elementary_linear_algebra_for_advanced_spectral_problems

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1、ELEMENTARYLINEARALGEBRAFORADVANCEDSPECTRALPROBLEMSJOHANNESSJOSTRANDANDMACIEJZWORSKI¨1.IntroductionThepurposeofthisarticleistodiscussasimplelinearalgebraictoolwhichhasproveditselfveryusefulinthemathematicalstudyofspectralproblemsarisinginelecromagnetismandquan-tummech

2、anics.Roughlyspeakingitamountstoreplacinganoperatorofinterestbyasuitablychoseninvertiblesystemofoperators.Thatapproachhasaverylongtraditionandappearsconstantlyunderdifferentnamesandguisesinmanyworksofpureandappliedmathematics.Ourpurposehereisnottoprovideahistoricalsur

3、veybuttopresentanaccountofaspecificapproachfromapersonalperspectiveoftheauthors.Ononehandwehopetoprovideasourceofsystematicreferencesforthepractitionersofourtypeofspectraltheoryand,hopefully,toconvinceothersoftheusefulnessofthismethod.Wedonotknow,butfindveryinteresting

4、,ifthemethodwhichhasproveditselfsosuccessfulintheoreticalstudieshasachanceofbeingusefulnumerically.Thekeyelementaryobservationgoesback–atleast–toSchurandhiscomplementformula:ifformatrices−1PR−EE+=,R+0E−E−+thenPisinvertibleifandonlyifE−+isinvertibleand(1.1)P−1=E−E

5、E−1E,E−1=−RP−1R.+−+−−+−+InfacttheequivalenceofinvertibilitiesofPandE−+holdsforsystemswithanonzerolowerrighthandcorner(seeLemma3.1)butsinceherewealwaysstartwithPandchooseR±wecannormallyconsiderthesesimplersystems.Sometimes,inthecontextofindextheoryoneconsidersoperator

6、sPwhichareneverinvertible.InthatcasetheindexofPisequaltheindexofE−+whichistrivialtocomputeifE−+isamatrix–see§2.4.arXiv:math/0312166v1[math.SP]8Dec2003InthestudyoflinearpartialdifferentialequationstheuseofenlargedsystemsappearedinGrushin’swork[7]onhypoellipticoperators

7、.Inadifferentcontexttheywereusedinthethesisofthefirstauthor[17]andthe±notationcomesfromthere–see§2.2foranexplanationinthecontextoflinearalgebra.Asisseenthereitisessentialthatthesystemisnon-self-adjoint†.Forthathistorical,ifsomewhatpersonalreason,werefertotheproblemPu+

8、R−u−=v(1.2)R+u=v+P:H1→H2,R−:H−→H2,R+:H1→H+,†ThatdistinguishesitfromtheKKT(forKarush-Kuhn-Tucker)systemspopularinnumericalstudies–se