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1、TRANSACTIONSOFTHEAMERICANMATHEMATICALSOCIETYVolume360,Number12,December2008,Pages6545–6554S0002-9947(08)04473-5ArticleelectronicallypublishedonJune26,2008APROBABILISTICAPPROACHTOBOUNDED/POSITIVESOLUTIONSFORSCHRODINGEROPERATORS¨WITHCERTAINCLASSESOFPOTENTIALSROSSG.PINSKYAbstract.Cons
2、idertheequation∗1d()∆u−Vu=0inR,2ford≥3.ForcertainclassesofpotentialsV,weuseprobabilistictoolstostudytheboundedsolutionsandthepositivesolutionsfor(*).Aprimarymotivationistoofferprobabilisticintuitionfortheresults.1.IntroductionandstatementofresultsInthispaperweuseprobabilistictoolsto
3、studytheboundedsolutionsandthepositivesolutionsforSchr¨odingeroperatorswithcertainclassesofpotentialsinRd.Werestricttod≥3becausethequestionsweaskhavetrivialanswersford=1,2.Aprimarymotivationhereistoofferprobabilisticintuitionfortheresults.Weconsiderclassicalsolutionstotheequation1d(
4、1.1)∆u−Vu=0inR,2whereVisH¨oldercontinuous.Asusual,weletV+=V∨0andV−=−(V∧0),sothatV=V+−V−.Toavoidtrivialities,weassumethatV≡0.LetX(t)bead-dimensionalBrownianmotion.WhentheBrownianmotionstartsfromx∈Rd,wedenoteprobabilitiesbyPxandthecorrespondingexpectationsbyEx.RecallthattheGreen’sfu
5、nctionfor1∆inRd,d≥3,isgivenbyG(x,y)=22
6、y−x
7、2−d,whereωisthesurfacemeasureoftheunitsphereinRd.Asis(d−2)ωddwellknown[3],foranynonnegativefunctionφ,onehas∞2φ(y)(1.2)Exφ(X(t))dt=G(x,y)φ(y)dy=dy.0Rd(d−2)ωdRd
8、y−x
9、d−2ThefollowingassumptionwillalwaysbemadeonthenegativepartV−ofthepotentia
10、lV:Assumption1.∞−−2V(y)supExV(X(t))dt=supdy<1.x∈Rd0(d−2)ωdx∈RdRd
11、y−x
12、d−2ReceivedbytheeditorsJune26,2006and,inrevisedform,January16,2007.2000MathematicsSubjectClassification.Primary60H30,35J10.Keywordsandphrases.Liouvilletheorem,boundedsolutions,positivesolutions,Schr¨odingerequati
13、on.c2008AmericanMathematicalSocietyRevertstopublicdomain28yearsfrompublication6545Licenseorcopyrightrestrictionsmayapplytoredistribution;seehttp://www.ams.org/journal-terms-of-use6546ROSSG.PINSKYByKhasminskii’slemma[14,LemmaB.1.2],Assumption1guaranteesthat∞(1.3)supEexp(V−(X(t))dt
14、)<∞.xx∈Rd0Remark.Actuallyi
