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We show that under minimal assumptions, the intrinsic metric induced by a strongly local Dirichlet form induces a length space. A main input is a dual characterization of length spaces in terms of the property that the 1-Lipschitz functions form a sheaf.
Comment: v3: final version, typos corrected; v2: references and some comments added; to appear in "Boundaries and spectra of random walks," proceedings of the Alp workshop Graz - St. Kathrein 2009, edited by D. Lenz, F. Sobieczky, W. Woess
We show that geometric disorder leads to purely singular continuous spectrum generically. The main input is a result of Simon known as the ``Wonderland theorem''. Here, we provide an alternative approach and actually a slight strengthening by showing that various sets of measures defined by regularity properties are generic in the set of all measures on a locally compact metric space. As a byproduct we obtain that a generic measure on euclidean space is singular continuous.
Comment: 16 pages, typos corrected, to appear in J. Funct. Anal
Let $M$ be a strongly pseudoconvex complex manifold which is also the total space of a principal $G$-bundle with $G$ a Lie group and compact orbit space $\bar M/G$. Here we investigate the $\bar\partial$-Neumann Laplacian on $M$. We show that it is essentially self-adjoint on its restriction to compactly supported smooth forms. Moreover we relate its spectrum to the existence of generalized eigenforms: an energy belongs to $\sigma(\square)$ if there is a subexponentially bounded generalized eigenform for this energy. Vice versa, there is an expansion in terms of these well-behaved eigenforms so that, spectrally, almost every energy comes with such a generalized eigenform.
The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Comment: 11 pages, submitted to Jounal of Integral Equations and Operator Theory
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