Table Of Content

Lecture Notes in Mathematics 2323 Alan Carey Galina Levitina Index Theory Beyond the Fredholm Case Lecture Notes in Mathematics Volume 2323 Editors-in-Chief Jean-MichelMorel,CMLA,ENS,Cachan,France BernardTeissier,IMJ-PRG,Paris,France SeriesEditors KarinBaur,UniversityofLeeds,Leeds,UK MichelBrion,UGA,Grenoble,France AnnetteHuber,AlbertLudwigUniversity,Freiburg,Germany DavarKhoshnevisan,TheUniversityofUtah,SaltLakeCity,UT,USA IoannisKontoyiannis,UniversityofCambridge,Cambridge,UK AngelaKunoth,UniversityofCologne,Cologne,Germany ArianeMézard,IMJ-PRG,Paris,France MarkPodolskij,UniversityofLuxembourg,Esch-sur-Alzette,Luxembourg MarkPolicott,MathematicsInstitute,UniversityofWarwick,Coventry,UK SylviaSerfaty,NYUCourant,NewYork,NY,USA László Székelyhidi , Institute of Mathematics, Leipzig University, Leipzig, Germany GabrieleVezzosi,UniFI,Florence,Italy AnnaWienhard,RuprechtKarlUniversity,Heidelberg,Germany This series reports on new developments in all areas of mathematics and their applications-quickly,informallyandatahighlevel.Mathematicaltextsanalysing newdevelopmentsinmodellingandnumericalsimulationarewelcome.Thetypeof materialconsideredforpublicationincludes: 1.Researchmonographs 2.Lecturesonanewfieldorpresentationsofanewangleinaclassicalfield 3.Summerschoolsandintensivecoursesontopicsofcurrentresearch. Textswhichareoutofprintbutstillindemandmayalsobeconsiderediftheyfall withinthesecategories.Thetimelinessofamanuscriptissometimesmoreimportant thanitsform,whichmaybepreliminaryortentative. Titles from this series are indexed by Scopus, Web of Science, Mathematical Reviews,andzbMATH. Alan Carey (cid:129) Galina Levitina Index Theory Beyond the Fredholm Case AlanCarey GalinaLevitina SchoolofMathematicsandApplied MathematicalSciencesInstitute Statistics AustralianNationalUniversity UniversityofWollongong Canberra,ACT,Australia Wollongong,NSW,Australia ISSN0075-8434 ISSN1617-9692 (electronic) LectureNotesinMathematics ISBN978-3-031-19435-1 ISBN978-3-031-19436-8 (eBook) https://doi.org/10.1007/978-3-031-19436-8 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface These notes are about extending index theory to some examples where non- Fredholm operators arise. They are far from comprehensive. We have focussed on one aspect of the problemof what replacesthe notion of spectral flow and the Fredholmindexwhentheoperatorsinquestionhavezerointheiressentialspectrum. Most work in this topic stems from the so-called Witten index that is discussed at length here. The new direction described in these notes is the introduction of ‘spectralflowbeyondtheFredholmcase’. Spectral flow was introduced by Atiyah-Patodi-Singer [APS76] for elliptic operators on odd dimensional compact manifolds. They argued that it could be computed from the Fredholm index of an elliptic operator on a manifold of one higher dimension. A general proof of this fact was produced by Robbin-Salamon [RS95]butwasrestrictedtooperatorswithdiscretespectrumasoccurinthestudy ofellipticoperatorsoncompactmanifolds. In [GLM+11], a start was made on extending the Robbin-Salamon theorem to operators with some essential spectrum as occurs on non-compact manifolds. Thenewingredientintroducedtherewastoexploitscatteringtheoryfollowingthe fundamental paper [Pus08]. These results do not apply to differential operators directly, only to pseudo-differentialoperators on manifolds, due to the restrictive assumption that spectral flow is only considered between an operator and its perturbationbyarelativelytrace-classoperator. In these lecture notes, we give an expository account of the main results of these earlier papers and their generalisation to the study of spectral flow between anoperatoranda perturbationsatisfyinga higherpth Schattenclassconditionfor 0 ≤ p < ∞, thus allowing differentialoperatorson manifolds of any dimension d < p +1. To achieve this, we establish an operator trace formula that does not assumeanyellipticityorFredholmpropertiesatall.Thisoperatortraceformulais motivatedbyBenameuretal.