# eigenfunction

Consider the Sturm-Liouville system given by

 $\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y+\lambda r(x)y=0\;\;\;\;\;\;a% \leq x\leq b$ (1)
 $a_{1}y(a)+a_{2}y^{\prime}(a)=0,\;\;\;\;\;\;b_{1}y(b)+b_{2}y^{\prime}(b)=0,$ (2)

where $a_{i},b_{i}\in\mathbb{R}$ with $i\in\{1,2\}$ and $p(x),q(x),r(x)$ are differentiable functions and $\lambda\in\mathbb{R}$. A non zero solution of the system defined by (1) and (2) exists in general for a specified $\lambda$. The functions corresponding to that specified $\lambda$ are called eigenfunctions.

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

Title eigenfunction Eigenfunction 2013-03-22 12:48:00 2013-03-22 12:48:00 tensorking (373) tensorking (373) 8 tensorking (373) Definition msc 34B24 characteristics function solution of system