## Abstract

Let H be the symmetric second-order differential operator on L_{2}(R) with domain C^{∞} _{c} and action H_{φ} = -(cφ^{′})′ where c ε W^{1,2} _{loc}(R) is a real function that is strictly positive on R\{0} but with c(0) = 0. We give a complete characterization of the self-adjoint extensions and the submarkovian extensions of H. In particular if ν = ν+ ∨ ν where ν±(x) = ± ∫^{±1} _{±x} c^{-1} then H has a unique self-adjoint extension if and only if ν ∉ L_{2}(0,1) and a unique submarkovian extension if and only if ν ∉ L_{∞}(0,1). In both cases, the corresponding semigroup leaves L_{2}(0,∞) and L_{2}(-∞,0) invariant. In addition, we prove that for a general non-negative c ε W^{1,∞} _{loc}(R) the corresponding operator H has a unique submarkovian extension.

Original language | English |
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Pages (from-to) | 731-759 |

Number of pages | 29 |

Journal | Journal of Evolution Equations |

Volume | 10 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2010 |