Sturm–Liouville Problems

1 sturm liouville ProblemsMore general eigenvalue problemsSo far all of our example PDEs have led to separated equationsof the formX + 2X= 0, with standard Dirichlet or Neumann boundary conditions. Notsurprisingly, more complicated equations often come up in practical Problems . Forexample, if the medium in a heat or wave problem is spatially inhomogeneous,* therelevant equation may look likeX V(x)X= 2 Xfor some functionV, or evena(x)X +b(x)X +c(x)X= , if the boundary in a problem is a circle, cylinder, or sphere, the solution of theproblem is simplified by converting to polar, cylindrical, or spherical coordinates,so that the boundary is a surface of constant radial coordinate.

2 This simplificationof the boundary conditions is bought at the cost of complicating the differentialequation itself: we again have to deal with ODEs with nonconstant coefficients,such asd2 Rdr2+1rdRdr n2r2R= good news is that many of the properties of Fourier seriescarry over tothese more general situations. As before, we can consider theeigenvalue problemdefined by such an equation together with appropriate boundary conditions: Findall functions that satisfy the ODE (foranyvalue of ) and also satisfy the bound-ary conditions. And it is still true (under certain conditions) that the set of alleigenfunctions iscomplete:Anyreasonably well-behaved function can be expandedas an infinite series where each term is proportional to one ofthe is what allows arbitrary data functions in the originalPDE to be matched toa sum of separated solutions!

3 Also, the eigenfunctions areorthogonalto each other;this leads to a simple formula for the coefficients in the eigenfunction expansion,and also to a Parseval formula relating the norm of the function to the sum of thesquares of the coefficients.*That is, the density, etc., vary from point to point. This is not the same as nonhomogeneous in the sense of the general theory of linear differential basesConsider an interval [a,b] and the real-valued (or complex-valued) functionsdefined on it. A sequence of functions{ n(x)}is calledorthogonalif ba n(x)* m(x)dx= 0 wheneverm6= is calledorthonormalif,in addition, ba| n(x)|2dx= normalization condition is merely a convenience; the important thing isthe orthogonality.

4 If we are lucky enough to have an orthogonal set, we can alwaysconvert it to an orthonormal set by dividing each function bythe square root of itsnormalization integral: n(x) n(x) ba| n(z)|2dz ba| n(x)|2dx= , in certain cases this may make the formula for nmore complicated, sothat the redefinition is hardly worth the effort. A prime example is the eigenfunc-tions in the Fourier sine series: n(x) sinnx 0| n(x)|2dx= 2;therefore, n(x) 2 sinnxare the elements of the orthonormal basis. (This is the kind of normalization oftenused for the Fourier sinetransform, as we have seen.) A good case can be made,however, that normalizing the eigenfunctions is more of a nuisance than a help inthis case; most people prefer to put the entire 2/ in one place rather than put halfof it in the Fourier series and half in the cofficient letf(x) be an arbitrary (nice) function on [a,b].

5 Iffhas an expansionas a linear combination of the s,f(x) = n=1cn n(x),then ba m(x)*f(x)dx= n=1cn ba m(x)* n(x)dx=cm ba| m(x)|2dx2by orthogonality. If the set is orthonormal, this just sayscm= ba m(x)*f(x)dx.( )(In the rest of this discussion, I shall assume that the orthogonal set is orthonor-mal. This greatly simplifies the formulas of the general theory, even while possiblycomplicating the expressions for the eigenfunctions in anyparticular case.)Remark:The integral on the right side of ( ) is called theinner productof mandfand is often written ash m,fi. It can be thought of as the generalizationto an infinite-dimensional and complex vector space of the familiar dot product ofvectors inR3.

6 Then the normalization integral | n|2dx=h n, ni k nk2isthe analog of the square of thelengthof a vector, and thedistancebetween twovectorsfandgiskf gk, as applied in the next two paragraphs. For more on thislinear-algebra analogy, see pp. 168 169 in Appendix B of thenotes (pp. 26 27 ). The two previous pages of that Appendix give a simple example of acomplete set of functions and a set that fails to be complete because some vectorshave been left can easily be shown (just as for Fourier series) that ba|f(x)|2dx= n=1|cn| is theParseval equationassociated to this orthonormal set. Furthermore, iffisnotof the form n=1cn n(x), then(1) n=1|cn|2< ba|f(x)|2dx(calledBessel s inequality), and (2) thebest approximationtof(x) of the form cn n(x) is the one where the coefficients are computed by formula ( ).

7 Theselast two statements remain true when{ n}is afiniteset in which case, obviously,the probability that a givenfwill not be exactly a linear combination of the sis greatly increased. The precise meaning of (2) is that the choice ( ) of thecnminimizes the integral ba f(x) n=1cn n(x) is, we are talking aboutleast squares approximation. It is understood in thisdiscussion thatfitself is square-integrable on [a,b]. Recall that the space of suchfunctions is calledL2(or, more specifically,L2(a,b)).3 Now suppose thateverysquare-integrablefis the limit of a series n=1cn n.(This series is supposed to converge in the mean that is, the least-squaresintegral ba f(x) M n=1cn n(x) 2dxfor a partial sum approaches 0 asM.)

8 Then{ n}is called acomplete setor anorthonormal basis. This is the analogue of the mean convergence theoremfor Fourier series. Under certain conditions there may alsobe pointwise or uni-form convergence theorems, but these depend more on the special properties of theparticular functions being far this is just a definition, not a theorem. To guarantee that our orthonor-mal functions form a basis, we have to know where they came from. The miracle ofthe subject is that the eigenfunctions that arise from variable-separation problemsdoform orthonormal bases: sturm liouville theoryTheorem:Suppose that the ODE that arises from some separation of variablesisL[X] = Xon (0,L),( )whereLis an abbreviation for a second-orderlinear differential operatorL[X] a(x)X +b(x)X +c(x)X,a,b, andcare continuous on [0,L], anda(x)>0 on [0,L].

9 Suppose further that L0(L[u](x))*v(x)dx= L0u(x)*(L[v](x))dx( )for all functionsuandvsatisfying the boundary conditions of the problem . (Interms of the inner product inL2, this condition is justhLu,vi=hu,Lvi. Anoperator satisfying this condition is calledself-adjointorHermitian. ) Then:(1) All the eigenvalues are real (but possibly negative).(2) The eigenfunctions corresponding to different s are orthogonal: L0 n(x)* m(x)dx= 0 ifn6=m. There is a technical distinction between these two terms, but it does not matter forregular sturm liouville (3) The eigenfunctions are complete. (This implies that thecorresponding PDEcan be solved for arbitrary boundary data, in precise analogy to Fourier seriesproblems.)

10 The proof that a givenLsatisfies ( ) (or doesn t satisfy it, as the case may be)involves integrating by parts twice. It turns out that ( ) will be satisfied ifLhasthe formddxp(x)ddx+q(x)(withpandqreal-valued and well-behaved, andp(x)>0) and the boundary con-ditions are of the type X (0) X(0) = 0, X (L) + X(L) = 0with , etc., real.* Such an eigenvalue problem is called aregular sturm proof of the conclusions (1) and (2) of the theorem is quite simple andis a generalization of the proof of the corresponding theorem for eigenvalues andeigenvectors of asymmetric matrix(which is proved in many physics courses andlinear algebra courses). Part (3) is harder to prove, like the convergence theoremsfor Fourier series (which are a special case of it).