Introduction: We are going to study a standard second order ODE called the Sturm-Liouville ODE. The solution of this ODE form a function space, or Hilbert space, which is similar to the vector space we have known. We will use similar concepts as in vector spaces, such as inner products, orthogonality, eigenvalues, eigenfunctions, and completeness. Sturm-Liouville ODE has been used to generate various special functions in physics. 1 December 5 Self-adjoint ODEs Chapter 10 Sturm-Liouville theory– Orthogonal functions

Self-adjoint operators: The most general form of a linear second-order differential operator L is Here to apply this operator to our former results of differential equations from the separation of variables, we assume p 0, p 1, and p 2 be real functions. Also p 0 > 0 in the range of interest a< x <b, while p 0 =0 can happen at the boundaries. We define the adjoint operator as 2

3 Theorem: A non-self-adjoint linear second-order differential operator can always be transformed into the self-adjoint form by L  w(x) L. Therefore the theory on linear second-order self-adjoint differential equations, which we are going to explore, is general.

4 Sturm-Liouville differential equation: The Sturm-Liouville differential equation is defined as an eigenvalue equation taking the following self-adjoint form: Examples of Sturm-Liouville differential equations:

5 Boundary conditions: Suppose u(x) and v(x) are solutions to the Sturm-Liouville differential equation If we assume to choose the boundary conditions so that for any two solutions u and v, then the Sturm-Liouville self-adjoint operator satisfies: The operator L is then a Hermitian operator. That is

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7 Weighted inner products: Suppose we define weighted inner products for our Sturm-Liouville problem as, then under the afore-mentioned boundary conditions, That is, L /w is Hermitian with respect to the weighted inner product. December 7 Sturm-Liouville theory Real eigenvalues and the orthogonality of eigenfunctions: Suppose we have the Sturm-Liouville differential equation We now have the important Sturm-Liouville theory: (at the chosen boundary conditions): 1)The eigenvalues of a Sturm-Liouville operator are real. 2)The eigenfunctions of a Sturm-Liouville operator are orthogonal with respect to the weighted inner product.

8 Completeness of eigenfunctions: The third part of the Sturm-Liouville theory is 3) The eigenfunctions of a Sturm-Liouville operator form a complete set. This means that any function f (x) satisfying the boundary conditions can be expanded as Mathematically  n (x) is complete if Complete relation of the eigenfunctions:

9 Example: Fourier series. Consider the Sturm-Liouville problem

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11 December 9 Green’s function-Eigenfunction expansion 10.5 Green’s function-Eigenfunction expansion Eigenfunction expansion of Green’s function: For a nonhomogeneous Sturm-Liouville ODE with a source term We know that by the Green’s function method, if we find G(x, x') that satisfies The solution of the nonhomogeneous equation is then Therefore the key point is how to find G(x, x'). We also know that according to the Sturm-Liouville theory, any function can be expanded by the eigenfunctions of the homogeneous equation. This means that u(x) and G(x, x') can be expanded by the eigenfunctions. Let us try to do it.

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13 Eigenfunction expansion of Green’s function: Example: Helmholtz equation

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