Given a differential equation

(1) |

(2) |

(3) |

(4) |

(5) |

(6) | |||

(7) |

The Legendre Differential Equation and the equation of Simple Harmonic Motion are self-adjoint, but the Laguerre Differential Equation and Hermite Differential Equation are not.

A nonself-adjoint second-order linear differential operator can always be transformed into a self-adjoint one using
Sturm-Liouville Theory. In the special case , (7) gives

(8) |

(9) |

(10) |

(11) |

A self-adjoint operator which satisfies the Boundary Conditions

(12) |

**References**

Arfken, G. ``Self-Adjoint Differential Equations.'' §9.1 in *Mathematical Methods for Physicists, 3rd ed.*
Orlando, FL: Academic Press, pp. 497-509, 1985.

© 1996-9

1999-05-26