eigenfunction

Consider the Sturm-Liouville system given by

$$\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y+\lambda r(x)y=0\mathit{\hspace{1em}\hspace{0.5em} }a\le x\le b$$(1)

$$a_{1}y(a)+a_2{y}^{\prime }(a)=0,b_{1}y(b)+b_2{y}^{\prime }(b)=0,$$(2)

where $a_i,b_i\in ℝ$ with $i\in \{1,2\}$ and$p(x),q(x),r(x)$ are differentiable functions and$\lambda \in ℝ$. A non zero solution of the system definedby (1) and (2) exists in general for aspecified $\lambda$. The functions corresponding to thatspecified $\lambda$ are called eigenfunctions.

More generally, if $D$ is some linear differential operator, and$\lambda \in ℝ$ and $f$ is a function such that$Df=\lambda f$ then we say $f$ is an eigenfunction of $D$ witheigenvalue $\lambda$.