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 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 $\lambda \in ℝ$ and $f$ is a function such that $Df=\lambda f$ then we say $f$ is an eigenfunction of $D$ with eigenvalue $\lambda$.

Title	eigenfunction
Canonical name	Eigenfunction
Date of creation	2013-03-22 12:48:00
Last modified on	2013-03-22 12:48:00
Owner	tensorking (373)
Last modified by	tensorking (373)
Numerical id	8
Author	tensorking (373)
Entry type	Definition
Classification	msc 34B24
Synonym	characteristics function
Defines	solution of system