左定离散Sturm-liouville算子的谱问题.pdf

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1、†½lÑSturm-LiouvilleŽf̯KSpectralProblemsofLeft-DeniteDiscreteSturm-LiouvilleOperatorsƉ;’µA^êÆïÄ)µÜæ•“µ¤IûÇU9ŒÆnÆ"˜c8ÕM5(²<(²¤¥ÆØ©´<3“•e?1ïÄóŠÚïĤJ,Ø©¥AO±I5Ú—ƒ?,Ø©¥Ø•¹Ù¦<®²uL½>LïĤJ,•Ø•¹•¼U9ŒÆ½Ù¦˜ÅƽyÖ¦^Lá.†·˜ÓóŠÓ“éïĤ‰?Ûzþ®3Ø©¥Š²(`²¿L«¿.ÆØ©Šö¶µiFÏ:cFÆ

2、Ø©‡¦^ÇÖÆØ©Šö)U9ŒÆk'3!¦^ÆØ©5½.AÇU9ŒÆŒ±òÆØ©ܽܩSN?k'êâ¥?1u¢,¿æ^K

3、f¥Ä‘XêÚ³¼êŒ±Ø´HermitianÝ,¼êŒ±´Ø½,>Š^‡•Œ±Ø÷vÏ~¿ÂegŠ^‡.3dœ/e,·‚©Žf'u˜„g.gŠ^‡,„y²,lÑSturm-Liouville¯Kdu©Žf¯K¿‡^‡´§AŠØ´¤kEê.1nÙ̇ïĆ½lÑSturm-LiouvilleŽf̯K,ÏL½Â#SÈ,E#SȘm,Žf3d˜mþgŠ5.?y²§AŠ•¢êÄÌ5Ÿ.1oÙ•(åŠ,o)©óŠ.'…cµlÑSturm-Liouville¯K;†½;gŠ;̯K.AbstractInthispaper,spectr

4、altheoryofleft-denitediscreteSturm-Liouvilleoperatorswillbeconsidered.Thepaperismainlydividedintofourchapters.Therstsectionistheintroductionofthewholepaper.Wetalkabouttheback-groundofthisproblem,andmakeplansfortheresearchoftheproblems.Nextsectionofthepapermainlyintroducesdierenceoperator

5、s,andtheirself-adjointnessandinvestigatesaclassofself-adjointSturm-Liouvilledierenceoperatorswitheitheranon-Hermitianleadingcoecientfunction,oranon-Hermitianpotentialfunction,oranon-deniteweightfunction,oranon-self-adjointboundarycondition.Theminimumconditionsforsuchdierenceoperatorstob

6、eself-adjointwithrespecttoanaturalquadraticformisobtained.ItisshownthatadiscreteSturm-Liouvilleproblemadmitsadierenceoperatorrealizationifandonlyifitdoesnothaveallcomplexnumbersaseigenvalues.Thethirdsectionmainlyinvestigatesthespectraltheoryofleft-denitediscreteSturm-Liouvilleoperators.So

7、mefundamentalspectralresults,whichincludetheresultthatitseigenvaluesarereal,areobtainedbymeansofthedenitionofnewinnerproductandthestructureofself-adjointsubspace.Atlast,wesummarizetheresultsofthewholepaper.Keywords:discreteSturm-Liouvilleproblem;left-de