## Abstract

Let - DpD be a differential operator on the compact interval [-b, b] whose leading coefficient is positive on (0, b] and negative on [b,0), with fixed, separated, self-adjoint boundary conditions at h and b and an additional interface condition at 0. The self-adjoint extensions of the corresponding minimal differential operator are non-semibounded and are related to non-semibounded sesquilinear forms by a generalization of Kato's representation theorems. The theory of non-semibounded sesquilinear forms is applied to this concrete situation. In particular, the generalized Friedrichs extension is obtained as the operator associated with the unique regular closure of the minimal sesquilinear form. Moreover, among all closed forms associated with the self-ad joint extensions, the regular closed forms are identified. As a consequence, eigenfunction expansion theorems are obtained for the differential operators as well as for certain indefinite Krein-Feller operators with a single concentrated mass.

Original language | English |
---|---|

Pages (from-to) | 291-318 |

Number of pages | 28 |

Journal | Proceedings of the royal society of edinburgh section a-Mathematics |

Volume | 140A |

Publication status | Published - 2010 |

## Keywords

- SPACES