Introduction to Sturm-Liouville Systems

Flashcards covering the introductory concepts of Sturm-Liouville boundary value problems, including ordinary differential equation classifications, eigenvalue/eigenfunction definitions, and worked boundary value problem examples on various intervals.

What constitutes an $n$-th order ordinary differential equation, and how many arbitrary constants does its general solution contain?

An $n$-th order ordinary differential equation is an equation involving a real-valued function $y(x)$ of a real variable $x$ along with its $n$ lowest derivatives, written in the form $y^{(n)} = F(y^{(n-1)}, \text{...}, y', y, x)$. Its general solution contains $n$ arbitrary constants.

What is the distinction between an initial value problem and a boundary value problem (separated end-point problem)?

In an initial value problem, all $n$ conditions are specified at a single value $x_0$ of the independent variable, giving rise to a unique solution. In a boundary value problem, conditions are specified at different values of the independent variable, such as at the boundaries $x = a$ and $x = b$ of an interval $[a, b]$.

What is the general solution to the differential equation $\frac{d^2y}{dx^2}(x) + \lambda y(x) = 0$ when $\lambda = -\alpha^2 < 0$ with $\alpha > 0$?

The general solution is $y_-(x) = A \cosh(\alpha x) + B \sinh(\alpha x)$, where $A$ and $B$ are arbitrary constants.

What are the general solutions to $\frac{d^2y}{dx^2}(x) + \lambda y(x) = 0$ for the cases $\lambda = 0$ and $\lambda = \omega^2 > 0$ ($\omega > 0$)?

For $\lambda = 0$, integrating twice yields $y_0(x) = cx + d$ for arbitrary constants $c$ and $d$. For $\lambda = \omega^2 > 0$, the general solution is $y_+(x) = E \sin(\omega x) + F \cos(\omega x)$ for arbitrary constants $E$ and $F$.

In the context of the differential equation $-\frac{d^2y}{dx^2} = \lambda y$, how are eigenvalues and eigenfunctions defined?

The values of the parameter $\lambda$ for which non-trivial solutions ($y(x) \not\equiv 0$) exist that satisfy the given boundary conditions are called eigenvalues, and the corresponding non-trivial solutions are called eigenfunctions.

What are the eigenvalues and eigenfunctions for $\frac{d^2y}{dx^2} + \lambda y = 0$ on $[0, \pi]$ subject to $y(0) = 0$ and $y(\pi) = 0$ (Example 1.3)?

There are no non-trivial solutions for $\lambda \le 0$. For $\lambda > 0$, the condition $\sin(\omega \pi) = 0$ yields countably infinitely many positive eigenvalues $\lambda_n = n^2$ with eigenfunctions $y_n(x) = E_n \sin(nx)$ for $n \in \mathbb{N}$.

In Example 1.5 ($\frac{d^2y}{dx^2} + \lambda y = 0$ on $[0, \pi]$ with $y(0) + y'(0) = 0$ and $y(\pi) = 0$), what transcendental equation and negative eigenvalue are represented by this plot?

The plot illustrates the transcendental equation $\tanh(\alpha \pi) = \alpha$ derived from the negative case $\lambda = -\alpha^2 < 0$. It shows a single strictly positive root $\alpha = \alpha^* > 0$, yielding exactly one negative eigenvalue $\lambda_{\alpha^*} = -(\alpha^*)^2$ with eigenfunction $y_{\alpha^*}(x) = \sinh(\alpha^*(x - \pi))$.

In Example 1.5, what transcendental equation is shown in this plot for $\lambda = \omega^2 > 0$, and where do the intersections occur?

The plot shows the transcendental equation $\tan(\omega \pi) = \omega$. The curves intersect once in each interval $(n, n + \frac{1}{2})$ for $n \in \mathbb{N}$, yielding countably infinitely many positive eigenvalues $\lambda_n = \omega_n^2$ with corresponding eigenfunctions $y_n(x) = \sin(\omega_n(\pi - x))$.

For $\frac{d^2y}{dx^2} + \lambda y = 0$ on the symmetric interval $[-\pi, \pi]$ with $y(-\pi) = 0$ and $y(\pi) = 0$ (Example 1.6), what condition on $\omega$ produces non-trivial solutions when $\lambda = \omega^2 > 0$?

Setting the determinant of the coefficients to zero yields $-2\sin(\omega \pi)\cos(\omega \pi) = -\sin(2\omega \pi) = 0$, which requires $\omega = \frac{n}{2}$ for $n \in \mathbb{N}$.

In Example 1.6 on $[-\pi, \pi]$ with $y(-\pi) = 0$ and $y(\pi) = 0$, what are the resulting sequences of eigenvalues and eigenfunctions for even and odd $n$?

For even $n = 2k$ ($k \in \mathbb{N}$), the eigenfunctions are odd: $y_k(x) = \sin(kx)$ with $\lambda_k = k^2$. For odd $n = 2k - 1$ ($k \in \mathbb{N}$), the eigenfunctions are even: $y_{k - 1/2}(x) = \cos((k - \frac{1}{2})x)$ with $\lambda_{k - 1/2} = (k - \frac{1}{2})^2$.

For the boundary value problem $\frac{d^2y}{dx^2} + \lambda y = 0$ on $[-\pi, \pi]$ with periodic boundary conditions $y(-\pi) = y(\pi)$ and $y'(-\pi) = y'(\pi)$ (Example 1.7), is $\lambda = 0$ an eigenvalue?

Yes. When $\lambda = 0$, the general solution $y(x) = cx + d$ satisfies $y(-\pi) = y(\pi) \implies -c\pi + d = c\pi + d$, forcing $c = 0$ and leaving $y(x) = d$. The condition $y'(-\pi) = y'(\pi)$ is automatically satisfied, giving eigenvalue $\lambda = 0$ with eigenfunction $y_0(x) = 1$.

In Example 1.7 with boundary conditions $y(-\pi) = y(\pi)$ and $y'(-\pi) = y'(\pi)$, what are the positive eigenvalues and the multiplicity of each eigenvalue for $n \in \mathbb{N}$?

The positive eigenvalues are $\lambda_n = n^2$ for $n \in \mathbb{N}$. Each has multiplicity two because there are two linearly independent eigenfunctions, $\sin(nx)$ and $\cos(nx)$, spanning the corresponding two-dimensional eigenspace.