📂Lebesgue Spaces

Sturm-Liouville Differential Equation

Definition¹

Let $p\in$, $C^{1}(\mathbb{R})$(../1594) and assume $q,r\in C(\mathbb{R})$, $\lambda \in \mathbb{R}$. The differential equation of the following form is called a Sturm-Liouville differential equation.

$$\begin{equation} \left[ p(x)u^{\prime}(x) \right]^{\prime}+\left[ q(x) +\lambda w(x) \right]u(x)=0 \end{equation}$$

or

$$p(x)u^{\prime \prime}(x)+p^{\prime}(x)u^{\prime}(x)+\left[ q(x)+\lambda w(x) \right]u(x)=0$$

Explanation

It is also referred to as the S-L problem.

Here, $w$ is called the weight function, because it becomes the weight for the inner product in the function space of solutions $u$ to the differential equation $(1)$.

$\lambda$ and $u$ are called the eigenvalue and eigenfunction, respectively. This is because when representing the above differential equation in the form of an eigenvalue equation, $\lambda$ becomes the eigenvalue. Also, the term multiplied by $u$ looking like $q + \lambda w$ is thought to further classify the differential equation according to the value of $\lambda$.

It is difficult to accurately find the definite integration interval and weight function in any function space. Furthermore, even if found, integrating is not always easy. At this point, such challenges can be addressed through the Sturm-Liouville problem. Handling the differential equation like $(1)$ without any conditions is difficult, so let’s consider the following conditions.

Regular Sturm-Liouville Problem

The differential equation $(1)$ is defined on the interval $[a,b]$ and is called a regular Sturm-Liouville problem when it satisfies the following two conditions:

$(\text{i})$ For all $x \in [a,b]$, $p(x)>0$, $w(x)>0$

$(\text{ii})$ Given $(c_{1},c_{2})\ne (0,0)$ and constants $(d_{1},d_{2})\ne (0,0)$, the following boundary conditions hold:

$$\begin{cases} c_{1}u(a) + c_{2}u^{\prime}(a) =0 \\ d_{1}u(b) + d_{2}u^{\prime}(b) =0 \end{cases}$$

The goal is to find solutions $u$ of the differential equation that belong to the following Hilbert space.

$$L_{w}^{2}(a,b) := \left\{ u : \mathbb{R} \to \mathbb{C} \bigg| \int_{a}^{b} \left| u(x) \right|^{2} w(x)dx <\infty \right\}$$

In such a weighted $L^{p}$ space, the inner product is given as follows:

$$\langle u, v \rangle _{L_{w}^{2}(a,b)} =\int_{a}^{b} u(x)\overline{v(x)}w(x)dx,\quad u,v\in L_{w}^{2}(\mathbb{R})$$