Sturm-Liouville

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1、1Sturm-LiouvilleSystem1.1Sturm-LiouvilleDierentialEquationDenition1.Asecondordereddierentialequationoftheformddp(x)y+q(x)y=!(x)y;x2[a;b](1)dxdxwithp;qand!speciedsuchthatp(x)>0and!(x)>0forx2(a;b),iscalledaSturm-Liouville(SL)dierentialequation.NotethatSLdierentialequation

2、isessentiallyaneigenvalueproblemsinceisnotspecied.WhilesolvingSLequation,bothandymustbedetermined.Example2.TheSchrodingerequation~2+V(x)=E2monaninterval[a;b]isaSLdierentialequation.1.2BoundaryConditionsBoundaryconditionsforasolutionyofadierentialequationoninterval[a;b]arecl

3、assiedasfollows:MixedBoundaryConditionsBoundaryconditionsoftheformcy(a)+dy0(a)=aacy(b)+dy0(b)=(2)bbwhere,ca;da;cb;db;andareconstants,arecalledmixedDirichlet-Neumannboundaryconditions.Whenboth==0theboundaryconditionsaresaidtobehomogeneous.SpecialcasesareDirichletBC(da=db=0)andNe

4、umannBC(ca=cb=0)PeriodicBoundaryConditionsBoundaryconditionsoftheformy(a)=y(b)y0(a)=y0(b)(3)arecalledperiodicboundaryconditions.Example3.Someexamplesare1.Stretchedvibratingstringclampedattwoends:DirichletBC2.Electrostaticpotentialonthesurfaceofavolume:DirichletBC3.Electrostatice

5、ldonthesurfaceofavolume:NeumannBC4.Aheat-conductingrodwithtwoendsinheatbaths:DirichletBC11.3Sturm-LiouvilleSystem(Problem)Denition4.TheSLdierentialequationonaniteinterval[a;b]withhomogeneousmixedboundaryconditions,thatis,ddp(x)y+q(x)y=!(x)y;x2[a;b]dxdxcy(a)+dy0(a)=0aacy(b)

6、+dy0(b)=0bbwithp(x)>0and!(x)>0forx2[a;b]iscalledasregularSturm-Liouvillesystem(orproblem).Aimistondallvaluesforwhichanontrivialsolutionyexists.Itisimplicitlyassumedthatyanditsderivativearecontinuouson[a;b],whichalsomeansthesearebounded.Example5.Quantumparticleina1Dbox:TheSchr

7、odingerequationandboundaryconditionsaregivenby~2(x)=E(x)x2[0;L]2m(0)=0(L)=0:ThisisaregularSturm-Liouvillesystem.Theeigenvaluesandeigenfunctionsare~22n2En=2mL2r2nxn(x)=sinLLwithn=1;2;:::.Denition6.ASLdierentialequationonaninterval[a;b]withperiodicboundaryconditionsandp(a)=p

8、(b)iscalledasperiodicSturm-Liouvillesyst