## 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

322 |
Schrödinger operators with application to quantum mechanics and global geometry. Text and Monographs in
- Cycon, Froese, et al.
- 1987
(Show Context)
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... |

247 |
Methods of modern mathematical physics. II. Fourier analysis, self-adjointness
- Reed, Simon
- 1975
(Show Context)
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... |

210 | Dirichlet forms and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathematics. Walter de Gruyter - Fukushima, Ōshima, et al. - 1994 |

202 |
Röckner: Introduction to the theory of (nonsymmetric) Dirichlet forms. Universitext
- Ma, M
- 1992
(Show Context)
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... |

158 | Solvable models in quantum mechanics - Albeverio, Geszetsky, et al. - 1988 |

100 |
F.: Dirichlet forms and Analysis on Wiener space, Walter de Gruyter
- Bouleau, Hirsch
- 1991
(Show Context)
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... |

88 |
Spectral theory and differential operators
- Davies
- 1995
(Show Context)
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... |

84 |
On Harnack’s theorem for elliptic differential equations
- Moser
- 1961
(Show Context)
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... |

78 |
Brownian motion and Harnack’s inequality for Schrödinger Hamiltonians
- Aizenman, Simon
- 1982
(Show Context)
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... |

70 |
Dirichlet Forms and Markov Processes
- Fukushima
- 1980
(Show Context)
Citation Context ...d wish him many more years of fun in analysis. 1. Basics 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 com... |

68 | Ultracontractivity and heat-kernel for Schrödinger operators and Dirichlet Laplacian - Davies, Simon - 1984 |

58 |
Local behavior of solutions of quasilinear elliptic equations
- Serrin
- 1964
(Show Context)
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... |

39 | Related aspects of positivity in Riemannian geometry - Sullivan - 1987 |

38 | Comparison theorems for the gap of Schrödinger operators - Kirsch, Simon - 1987 |

32 | Seba: Schrödinger operators with singular interactions - Brasche, Exner, et al. - 1994 |

23 |
Analysis on local Dirichlet spaces. I: Recurrence, conservativeness and L p -Liouville properties
- Sturm
(Show Context)
Citation Context ...he intrinsic balls by B(x,r) := {y ∈ X|ρ(x,y) ≤ r}. An important consequence of the latter assumption is that the distance function ρx(·) := ρ(x, ·) itself is a function in Dloc with dΓ(ρx) ≤ dm, see =-=[35]-=-. This easily extends to the fact that for every closed E ⊂ X the function ρE(x) := inf{ρ(x,y)|y ∈ E} enjoys the same properties (see the Appendix of [14]). This has a very important consequence. When... |

22 |
Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality
- Sturm
- 1996
(Show Context)
Citation Context |

21 | Perturbation of Dirichlet forms by measures
- Stollmann, Voigt
- 1996
(Show Context)
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 ... |

20 |
A Saint-Venant type principle for Dirichlet forms on discontinuous
- Biroli, Mosco
- 1995
(Show Context)
Citation Context ...ny situations are known in which the Harnack principle is satisfied: Remark 3.2. (1) For ν ≡ 0 and E = 0 a Harnack inequality holds, whenever E satisfies a Poincaré and a volume doubling property; cf =-=[9]-=- and the discussion there. (2) The most general results for H0 = −∆ in terms of the measures ν that are allowed seem to be found in [17], which also contains a thorough discussion of the literature pr... |

17 |
On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds, in Methods of functional analysis and theory of elliptic equations
- Agmon
- 1982
(Show Context)
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 ... |

16 | Lifschitz singularities for periodic operators plus random potentials - Mezincescu - 1987 |

15 | An estimate of the gap of the first two eigenvalues in the Schrödinger operator - Singer, Wong, et al. |

14 |
Perturbation of translation invariant positivity preserving semigroups
- Herbst, Sloan
- 1978
(Show Context)
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... |

13 |
Positive solutions of elliptic equations
- Moss, Piepenbrink
- 1978
(Show Context)
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 ... |

11 |
Perturbation of Dirichlet forms—lower semiboundedness, closability, and form cores
- Albeverio, Ma
- 1991
(Show Context)
Citation Context ...ls 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 U of X and by MR,0(U) the subset of measures ν that do not charge sets of capacity 0, i.e., those measures wi... |

