Sturm–Liouville theory

Definition

Let $a < b$ be real numbers. Let $p \in C^1([a, b])$ and $q, w \in C([a, b])$ be real functions with $p(x) > 0$ and $w(x) > 0$ for all $x \in [a, b]$. Let $\alpha_1, \alpha_2, \beta_1, \beta_2$ be real numbers with $(\alpha_1, \alpha_2) \ne (0, 0)$ and $(\beta_1, \beta_2) \ne (0, 0)$. The regular Sturm–Liouville problem with these data is the boundary value problem $$-(p y')' + q y = \lambda w y \quad \text{on } [a, b],$$ $$\alpha_1 y(a) + \alpha_2 y'(a) = 0, \qquad \beta_1 y(b) + \beta_2 y'(b) = 0,$$ where $\lambda$ is a complex parameter. A number $\lambda$ is an eigenvalue of the problem if the problem has a solution $y \in C^2([a, b])$ that is not identically zero. Such a $y$ is an eigenfunction for $\lambda$. Sturm–Liouville theory is the study of the eigenvalues and eigenfunctions of such problems. It also studies singular versions, where the interval is unbounded or the coefficients are singular at an end.