17 Sturm-Liouville problems, part 2

Separated boundary conditions

  • We recall the setup from the previous slides. We consider the linear second-order differential operator \(L\) given by \[ L(f) = r f'' + r' f' + pf. \]

    1. We suppose that \(r\) and \(p\) are real functions on \([a,b]\), each with continuous second derivatives.
    2. We suppose that \(r(x)\neq 0\) for all \(x\in [a,b]\).
    3. We suppose that the domain of \(L\) is the set of all functions \(f\) on \([a,b]\) (possibly complex valued) with continuous second derivatives.

  • By Langrange’s identity, we saw that (provided \(r(a)=r(b)\)) if we restrict \(L\) to those functions \(f\) that satisfy the periodic boundary conditions \[ f(a) =f(b) \quad \text{and} \quad f'(a) =f'(b), \] then \(L\) is self-adjoint.

  • We may also consider separated boundary conditions, which are of the form \[ \alpha_1 f(a) + \alpha_2 f'(a) =0 \quad \text{and} \quad \beta_1 f(b) + \beta_2 f'(b) =0, \] for some \(\alpha_1,\alpha_2,\beta_1,\beta_2\in \mathbb{R}\), where not both \(\alpha_1\) and \(\alpha_2\) are \(0\) simultaneously, and similarly for \(\beta_1\) and \(\beta_2\).

Exercise 1: Separated boundary conditions imply self-adjoint

Suppose we restrict \(L\) to those functions that satisfy separated boundary conditions. Show that \(L\) is self-adjoint.

Regular Sturm-Liouville problems

Definition

A regular Sturm-Liouville problem is given by the following:

  1. A second-order linear differential operator \(L\) of the previous form.
  2. A set of boundary conditions, either periodic or separated, with respect to which \(L\) is self-adjoint.
  3. A (real) positive, continuous weight function \(w\) on \([a,b]\).

Then the object is to find all solutions \(f\) of the boundary value problem \[ L(f) + \lambda w f = 0, \] where \(\lambda\) is an arbitrary constant and \(f\) satisfies the given boundary conditions.

  • Clearly a (regular) Sturm-Liouville problem always has the trivial solution \(f=0\).

  • If \(f\) is a nontrivial solution (i.e., \(f\neq 0\)), then \(f\) is called an eigenfunction and the corresponding \(\lambda\) is called an eigenvalue.

  • Linear combinations of eigenfunctions with the same eigenvalue are again eigenfunctions, so together with the trivial solution every eigenvalue \(\lambda\) has a corresponding subspace of eigenfunctions called its eigenspace.

The “Spectral Theorems” of regular Sturm-Liouville problems

Theorem 3.9

Let a regular Sturm-Liouville problem be given.

  1. All eigenvalues are real.

  2. Eigenfunctions corresponding to distinct eigenvalues are orthogonal in \(L^2_w(a,b)\).

  3. The eigenspace for any eigenvalue \(\lambda\) is at most \(2\)-dimensional. If the boundary conditions are separated, it is always \(1\)-dimensional.

Theorem 3.10

For every regular Sturm-Liouville problem on \([a,b]\):

  1. The set of eigenvalues is countably infinite.
  2. There is an orthonormal basis \(\{\phi_n\}_{n=1}^\infty\) of \(L^2_w(a,b)\) consisting of eigenfunctions.

Exercise 1: A simple Sturm-Liouville problem

Consider the boundary value problem

\[ f'' + \lambda f= 0, \quad f(0) = f(\ell) = 0, \]

for some \(\ell>0\).

  1. Identify this as a regular Sturm-Liouville problem. What types of boundary conditions are these?

  2. Find all eigenvalues and eigenfunctions of the problem.