## The Allegretto-Piepenbrink Theorem for Strongly Local Dirichlet Forms (2009)

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Venue: | DOCUMENTA MATH. |

Citations: | 6 - 5 self |

### BibTeX

@MISC{Lenz09theallegretto-piepenbrink,

author = {Daniel Lenz and Peter Stollmann and Ivan Veselić},

title = { The Allegretto-Piepenbrink Theorem for Strongly Local Dirichlet Forms },

year = {2009}

}

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### Abstract

The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.

### Citations

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Citation Context ...ties are well established in the classical case. Given these tools, we prove this part of the Allegretto-Piepenbrink theorem in Section 3 with arguments reminiscent of the corresponding discussion in =-=[12]-=-. For somewhat complementary results we refer to [14] where it is shown that existence of a nontrivial subexponentially bounded solution of HΦ = EΦ yields that E ∈ σ(H). This implies, in particular, t... |

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Citation Context ...unded with bound less than one; i.e. measures for which there is a κ < 1 and a cκ such that µ[u,u] ≤ κE[u] + cκ‖u‖ 2 . The set MR,1 can easily be seen to be a subset of MR,0. By the KLMN theorem (see =-=[29]-=-, p. 167), the sum E + ν given by D(E + ν) = {u ∈ D | ũ ∈ L2 (X,ν+)} is closed and densely defined (in fact D ∩ Cc(X) ⊂ D(E + ν)) for ν+ ∈ MR,0(X) and ν− ∈ MR,1. We denote the associated selfadjoint o... |

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Citation Context ...sics and notation concerning strongly local Dirichlet forms and measure perturbations Dirichlet forms. We will now describe the set-up; we refer to [15] as the classical standard reference as well as =-=[10, 13, 16, 22]-=- for literature on Dirichlet forms. Let us emphasize that in contrast to most of the work done on Dirichlet forms, we treat real and complex function spaces at the same time and write K to denote eith... |

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Citation Context ...sics and notation concerning strongly local Dirichlet forms and measure perturbations Dirichlet forms. We will now describe the set-up; we refer to [15] as the classical standard reference as well as =-=[10, 13, 16, 22]-=- for literature on Dirichlet forms. Let us emphasize that in contrast to most of the work done on Dirichlet forms, we treat real and complex function spaces at the same time and write K to denote eith... |

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Citation Context ...sics and notation concerning strongly local Dirichlet forms and measure perturbations Dirichlet forms. We will now describe the set-up; we refer to [15] as the classical standard reference as well as =-=[10, 13, 16, 22]-=- for literature on Dirichlet forms. Let us emphasize that in contrast to most of the work done on Dirichlet forms, we treat real and complex function spaces at the same time and write K to denote eith... |

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Citation Context ...and the uniformity of the estimates from [17] immediately gives that the uniform Harnack principle is satisfied in that context. Of the enormous list of papers on Harnack’s inequality, let us mention =-=[2, 11, 17, 19, 20, 23, 31, 32, 37, 38]-=-THE ALLEGRETTO-PIEPENBRINK THEOREM 9 Apart from the Harnack principle there are two more properties that will be important in the proof of existence of positive general eigensolutions at energies bel... |

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Citation Context ...and the uniformity of the estimates from [17] immediately gives that the uniform Harnack principle is satisfied in that context. Of the enormous list of papers on Harnack’s inequality, let us mention =-=[2, 11, 17, 19, 20, 23, 31, 32, 37, 38]-=-THE ALLEGRETTO-PIEPENBRINK THEOREM 9 Apart from the Harnack principle there are two more properties that will be important in the proof of existence of positive general eigensolutions at energies bel... |

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Citation Context .... We will be dealing with Schrödinger type operators, i.e., perturbations H = H0 + V for suitable potentials V . In fact, we can even include measures as potentials. Here, we follow the approach from =-=[33, 34]-=-. Measure perturbations have been regarded by a number of authors in different contexts, see e.g. [3, 17, 36] and the references there. We denote by MR(U) the signed Radon measures on the open subset ... |

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Citation Context ...corresponding partial differential operator. Introduction The Allegretto-Piepenbrink theorem relates solutions and spectra of 2nd order partial differential operators H and has quite some history, cf =-=[1, 4, 5, 6, 24, 25, 26, 27, 28]-=-. One way to phrase it is that the supremum of those real E for which a nontrivial positive solution of HΦ = EΦ exists coincides with the infimum of the spectrum of H. In noncompact cases this can be ... |

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Citation Context ...re accretive. In the context of PT–symmetric operators there is recent interest in this type of Schrödinger operators, see [7] (2) Instead of measures also certain distributions could be included. Cf =-=[18]-=- for such singular perturbations. 3. The existence of positive weak solutions below the spectrum As noted in the preceding section, we find that H0 + ν ≥ E whenever E + ν is closable and admits a posi... |

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Citation Context ...corresponding partial differential operator. Introduction The Allegretto-Piepenbrink theorem relates solutions and spectra of 2nd order partial differential operators H and has quite some history, cf =-=[1, 4, 5, 6, 24, 25, 26, 27, 28]-=-. One way to phrase it is that the supremum of those real E for which a nontrivial positive solution of HΦ = EΦ exists coincides with the infimum of the spectrum of H. In noncompact cases this can be ... |

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Citation Context ...n these tools, we prove this part of the Allegretto-Piepenbrink theorem in Section 3 with arguments reminiscent of the corresponding discussion in [12]. For somewhat complementary results we refer to =-=[14]-=- where it is shown that existence of a nontrivial subexponentially bounded solution of HΦ = EΦ yields that E ∈ σ(H). This implies, in particular, that the positive solutions we construct for energies ... |

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Citation Context .... We will be dealing with Schrödinger type operators, i.e., perturbations H = H0 + V for suitable potentials V . In fact, we can even include measures as potentials. Here, we follow the approach from =-=[33, 34]-=-. Measure perturbations have been regarded by a number of authors in different contexts, see e.g. [3, 17, 36] and the references there. We denote by MR(U) the signed Radon measures on the open subset ... |

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Citation Context ...corresponding partial differential operator. Introduction The Allegretto-Piepenbrink theorem relates solutions and spectra of 2nd order partial differential operators H and has quite some history, cf =-=[1, 4, 5, 6, 24, 25, 26, 27, 28]-=-. One way to phrase it is that the supremum of those real E for which a nontrivial positive solution of HΦ = EΦ exists coincides with the infimum of the spectrum of H. In noncompact cases this can be ... |

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