Some examples in one-dimensional geometric scatter

Someexamplesinone-dimensional\geometricscatteringonmanifoldsAlexanderKiselevDivisionofMathematics,PhysicsandAstronomyCaliforniaInstituteofTechnology,253-37Pasadena,CA91125AbstractWeconsider\geometricscatteringforaLaplace-BeltramioperatoronacompactRiemannianmanifoldinsertedbetweentwohalf-lines.Wediscussapplicabilityandcorrectnessofthismodel.Withanexample,weshowthatsuchscatteringproblemmayexhibitunusualproperties:thetransitioncoecienthasasequenceofsharppeakswhichbecomemoreandmoredistantathighenergyandotherwiseturnstozero.IntroductionInthispaperweconsidercertainboundaryvalueproblems,namely,theone-dimensionalscatteringontwo-orthree-dimensionalcompactRiemannianmanifolds.Themotivationforstudyingtheseproblemsistwofold.TherstcomesfromthepaperofJ.Avron,P.ExnerandY.Last[2],whodicussedtheproblemofapproximatingthescatteringpropertiesof0-potentialbycertaingraphs.By0-potentialwemeantheboundaryconditionofthetypeu0(+a)=u0(a);u(+a)u(a)=u0(a)imposedonthefunctionsfromthedomainoftheoperatoratthegivenpointa.Werecallthatforone-dimensionalscatteringonthelineonasucientlyrapidlydecayingpotential,itistypicalthatthetransitioncoecient1tendstooneathighenergies.Ontheotherhand,the0-potentialhasinthisrespectverydierentproperties:thetransitioncoecientt()turnsto0as!1attherate1=2:Thisfeatureleadstoimportantspectralconsequencesfor0Wannier-Starkladders.Namely,theabsolutelycontinuousspectruminasystemofperiodic0interactionswithelectriceldisvoid[4]underverygeneralassumptionsonthecouplingconstants.Thisisaveryspecialphenomenainthesensethatonecannothavesuchaneectif0isreplacedbysomesmoothpotentialwithcompactsupport.Avron,ExnerandLastsuggestedthatcertaingeometricscatterersmaypossesstheprop-ertiessimilartothoseof0:In[2],theyshowedthatthetransitioncoecientforan\onion{Nsegmentsofequallengthlgluedtogetherattwopoints,towhichtwohalf-axesareattached{mayapproximatethebehaviorof0transitioncoecientonanarbitrarylargescaleofenergy,ifoneadjustsNandlinasuitableway.Eventually,however,asistypicalforallgraphsofthistype[6],thebehaviorofthetransitioncoecientisperiodicinenergy.Theexamplewewillconsiderinthispapershowsthatonecanndsomegeometricstructuresforwhichthetransitioncoecientwillingeneraldecayatinnity,withoutbeingperiodic.Therewillbepresentasequenceofsharpresonances,however{aninterestingphenomenaonitsown.Thesecondmotivationcomesfromthefactthatinsomesituations,theboundaryvalueproblemsweconsiderhereareknowntoapproximate,inacertainsense,thecorresponding\realboundaryvalueproblems.InoursituationitwouldbeascatteringproblemforaLaplace-Beltramioperatoronthemanifoldwithtwothinhalf-innitetubesattachedtoit.WewilldiscusstheknownresultsinmoredetailinSection1,whenweexplainthechoiceoftheboundaryconditionsatthecontactpoints.Wedonotexpect,however,astraightforwardanalogyinthecasewestudyhere,sincebytakingalineinsteadofatubeweignoretheeectsduetothetransversemodesexistinginthetube,whichisessentialforhigh-energylimit.Itisreasonable,however,toexpectthatthemodelproblemwillapproximatethereal2oneonacertainscaleofenergiesrelatedtothesizeofthechannelinarealproblem.Thepaperisorganizedasfollows:inSection1,weformallydenetheLaplaceoperatoronourdomainanddiscussnaturalrestrictionsandchoiceoftheboundaryconditions.InSection2,wecomputethetransitioncoecientinageneralsituation.Section3isdevotedtoaspecicexample,whenwetakeasphereasamanifold.SetupoftheproblemWewouldliketoconsiderascatteringproblemforaLaplacianonthecompactsmoothRiemannianmanifoldMinsertedbetweentwohalf-linesR+andR:Wewillassume,unlessstatedotherwise,thatMisaC1manifoldwithoutboundary.Letx1;x2bethepointsatwhichthehalf-linesjointhemanifold:x1=R\Mandx2=R+\M:WedenotebytheunionR[R+[M:MwillstayfortheLaplace-BeltramioperatoronMandDfortheoperatorsofdoubledierentiationonRwithDirichletboundaryconditionsatx1orx2correspondingly.TodeneaLaplaceoperatoronthedomain;weproceedinaclassicway:rstdenesomesymmetricoperatorgivenbythedierentialexpressionofLaplacianonthesetofinnitelydierentiablefunctionsvanishingintheneighborhoodofthespecialpointsx1;x2andthenconsideritsself-adjointextensions.Namely,onthesetoffunctionsC10(D;x1;x2)=C10(R+)[C10(R)[C10(M;x1;x2)describedabove,wedeneanoperator0;whichactsonthefunctionfasfollows:0f(x)=8:d2dx2f(x);x2R+orR;Mf(x);x2M:Hence,0actsasadoubledierentiationoperatoronhalf-axesandasaLaplace-BeltramioperatoronM:Theclosureofthisoperator0isclearlysymmetricbutnotself-adjoint.Amongtheobviousself-adjointextensionsof0istheoperatorDMD+;andtheoperatorswithotherthanDirichletboundaryconditionsatx1;x2;butstillwithoutinterac-tionbetweenhalf-linesandthemanifold.WeareofcourseinterestedintheLaplacianson3notsplittingintoadirectsumwithsummandsactingondierentgeometriccomponents.Therefore,welookforsomeboundaryconditionsatthepointsx1;x2otherthanleadingtoorthogonalsums.TherangeoftheapplicabilityofourmodelisgivenbythefollowingProposition1.1.SupposethatdimM=2or3:Thentheclosure0isasymmetricopera-torwithdeciencyindices(4;4):IfMhashigherdimension,thedeciencyindicesare(2;2):Proof.Thefunctio