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Operator theory of electrical resistance networks | Palle E. T. Jorgensen
; Erin P. J. Pearse
; | Date: |
23 Jun 2008 | Abstract: | An electrical resistance network (ERN) is a weighted graph $(G,mu)$ with
intrinsic (resistance) metric $R$. We embed the ERN into a Hilbert space
${mathcal H}_{mathcal E}$ of functions of finite energy, in a way that
respects the structure of the Laplace operator $Delta$ and the Dirichlet
energy form $mathcal E$ on the ERN. We study ${mathcal H}_{mathcal E}$ and
show that the embedded images of the vertices ${v_x}$ form a reproducing
kernel for this Hilbert space. We also show that ERNs which support nonconstant
harmonic functions of finite energy have a certain type of emph{boundary}. We
obtain an analytic boundary representation for the harmonic functions of finite
energy in a sense analogous to the Poisson or Martin boundary representations,
but with entirely different hypotheses. In the process, we construct a dense
space of ’’smooth’’ functions of finite energy and obtain a Gel’fand triple for
${mathcal H}_{mathcal E}$, thus providing a method for performing a mildly
restricted form of Fourier analysis on a general ERN.
We also study the spectral representation for $Delta$ on ${mathcal
H}_{mathcal E}$ and show how the boundary of an ERN corresponds to the
deficiency indices of $Delta$, and hence how boundaries (and many other
interesting phenomena) are detected by the operator theory of ${mathcal
H}_{mathcal E}$ but not $ell^2$.
Our results apply to the Heisenberg model for the isotropic ferromagnet,
improving earlier results of R. T. Powers on the problem of long-range order
(in reference to KMS states on the $C^ast$-algebra of the model). | Source: | arXiv, 0806.3881 | Services: | Forum | Review | PDF | Favorites |
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