|
| | | | |
| | | |
| Stat |
Members: 3645
Articles: 2'500'096
Articles rated: 2609
18 April 2024 |
|
| | | | |
|
Article overview
| |
|
|
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 | |
|
|
No review found.
Did you like this article?
Note: answers to reviews or questions about the article must be posted in the forum section.
Authors are not allowed to review their own article. They can use the forum section.
browser claudebot
|
| |
|
|
|
|
News, job offers and information for researchers and scientists:
| |
REGISTER