283 research outputs found
Spectral analysis on infinite Sierpinski fractafolds
A fractafold, a space that is locally modeled on a specified fractal, is the
fractal equivalent of a manifold. For compact fractafolds based on the
Sierpinski gasket, it was shown by the first author how to compute the discrete
spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian.
A similar problem was solved by the second author for the case of infinite
blowups of a Sierpinski gasket, where spectrum is pure point of infinite
multiplicity. Both works used the method of spectral decimations to obtain
explicit description of the eigenvalues and eigenfunctions. In this paper we
combine the ideas from these earlier works to obtain a description of the
spectral resolution of the Laplacian for noncompact fractafolds. Our main
abstract results enable us to obtain a completely explicit description of the
spectral resolution of the fractafold Laplacian. For some specific examples we
turn the spectral resolution into a "Plancherel formula". We also present such
a formula for the graph Laplacian on the 3-regular tree, which appears to be a
new result of independent interest. In the end we discuss periodic fractafolds
and fractal fields
A Non-Riemannian Metric on Space-Time Emergent From Scalar Quantum Field Theory
We show that the two-point function
\sigma(x,x')=\sqrt{} of a scalar quantum field theory
is a metric (i.e., a symmetric positive function satisfying the triangle
inequality) on space-time (with imaginary time). It is very different from the
Euclidean metric |x-x'| at large distances, yet agrees with it at short
distances. For example, space-time has finite diameter which is not universal.
The Lipschitz equivalence class of the metric is independent of the cutoff.
\sigma(x,x') is not the length of the geodesic in any Riemannian metric.
Nevertheless, it is possible to embed space-time in a higher dimensional space
so that \sigma(x,x') is the length of the geodesic in the ambient space.
\sigma(x,x') should be useful in constructing the continuum limit of quantum
field theory with fundamental scalar particles
Leibnizian, Galilean and Newtonian structures of spacetime
The following three geometrical structures on a manifold are studied in
detail: (1) Leibnizian: a non-vanishing 1-form plus a Riemannian
metric \h on its annhilator vector bundle. In particular, the possible
dimensions of the automorphism group of a Leibnizian G-structure are
characterized. (2) Galilean: Leibnizian structure endowed with an affine
connection (gauge field) which parallelizes and \h. Fixed
any vector field of observers Z (), an explicit Koszul--type
formula which reconstruct bijectively all the possible 's from the
gravitational and vorticity fields
(plus eventually the torsion) is provided. (3) Newtonian: Galilean structure
with \h flat and a field of observers Z which is inertial (its flow preserves
the Leibnizian structure and ). Classical concepts in Newtonian
theory are revisited and discussed.Comment: Minor errata corrected, to appear in J. Math. Phys.; 22 pages
including a table, Late
Noncommutative Figa-Talamanca-Herz algebras for Schur multipliers
We introduce a noncommutative analogue of the Fig\'a-Talamanca-Herz algebra
on the natural predual of the operator space of
completely bounded Schur multipliers on Schatten space . We determine the
isometric Schur multipliers and prove that the space of bounded
Schur multipliers on Schatten space is the closure in the weak operator
topology of the span of isometric multipliers.Comment: 24 pages; corrected typo
Nonexistence of extremals for the adjoint restriction inequality on the hyperboloid
We study the problem of existence of extremizers for the to
adjoint Fourier restriction inequalities on the hyperboloid in dimensions 3 and
4, in which cases is an even integer. We will use the method developed by
Foschi to show that extremizers do not exist.Comment: 32 pages. Correction for Theorem 1.2 and Proposition 7.5 and addition
of Remark 1.
Generalized and weighted Strichartz estimates
In this paper, we explore the relations between different kinds of Strichartz
estimates and give new estimates in Euclidean space . In
particular, we prove the generalized and weighted Strichartz estimates for a
large class of dispersive operators including the Schr\"odinger and wave
equation. As a sample application of these new estimates, we are able to prove
the Strauss conjecture with low regularity for dimension 2 and 3.Comment: Final version, to appear in the Communications on Pure and Applied
Analysis. 33 pages. 2 more references adde
Dirac Operators and the Calculation of the Connes Metric on arbitrary (Infinite) Graphs
As an outgrowth of our investigation of non-regular spaces within the context
of quantum gravity and non-commutative geometry, we develop a graph Hilbert
space framework on arbitrary (infinite) graphs and use it to study spectral
properties of graph-Laplacians and graph-Dirac-operators. We define a spectral
triplet sharing most of the properties of what Connes calls a spectral triple.
With the help of this scheme we derive an explicit expression for the
Connes-distance function on general directed or undirected graphs. We derive a
series of apriori estimates and calculate it for a variety of examples of
graphs. As a possibly interesting aside, we show that the natural setting of
approaching such problems may be the framework of (non-)linear programming or
optimization. We compare our results (arrived at within our particular
framework) with the results of other authors and show that the seeming
differences depend on the use of different graph-geometries and/or Dirac
operators.Comment: 27 pages, Latex, comlementary to an earlier paper, general treatment
of directed and undirected graphs, in section 4 a series of general results
and estimates concerning the Connes Distance on graphs together with examples
and numerical estimate
Strichartz estimates on Schwarzschild black hole backgrounds
We study dispersive properties for the wave equation in the Schwarzschild
space-time. The first result we obtain is a local energy estimate. This is then
used, following the spirit of earlier work of Metcalfe-Tataru, in order to
establish global-in-time Strichartz estimates. A considerable part of the paper
is devoted to a precise analysis of solutions near the trapping region, namely
the photon sphere.Comment: 44 pages; typos fixed, minor modifications in several place
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