[BCP+06]andCareyetal.[CGK16].Wethenexplain someapplicationstoindextheoryandspectralflow.Examplesmaybeobtainedby using Dirac type operators on L (Rd) for arbitrary d ∈ N (see Sect.7.1). In this 2 setting, Theorem 6.2.2 substantially extends [CGG+16, Theorem 3.5], where the cased =1wastreated. v vi Preface We now briefly explain the central point of these notes using notation listed in the Notations. Our discussion focuses on a family {B(t)}t∈R of bounded self- adjointoperatorsontheHilbertspaceHthatarep-relativetrace-classperturbations of an unbounded self-adjoint operator A− (see Hypothesis 3.2.5 for the precise assumptions).DenotebyB+ theuniformnormasymptoteast → ∞ofthefamily {B(t)}t∈R. Introduce the model operator DA in L2(R;H) by first defining A in L2(R;H)as(Af)(t)=(A−+B(t))f(t)andthensetting d DA = +A, dom(DA)=W1,2(R;H)∩dom(A−). dt ThenwithA+ = A−+B+,andAs = A−+sB+,s ∈ [0,1],weobtainthemain fact,namely,whatwecallthe‘principaltraceformula’forthesemigroupdifference forallu>0, (cid:2) (cid:3) (cid:2) (cid:3) ˆ (cid:2) (cid:3) tr e−uDAD∗A −e−uD∗ADA =− u 1/2 1tr e−uA2s(A+−A−) ds, π 0 where,asaresultofourassumptions,theoperatordifferenceunderthetraceonthe left-handsideisindeedtrace-classandtheintegralontheright-handsideconverges. Toappreciatethesignificanceofthisprincipaltraceformula,thesenotesdevelop severalapplications.Forthisweexploitsomerecentdevelopmentsinthetheoryof Krein’s spectral shift function from quantum mechanicalscattering theory. Going backtothemotivatingworkofPushnitski,wediscusstheproofofageneralisation ofhisresultthatrelatesthespectralshiftfunctionofthepair(A+,A−)withthatfor the pair (H ,H ) (see Theorem6.3.1).Thenwe proveresults that relate spectral 2 1 flow along the path {A− + B(t)}∞t=−∞ to the Fredholm or Witten index of DA in the sense that we establish a theorem of Robbin-Salamon type for sufficiently regular paths of self-adjoint Fredholm operators joining operators A± that have some essential spectrum outside zero. An exposition of the theory of the spectral shiftfunctionisincluded. As the principaltrace formulaappliesalso to paths of non-Fredholmoperators joiningA±,weobtainaformulafortheWittenindexunderregularityassumptions ontherespectivespectralshiftfunctions.Thisleadsustointroduceanotionof‘gen- eralised spectralflow’ for such pathsand to investigateits properties.Importantly weconsiderarangeofexamplestowhichthesemoreabstractconsiderationsapply. Properties of this extended notion of spectral flow are described, and their proofs exploitthedoubleoperatorintegraltechniquedueoriginallyto[BS66].Weinclude inthesenotessomebackgroundondoubleoperatorintegralsincludingasimplified introduction. Onemaythinkoftheseresultsinthefollowingterms.TheAtiyah-Singerindex theoremgivesatopologicalexpressionfortheFredholmindexofellipticoperators. Forthe case of Diracoperatorson evendimensionalcompactmanifolds,one may formulate the index theorem using a model operator of the same general form as DA.Thusthe choiceofthis‘modeloperator’isnotarbitrarybutisadaptedtothe Preface vii classicalsetting.Wealsonotethatinsupersymmetricquantummechanicsasstudied byWitten[Wit82],theHamiltoniansinvolvethesametypeofmodeloperator. Forthegeneraloperatorsweconsiderhere,thatmayhavezerointheiressential spectrum,thereisnotopologicalformulaforeithertheWittenindexorgeneralised spectral flow. Rather, what we find is that