10 | Schnol’s theorem for strongly local forms
- Monvel, Lenz, et al.
- 2008
(Show Context)
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 ... |

10 | Interior Hölder estimates for solutions of Schrödinger equations and the regularity of nodal sets
- Hoffmann-Ostenhof, Hoffmann-Ostenhof, et al.
- 1995
(Show Context)
Citation Context |

10 |
Parabolic Harnack inequality for divergence-form second-order differential operators
- Saloff-Coste
- 1995
(Show Context)
Citation Context |

10 |
Smooth perturbations of regular Dirichlet forms
- Stollmann
- 1992
(Show Context)
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 ... |

9 |
Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations. In Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, volume 76
- Pinchover
- 2007
(Show Context)
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 ... |

9 |
Positive harmonic functions and diffusion, volume 45 of Cambridge
- Pinsky
- 1995
(Show Context)
Citation Context |

8 | An isoperimetric problem for leaky loops and related mean-chord inequalities - Exner |

7 |
Harnack’s inequality for Schrödinger operators and the continuity of solutions
- Chiarenza, Fabes, et al.
- 1986
(Show Context)
Citation Context |

7 |
Measures charging no polar sets and additive functionals of Brownian motion
- Sturm
- 1992
(Show Context)
Citation Context ...ls 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 U of X and by MR,0(U) the subset of measures ν that do not charge sets of capacity 0, i.e., those measures wi... |

6 |
Nonoscillatory elliptic equations
- Piepenbrink
- 1974
(Show Context)
Citation Context |

6 |
Harnack’s inequality for parabolic operators with singular low order terms
- Sturm
- 1994
(Show Context)
Citation Context |

5 |
On the equivalence of two types of oscillation for elliptic operators
- Allegretto
- 1974
(Show Context)
Citation Context |

5 | Schrödinger type and relaxed Dirichlet problems for the subelliptic pLaplacian - Biroli - 2001 |

4 | Lower bounds on the lowest spectral gap of singular potential Hamiltonians - Kondej, Veselić |

4 |
Harnack inequalities for Schrödinger operators
- Hansen
- 1999
(Show Context)
Citation Context ...ls 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 U of X and by MR,0(U) the subset of measures ν that do not charge sets of capacity 0, i.e., those measures wi... |

3 |
Positive solutions and spectral properties of second order elliptic operators
- Allegretto
- 1981
(Show Context)
Citation Context |

2 |
Spectral estimates and oscillation of singular differential operators
- Allegretto
- 1979
(Show Context)
Citation Context |

2 |
PT symmetric quantum mechanics. J.Math.Phys
- Bender, Boettcher, et al.
- 1999
(Show Context)
Citation Context ...for complex measures ν without problems, as long as the corresponding forms are 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... |

2 | Harnack inequality for the Schrödinger problem relative to strongly local Riemannian p-homogeneous forms with a potential in the Kato class - Biroli, Marchi |

2 | On eigenvalues and eigensolutions of the Schrödinger equation on the complement of a set with classical capacity zero Methods Funct - Brasche - 2003 |

2 |
Irreducibility and connectedness for Dirichlet forms
- Lenz, Stollmann, et al.
(Show Context)
Citation Context ...igroup (Tt;t ≥ 0). We denote by H the associated operator. Actually, the cases of interest in this paper are h = E or h = E + ν with ν ∈ MR,0 − MR,1. We refer to [30], XIII.12 and a forthcoming paper =-=[21]-=- for details. We say that h is reducible, if there is a measurable set M ⊂ X such that M and its complement M c are nontrivial (have positive measure) and L 2 (M) is a reducing subspace for M, i.e., 1... |

2 |
A conjecture of Glazman
- Piepenbrink
- 1977
(Show Context)
Citation Context |

2 | A lower bound on the first spectral gap of Schrödinger operators with Kato class measures. to appear: Ann. Henri Poincaré. Documenta Mathematica 14 (2009) 235–257 Allegretto-Piepenbrink Theorem 257 Daniel Lenz Mathematisches Institut Friedrich-Schiller Un - Vogt - 2009 |

1 | inequalities for Schrödinger operators. Ann. Scuola Norm - Harnack - 1999 |

1 |
inequalities: an introduction
- Harnack
(Show Context)
Citation Context |