5,547 research outputs found
Unbounded Laplacians on Graphs: Basic Spectral Properties and the Heat Equation
We discuss Laplacians on graphs in a framework of regular Dirichlet forms. We
focus on phenomena related to unboundedness of the Laplacians. This includes
(failure of) essential selfadjointness, absence of essential spectrum and
stochastic incompleteness
On the ubiquity of Beutler-Fano profiles: from scattering to dissipative processes
Fano models - consisting of a Hamiltonian with discrete-continuous spectrum -
are one of the basic toy models in spectroscopy. They have been succesfull in
explaining the lineshape of experiments in atomic physics and condensed matter.
These models however have largely been out of the scope of dissipative
dynamics, with ony a handful of works considering the effect of a thermal bath.
Yet in nanostructures and condensed matter systems, dissipation strongly
modulates the dynamics. In this article, we present an overview of the
theoretical works dealing with Fano interferences coupled to a thermal bath and
compare them to the scattering formalism. We provide the solution to any
discrete-continuous Hamiltonian structure within the wideband approximation
coupled to a Markovian bath. In doing so, we update the toy models that have
been available for unitary evolution since the 1960s. We find that the Fano
lineshape is preserved as long as we allow a rescaling of the parameters, and
an additional Lorentzian contribution that reflects the destruction of the
interference by dephasings. We discuss the pertinence of each approach -
dissipative and unitary - to different experimental setups: scattering,
transport and spectroscopy of dissipative systems. We finish by discussing the
current limitations of the theories due to the wideband approximation and the
memory effects of the bath.Comment: Expanded bibliography, minor typos correcte
Cauchy conformal fields in dimensions d>2
Holomorphic fields play an important role in 2d conformal field theory. We
generalize them to d>2 by introducing the notion of Cauchy conformal fields,
which satisfy a first order differential equation such that they are determined
everywhere once we know their value on a codimension 1 surface. We classify all
the unitary Cauchy fields. By analyzing the mode expansion on the unit sphere,
we show that all unitary Cauchy fields are free in the sense that their
correlation functions factorize on the 2-point function. We also discuss the
possibility of non-unitary Cauchy fields and classify them in d=3 and 4.Comment: 45 pages; v2: references adde
Diffusion determines the recurrent graph
We consider diffusion on discrete measure spaces as encoded by Markovian
semigroups arising from weighted graphs. We study whether the graph is uniquely
determined if the diffusion is given up to order isomorphism. If the graph is
recurrent then the complete graph structure and the measure space are
determined (up to an overall scaling). As shown by counterexamples this result
is optimal. Without the recurrence assumption, the graph still turns out to be
determined in the case of normalized diffusion on graphs with standard weights
and in the case of arbitrary graphs over spaces in which each point has the
same mass. These investigations provide discrete counterparts to studies of
diffusion on Euclidean domains and manifolds initiated by Arendt and continued
by Arendt/Biegert/ter Elst and Arendt/ter Elst. A crucial step in our
considerations shows that order isomorphisms are actually unitary maps (up to a
scaling) in our context.Comment: 30 page
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