scattering theory, which is naturally relevantinunderstandingtheessentialspectrum,providesthroughthespectralshift function,expressionsforbothofthese quantitiesandalso connectsthem.Inorder to understandthispointof view,one also needsthe backgroundmaterialincluded inthesenotesonthespectralshiftfunction,theanalyticapproachtospectralflow, theWittenindexandthedoubleoperatorintegraltechnique. These lecture notes are based in part on our recent papers [CLPS22] and [CGL+22] and also on lectures given at meetings in BIRS, Santiago and Mün- ster. We have, however, incorporated background from other sources. There are expository sections in each chapter that also include some survey material from the literature going back to the origins of the topics covered. Conversely some technical details are omitted where they may be found in [CLPS22] and do not addsubstantiallytothediscussion. Wollongong,NSW,Australia AlanCarey Canberra,ACT,Australia GalinaLevitina Acknowledgements A.C.thankstheErwinSchrödingerInternationalInstituteforMathematicalPhysics (ESI), Vienna, Austria, for fundingsupportfor this collaborationin the form of a Research-in Teams project, ‘Scattering Theory and Non-Commutative Analysis’, forthedurationofJune22toJuly22,2014.G.L.isindebtedtoGeraldTeschland theESIforakindinvitationtovisittheUniversityofVienna,Austria,foraperiod of 2 weeks in June/July of 2014. The authors thank BIRS for funding a focussed researchgrouponthetopicoftheselecturenotesinJune2017andalsogratefully acknowledgefinancialsupportfromtheAustralianResearchCouncil.A.C. thanks the Humboldt Stiftung and the University of Münster Mathematics Institute for support.G.L.acknowledgesthesupportofAustraliangovernmentscholarships.We thank Fritz Gesztesy, Jens Kaad, Harald Grosse, Denis Potapov, Fedor Sukochev and Dmitriy Zanin for many conversations on the subject matter of these notes. Finallywearegratefultotherefereesforcarefullyreadingthemanuscript. ix Notations Herewesummarisenotationthatisusedwithoutcommentinthetext. For a Banach space X, we denote by L(X) the algebra of all linear bounded operatorsonX.Wedenoteby1theidentitymappingonX. In case whenX = H isa separablecomplexHilbertspace H, we usenotation (cid:8)·(cid:8)fortheuniformnorm. For an operatorA ∈ L(H) we denote by Re(A) (respectively,Im(A)) the real (respectively,imaginary)partofA. We write K(H) for the compact operators on H. The Calkin algebra is the quotientalgebraL(H)/K(H). Thecorresponding(cid:3) -basedSchatten–vonNeumannidealsonHaredenotedby p L (H),withassociatednormabbreviatedby(cid:8)·(cid:8) ,p≥1.Moreover,tr(A)denotes p p theclassicaltraceofatrace-classoperatorA∈L (H). 1 We use the symbols n-lim and s-lim to denote the operator norm limit (i.e., convergenceinthetopologyofL(H)),andtheoperatorstronglimit. If T is a linear operator mapping (a subspace of) a Hilbert space into another, thendom(T)andker(T)denotethedomainandkernel(i.e.,nullspace)ofT.The closureofaclosableoperatorS isdenotedbyS. ThespectrumandresolventsetofaclosedlinearoperatorinHwillbedenotedby σ(·),andρ(·),respectively.Theessential(respectively,point)spectrumofaclosed linearoperatorinHisdenotedbyσ (·)(respectively,byσ (·)). ess p The spectral projectionsof a self-adjointoperatorS in H we denote by E (·). S The space of all (essentially) bounded dE -measurable functions is denoted by S L∞(R,dES). Thenotation[·,·]standsforcommutatoroftwooperators,thatis [A,B]=AB−BA, wheneveritisclearhowtodealwithdomainsofpossibleunboundedoperators. ForaFredholmoperatorT,wedenotebyindex(T)itsFredholmindex. Unlessexplicitlystatedotherwise,wheneverwewriteL (Rd)(L (0,∞)etc.) p p we assume the classical Lebesgue measure on Rd ((0,∞) etc.). The space of all xi