Sturm–Liouville problems

Consider −(p y′)′+q y=λw y on a finite interval [a,b]. For this lesson take real p continuously differentiable and strictly positive, and real continuous q and w with w strictly positive. Dividing by w defines an operator in L2([a,b],w dx), where the weight appears in the inner product. An eigenvalue problem written with λw y on the right is therefore naturally symmetric in the weighted space, rather than automatically in the unweighted one.

The two endpoint conditions complete the problem. Dirichlet conditions fix y to zero, Neumann conditions fix the endpoint flux p y′ to zero, and real separated Robin conditions prescribe a linear combination at each endpoint. These are examples, not the full classification of self-adjoint boundary conditions; coupled periodic conditions also exist. Coefficients that vanish or blow up, an infinite interval, or a nonintegrable endpoint require singular Sturm–Liouville theory with additional domain analysis. The regular results should not be transported there unchanged.

For sufficiently regular f and g, integration by parts cancels the interior terms in inner-product(f,Lg)−inner-product(Lf,g), leaving an endpoint expression. Dirichlet conditions make both function values vanish, so the boundary form is zero. Real separated Robin conditions also cancel it because both functions obey the same real relation between value and flux. Vanishing of the form on a proposed domain establishes symmetry. Self-adjointness additionally requires that the adjoint domain impose exactly the same endpoint restrictions.

For regular coefficients and self-adjoint endpoint conditions, this domain analysis gives real eigenvalues, orthogonality of eigenfunctions at distinct eigenvalues, and a discrete spectrum with eigenvalues tending upward without bound. The corresponding inverse after a suitable spectral shift is compact. These statements support expansion in normal modes. Orthogonality is with the weight w, and normalization must use that same weight. Changing the boundary conditions changes the eigenfunctions and can change whether a zero-frequency mode is present.

With p=w=1 and q=0, the Dirichlet problem on [0,L] is −y″=λy with y(0)=y(L)=0. A nontrivial solution has positive λ, since integrating against y gives a nonnegative derivative energy and a zero eigenvalue would force a constant that vanishes at both ends. Writing λ=k² leaves a sine after imposing the left endpoint, and the right endpoint requires kL to be a positive integer multiple of π. Boundary conditions quantize the allowed spatial scales.

For Neumann endpoints, the constant function is allowed and gives eigenvalue zero. This difference matters in diffusion: a constant temperature survives when the boundaries prevent flux, while zero Dirichlet boundaries remove it. The mode expansion still needs a declared convergence sense, such as L2 convergence for square-integrable initial data. Termwise differentiation or pointwise boundary recovery may require stronger assumptions. A finite sum can demonstrate shapes without proving those convergence and regularity claims for arbitrary data.