# eigenfunction

Consider the Sturm-Liouville system given by

$$\frac{d}{dx}\left[p(x)\frac{dy}{dx}\right]+q(x)y+\lambda r(x)y=0\mathit{\hspace{1em}\hspace{0.5em}\u2006}a\le x\le b$$ | (1) |

$${a}_{1}y(a)+{a}_{2}{y}^{\prime}(a)=0,{b}_{1}y(b)+{b}_{2}{y}^{\prime}(b)=0,$$ | (2) |

where ${a}_{i},{b}_{i}\in \mathbb{R}$ 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 |
---|---|

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 |