16 Sturm-Liouville problems, part 1

Orthogonal bases and differential equations

  • Theorem 3.5 tells us that \(\{e^{inx}\}_{n=-\infty}^\infty\) is an orthogonal basis of \(L^2(-\pi,\pi)\).

  • Similar arguments can be used to show that both \(\{\cos{nx}\}_{n=0}^\infty\) and \(\{\sin{nx}\}_{n=1}^\infty\) are orthogonal bases for \(L^2(0,\pi)\).

  • Recall that these three sets of functions arose originally in our solutions to partial differential equations. In particular, the latter two sets arose during separation of variables when we attempted to solve a differential equation of the form \[ u''(x) + \lambda^2 u(x)=0 \] subject to either the conditions \(u(0)=u(\pi)=0\) or \(u'(0)=u'(\pi)=0\).

  • Our goal in this section is to show that second-order ordinary differential equations of particular forms lead to orthogonal bases of \(L^2\)-spaces.

Exercise 1: Background from linear algebra

A linear transformation \(T: \mathbb{C}^k \to \mathbb{C}^k\) is called self-adjoint if \[ \langle T(\mathbf{u}),\mathbf{v} \rangle = \langle \mathbf{u}, T(\mathbf{v}) \rangle \] for all \(u,v \in \mathbb{C}^k\).

  1. Consider two linear transformations \(S,T: \mathbb{C}^2 \to \mathbb{C}^2\) with matrices \[ A = \begin{bmatrix} 1 & 1+i \\ 1-i & 0 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}, \] respectively, with respect to the standard bases on \(\mathbb{C}^2\). Which of \(S\) and \(T\) is self-adjoint?

  2. Can you formulate a condition on the matrix of a general linear transformation \(T: \mathbb{C}^k \to \mathbb{C}^k\) that characterizes the self-adjoint property? Can you prove it?

  3. For the linear transformations in part (a), compute their eigenvalues and eigenvectors. Any particular things that you notice?

The spectral theorem

Theorem (The spectral theorem for self-adjoint operators on \(\mathbb{C}^k\))

Let \(T: \mathbb{C}^k \to \mathbb{C}^k\) be a self-adjoint linear operator.

  1. There is an orthonormal basis of \(\mathbb{C}^k\) consisting of eigenvectors of \(T\).

  2. All eigenvalues of \(T\) are real.

Moving toward self-adjoint linear differential operators

  • The goal now is to use our knowledge of self-adjoint linear operators on \(\mathbb{C}^k\) and the spectral theorem and adapt it to intuition for linear differential operators on \(L^2(a,b)\).

  • We begin with the following definition, which is “almost” correct. (See the comment afterwards.)

Definition

A linear operator \(T: L^2(a,b) \to L^2(a,b)\) is self-adjoint if \[ \langle T(f), g \rangle = \langle f, T(g) \rangle \] for all \(f,g\in L^2(a,b)\).

  • Technically, in order for this definition to be correct, we need to restrict the domain of \(T\) to a proper subspace of the \(L^2\)-space. We will not be very careful about this.

A second-order linear differential operator, part 1

  • Consider the second-order linear differnetial operator \(L\) given by \[ L(f) = r f'' + r' f' + pf. \]
    • We suppose that \(r\) and \(p\) are real functions on \([a,b]\), each with continuous second derivatives.
    • We suppose that \(r(x)\neq 0\) for all \(x\in [a,b]\).
    • We suppose that the domain of \(L\) is the set of all functions on \([a,b]\) (possibly complex valued) with continuous second derivatives.
  • We want to investigate when \(L\) is self-adjoint.

Lemma

  1. We have \[ \int_a^b (rf'')\overline{g} \, dx = \int_a^b f(r\overline{g})'' \, dx + \left[ rf'\overline{g} - f(r\overline{g})'\right]_a^b. \]

  2. We have \[ \int_a^b (r'f') \overline{g} \, dx = - \int_a^b f(r'\overline{g})'\, dx + r' f\overline{g}\big|_a^b. \]

A second-order linear differential operator, part 2

Theorem (Lagrange’s Identity)

We have

\[ \langle L(f), g \rangle = \langle f, L(g) \rangle + \left[ r(f' \overline{g}-f\overline{g}') \right]_a^b. \]

  • Now, suppose we are interested in solving the second-order linear ODE \[ L(f) = r f'' + r' f' + pf = 0, \] subject to the periodic boundary conditions \[ f(a)=f(b) \quad \text{and} \quad f'(a) = f'(b). \] Suppose also that \(r(a)=r(b)\).

Theorem (Periodic boundary conditions \(\Rightarrow\) \(L\) self-adjoint)

If we restrict the linear operator \(L(f) = rf'' + r'f' + pf\) to the set of functions satisfying the periodic boundary conditions above, then \(L\) is self-adjoint.