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1. CJM 2000 (vol 52 pp. 1057)
The Spectrum of an Infinite Graph In this paper, we consider the (essential) spectrum of the discrete
Laplacian of an infinite graph. We introduce a new quantity for an
infinite graph, in terms of which we give new lower bound estimates of
the (essential) spectrum and give also upper bound estimates when the
infinite graph is bipartite. We give sharp estimates of the
(essential) spectrum for several examples of infinite graphs.
Keywords:infinite graph, discrete Laplacian, spectrum, essential spectrum Categories:05C50, 58G25 |
2. CJM 2000 (vol 52 pp. 119)
Corrigendum to ``Spectral Theory for the Neumann Laplacian on Planar Domains with Horn-Like Ends'' Errors to a previous paper (Canad. J. Math. (2) {\bf 49}(1997),
232--262) are corrected. A non-standard regularisation of the
auxiliary operator $A$ appearing in Mourre theory is used.
Categories:35P25, 58G25, 47F05 |
3. CJM 1999 (vol 51 pp. 952)
On Limit Multiplicities for Spaces of Automorphic Forms Let $\Gamma$ be a rank-one arithmetic subgroup of a
semisimple Lie group~$G$. For fixed $K$-Type, the spectral
side of the Selberg trace formula defines a distribution
on the space of infinitesimal characters of~$G$, whose
discrete part encodes the dimensions of the spaces of
square-integrable $\Gamma$-automorphic forms. It is shown
that this distribution converges to the Plancherel measure
of $G$ when $\Ga$ shrinks to the trivial group in a certain
restricted way. The analogous assertion for cocompact
lattices $\Gamma$ follows from results of DeGeorge-Wallach
and Delorme.
Keywords:limit multiplicities, automorphic forms, noncompact quotients, Selberg trace formula, functional calculus Categories:11F72, 22E30, 22E40, 43A85, 58G25 |
4. CJM 1999 (vol 51 pp. 266)
Spectral Estimates for Towers of Noncompact Quotients We prove a uniform upper estimate on the number of cuspidal
eigenvalues of the $\Ga$-automorphic Laplacian below a given bound
when $\Ga$ varies in a family of congruence subgroups of a given
reductive linear algebraic group. Each $\Ga$ in the family is assumed
to contain a principal congruence subgroup whose index in $\Ga$ does
not exceed a fixed number. The bound we prove depends linearly on the
covolume of $\Ga$ and is deduced from the analogous result about the
cut-off Laplacian. The proof generalizes the heat-kernel method which
has been applied by Donnelly in the case of a fixed lattice~$\Ga$.
Categories:11F72, 58G25, 22E40 |
5. CJM 1997 (vol 49 pp. 232)
Spectral theory for the Neumann Laplacian on planar domains with horn-like ends The spectral theory for the Neumann Laplacian on planar domains with
symmetric, horn-like ends is studied. For a large class of such domains,
it is proven that the Neumann Laplacian has no singular continuous
spectrum, and that the pure point spectrum consists of eigenvalues
of finite multiplicity which can accumulate only at $0$ or $\infty$.
The proof uses Mourre theory.
Categories:35P25, 58G25 |