Path Integral Solution of Linear Second Order Part

arXiv:math-ph/0405032v110May2004PathIntegralSolutionofLinearSecondOrderPartialDifferentialEquationsII.Elliptic,ParabolicandHyperbolicCasesJ.LaChapelleAbstractAtheoremthatconstructsapathintegralsolutionforgeneralsecondorderpartialdifferentialequationsisspecializedtoobtainpathintegralsthataresolutionsofelliptic,parabolic,andhyperboliclinearsecondorderpartialdifferentialequationswithDirichlet/Neumannboundaryconditions.Theconstructionischeckedbyeval-uatingseveralknownkernelsforregionswithplanarandsphericalboundaries.Somenewcalculationaltechniquesareintroduced.Keywords:partialdifferentialequations,pathintegrals,functionalintegrationPACS:2.30.Cj,2.30.Jr1IntroductionThegeneralpathintegraldevelopedin[1]isspecializedtoyieldpathintegralsolutionstoelliptic,parabolic,andhyperboliclinearsecondorderPDEs.TheexamplesinSection3provideexplicitrealizationsoftheconstructionandcanbecomparedtoknownsolutions.Somenewcalculationaltechniquesareintroducedwhichmayoffersomeadvantageinnumericalmethods.Materialandnotationpresentedin[1]willbeassumedhere.Thereaderrequiringmotivationtofullydigest[1]maywishtobeginwithperhapsmorefamiliarmaterialintheexamplespresentedhereinSubsections3.1(thePoisson,diffusion/Schr¨odinger,andwaveequationsinunboundedspace),3.2.1(thediffusionequationinthehalf-plane),and3.3.1(theLaplace/PoissonequationsforaballinRn).Asin[1]theissuesofexistenceanduniquenessarenotaddressed;andfunctions,distributions,boundaries,etc.aregenerallyassumedtobewell-definedinanygivencase.Emailaddress:[e-mail](J.LaChapelle).PreprintsubmittedtoAnnalsofPhysics4February20082PathIntegralSolutionofPDEs2.1GeneralsolutionForreferencepurposes,thetheoremof[1]isstatedwithoutproof.Relevantdefinitionsandnotationcanbefoundin[1].Theorem2.1LetMbeareal(complex)m-dimensional(m≥2)paracompactdif-ferentiablemanifoldwithalinearconnection,andletUbeaboundedorientableopenregioninMwithboundary∂U.1Letfandϕbeelementsofthespaceofsectionsorsectiondistributionsofthe(r,s)-tensorbundleoverM.Assumegiventhefunc-tionalS(x(τa′,z))whoseassociatedbilinearformQsatisfiesRe(Q(x(τa′,z))0for(x(τa′,z))6=0.Ifχ(xa·Σ(hτ′,τi,z))∈FR(Ω)whereχ(xa·Σ(hτ′,τi,z)):=ZC+θ(hτ′,τi−τa′)f(xa·Σ(τa′,z))exp{−S(x(τa′,z))}dτa′+ϕ(xa·Σ(hτ′,τi,z))exp{−S(x(hτ′,τi,z))};(2.1)then,forxa=x(τa)∈U,Ψ(xa)=ZΩχ(xa·Σ(τ⊥xa,z))DΩ(2.2)isasolutionoftheinhomogeneousPDEGαβ4πLX(α)LX(β)+LY+V(x)#x=xaΨ(xa)=−f(xa)(2.3)withboundaryconditionΨ(xB)=ϕ(xB).(2.4)2.2EllipticPDEsHenceforth,restricttothecasewhereC+=R+orC+=iR.Writeτ=sbτwiths∈{1,i}andbτ∈R+orbτ∈Rwhicheverthecasemaybe.ForsimplicityIwillcontinuetowriteτinplaceofbτwiththeunderstandingthatτisrealinthiscontext.AlthoughTheorem2.1holdsmoregenerally,itisappropriatetorendertheconstructionmore1Theboundary∂Uisassumedtobesufficientlyregular.2accessibletostandardapplicationsandtoconnectwiththenotationof[6],[9],and[7].EllipticPDEsarecharacterizedbyclosedboundarieswithDirichlet/Neumannbound-aryconditionsandapositivedefiniteQ.ItisasimplemattertospecializeTheorem2.1forthiscase:Corollary2.1GiventhehypothesesofTheorem2.1,maketwoqualifications:LetUbeaboundedorientableregioninMwithclosedboundary∂UandassumethematrixGαβhasindex(d,0).Ifχ(E)(xa·Σ(hτ′,τi,z))∈FR(Ω)whereχ(E)(xa·Σ(hτ′,τi,z)):=ZC+θ(s[hτ′,τi−τa′])f(xa·Σ(τa′,z))exp{−s−1S(x(τa′,z))}dτa′+φ(xa·Σ(hτ′,τi,z))exp{−s−1S(x(hτ′,τi,z))};(2.5)then,forxa∈U,Ψ(E)(xa)=ZΩχ(E)(xa·Σ(τ⊥xa,z))DΩ,(2.6)isasolutionoftheinhomogeneousellipticPDEs24πGαβLX(α)LX(β)+sLY+V(x)#x=xaΨ(E)(xa)=−f(xa)(2.7)withboundaryconditionΨ(E)(xB)=φ(xB).(2.8)ThecorollaryclearlyfollowsfromTheorem2.1.Theonlysubtletyiskeepingtrackoffactorsofs.ThedomainofintegrationC+isnowR+orRdependingonwhethers=1ors=i.ThekernelsforDirichlet/NeumannboundaryconditionscanbeobtaineddirectlyfromSection3.2of[1]andwillnotberepeatedhere.Asanexample,iftheboundaryofUisatinfinity,V(x)→V(x)+2πEandφ=0,then(2.6)canbewrittenastheFourier/LaplacetransformofapathintegralsolutionofaninhomogeneousparabolicPDE.Thekernelof(2.7),forthiscasewithvanishingDirichletboundaryconditions,isK(D)U(xa,xa′;E)=ZZdaZC+δ(xa·Σ(τa′,z),xa′)expn−s−1S(x(τa′,z);E)odτa′DQ,Wz.(2.9)Inquantumphysics,(2.9)withs=iisthefixed-energyGreen’sfunctionofthetime-independentinhomogeneousSchr¨odingerequationwhentheboundaryisatinfinity.ItistheFourier/Laplacetransformoftheposition-to-positiontransitionamplitude3K(xa′,τa′;xa,τa)associatedwiththetime-dependentSchr¨odingerequation.Asprevi-ouslymentioned,theFourier/LaplacetransforminterpretationbecomesaLagrangemultiplierinterpretationforphasespaceconstructions.Thatis,(2.9)canbeused([7])togiveaphasespacefixed-energyGreen’sfunction.Remark:In[7],theintegratorDτwaschosentobeGaussian,whichinhindsightwasnotagoodchoice.However,comparing[7]withthepresentconstructionshowsthatthetheoremin[7]stillholdsprovidedDτisagammaintegrator.InlightofCorollary2.1,thepathintegralinthetheoremin[7]solvestheinhomogeneousellipticPDEwithvanishingboundaryconditionsatinfinity.Consequently,thefixed-energyGreen’sfunctioncalculatedin[7]istheelementarykernelwithDirichletboundaryconditionsatinfinityandnottheboundarykernel.2.3ParabolicPDEsThepathintegralsol