具有转移条件且边界条件含特征参数的奇异微分算子的谱分析(英文).docx

具有转移条件且边界条件含特征参数的奇异微分算子的谱分析(英文)
Introduction
Singulardifferentialoperatorsareessentialtoolsinthestudyofnon-smoothandsingularproblemsinmathematicalphysicsandengineering.Theseoperatorshavetheuniquepropertyofbeingabletocapturethesingularitiesoftheunderlyingphysicalsystems,andasaresult,theyprovideimportantinsightsintotheirbehavior.Thespectralanalysisofsingulardifferentialoperatorshasbeenanactiveresearchareaforseveraldecades;inthispaper,wefocusonthespectralanalysisofsingulardifferentialoperatorswithtransferconditionsandboundaryconditionscontainingcharacteristicparameters.
Background
Singulardifferentialoperatorsaredefinedaslineardifferentialoperatorswhosecoefficientsarenotsmoothintheneighborhoodofsomepoint(s).Theseoperatorshavevariousapplicationsinseveralfieldssuchasmechanics,electromagnetism,andmathematicalphysics.Singularoperatorsplayapivotalroleintheanalysisofdifferenttypesofsingularities,suchasconicalpoints,corners,edges,andcracks.Ingeneral,thesingularproblemscanbeclassifiedintothreecategories:interior,exterior,andboundary.
Thespectralanalysisofsingularoperatorsisachallengingtaskduetotheirnon-smoothnessandnon-differentialbehavior.Thepresenceofcharacteristicparametersinthetransferandboundaryconditionsmakestheproblemevenmorecomplex,astheycanchangethepropertiesofthespectrumoftheoperator.Thespectralanalysisofsingulardifferentialoperatorsisacrucialstepinsolvingthecorrespondingdifferentialequationsandunderstandingtheirphysicalimplications.
Methodology
Theanalysisofsingulardifferentialoperatorswithtransferconditionsandboundaryconditionscontainingcharacteristicparametersinvolvesseveralsteps.First,weneedtotransformthesingularoperatorintoaself-adjointornormaloperatortodetermineitsspectralproperties.Thetransformationcanbeachievedbymultiplyingtheoperatorsbyasuitableweightfunctionorbyintroducingauxiliaryvariables.Next,weneedtoidentifytheboundaryconditionsthatsatisfythetransferconditionsandthecharacteristicparameters.Theboundaryconditionsshouldensurethatthesolutionsarewell-defined