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1、RESEARCHANNOUNCEMENTSBULLETIN(NewSeries)OFTHEAMERICANMATHEMATICALSOCIETYVolume8,Number2,March1983INSTANTONS,DOUBLEWELLSANDLARGEDEVIATIONSBARRYSIMON1ABSTRACT.Wefindtheleadingasymptoticsoftheexponentiallysmallsplittingofthetwolowesteigenvaluesof—^A-f2VinthelimitasX—•oowhereVisanonneg
2、ativepotentialwithtwozeros.Inthisnote,weconsidereigenvaluesofaSchrödingeroperator—
3、A+2yinthelimitX—>oo(whichisquasiclassicalsinceuptoafactorofti2,—h2A+Vhasthisformwithh=X-1goingtozero).Themethodcandealwitheigenvaluesotherthanthelowest,withmultipleminima(evenmanifoldsofminima)ordege
4、nerateminima.Inthisnote,forsimplicitywediscussonlythetwolowesteigenvalues,E0()and£i(X),withcorrespondingnormalizedeigenvectorsHo(X),Qi(X).Moregeneralsituationsanddetailedproofswillappearelsewhere[13].WewillsupposethefollowingaboutV.(i)VisC00,(ii)V(x)>0forallxand^rnLx^00V(x)>0,(ii
5、i)Vvanishesatexactlytwopointsaandbandatthesepointsd2V/dXidXjisstrictlypositivedefinite.Underthesecircumstances,onecanprove(seee.g.[12])thatipoiscon•centratedasX—•oonearthepointsa,bandthatEi()/hasafinitenonzerolimit.Wewantalsotosupposethatforallesmallandfory=aandfory=b(1)Urnfipo{
6、x)2dx>0.Onecasewhere(1)holdsiswhenthereisasymmetryoforder2(suchasreflection)whichleavesVand—Ainvariant,sothatthelimitisfory=aorb.Underthesecircumstances,oneexpectsthatthesplittingofEandEQwillbegovernedbytunnelingandonegoalhereistoobtainmultidimensionalReceivedbytheeditorsOctober2
7、1,1982and,inrevisedform,November29,1982.1980MathematicsSubjectClassification.Primary35P15,81H99;Secondary60J65.ResearchpartiallysupportedbyUSNSFundergrantMCS-81-20833.©1983AmericanMathematicalSociety0273-0979/82/0000-1204/$01.75323324BARRYSIMONtunnelingresults.Abasicgeometricobjectw
8、hichwillenteristheAgmonmetricdefinedby(2)fav)=nf(f0^VHs))^s)dJci{0)=xtl(l)=y)thatisthegeodesicdistanceinthemetric2V(x)
9、dx
10、2.Itisanelementaryfact(seee.g.[3])that(3)^(x,y)=infQ/o%s)2ds+JoV(1(s))ds1(0)=x,1(T)=y).OnebasicresultisTHEOREM1.UndertheabovehypothesesonV(including(1))(4)xli
11、mj-iInEt()-B0(X)
12、
