemaybeobtainedbyusingthevirialtheorem[12,13].Wepresenttheanalysistoestimate
6、thethresholdresonanceenergyfortheEckartpotentialthatroughlyresemblesthelinearH+Hreaction,2andforanunsymmetricmodiedEckartpotentialthatresemblestherstvibrationallyexcitedadiabaticonedimensionalpotentialenergysurfacefortheO+Dreaction.Theseapplicationsarephysicallymeaningf
7、ul.Ithas2beenconcludedrecentlythat"chemicalreactionthresholdsassociatedwithbarriersarequantummechanicalresonances.thatthestudyofreactivitymayberecastasastudyofpolesofthescatteringmatrix"[14,15].Brie
y,thecomplexscalingisbasedonthedilationcontinuationoftheHamiltonian
8、withthescalingtransformationontheradialcoordinatesir!re(1)whereisarotationalangle.Forone-dimen