We identify the quantum limits of scattering states for the modular surface. This is obtained through the study of quantum measures of non-holomorphic Eisenstein series away from the critical line. We provide a range of stability for the quantum unique ergodicity theorem of Luo and Sarnak.

We provide a proof of a conjecture by Jakobson, Nadirashvili, and Toth stating that on an $n$-dimensional flat torus ${{\mathbb{T}}^{n}}$, and the Fourier transform of squares of the eigenfunctions ${{\left| \varphi \lambda \right|}^{2}}$ of the Laplacian have uniform ${{l}^{n}}$ bounds that do not depend on the eigenvalue $\lambda $. The proof is a generalization of an argument by Jakobson, et al. for the lower dimensional cases. These results imply uniform bounds for semiclassical limits on ${{\mathbb{T}}^{n+2}}$. We also prove a geometric lemma that bounds the number of codimension-one simplices satisfying a certain restriction on an $n$-dimensional sphere ${{S}^{n}}\,\left( \text{ }\!\!\lambda\!\!\text{ } \right)$ of radius $\sqrt{\lambda }$, and we use it in the proof.

In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a high honour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to a mathematician, and the choice of Arthur, although certainly a recognition of his greatmerits, is also a recognition of the vigour of contemporary Canadian mathematics.

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.

The growth properties at infinity for eigenfunctions corresponding to embedded eigenvalues of the Neumann Laplacian on horn-like domains are studied. For domains that pinch at polynomial rate, it is shown that the eigenfunctions vanish at infinity faster than the reciprocal of any polynomial. For a class of domains that pinch at an exponential rate, weaker, ${{L}^{2}}$ bounds are proven. A corollary is that eigenvalues can accumulate only at zero or infinity.

Errors to a previous paper (Canad. J. Math. (2) 49(1997), 232–262) are corrected. A non-standard regularisation of the auxiliary operator $A$ appearing in Mourre theory is used.

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 $\Gamma $ 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.

We prove a uniform upper estimate on the number of cuspidal eigenvalues of the $\Gamma$-automorphic Laplacian below a given bound when $\Gamma$ varies in a family of congruence subgroups of a given reductive linear algebraic group. Each $\Gamma$ in the family is assumed to contain a principal congruence subgroup whose index in $\Gamma$ does not exceed a fixed number. The bound we prove depends linearly on the covolume of $\Gamma$ 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 $\Gamma$.

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 ∞. The proof uses Mourre theory.

We use the eta invariant to show every non-simply connected spherical space form of dimension m ≥ 5 has a countable family of non bordant metrics of positive scalar curvature.

The purpose for this short note is to describe the space of harmonic spinors on hyperbolic n-space Hn. This is a natural continuation of the study of harmonic functions on Hn in [Minemura] and [Helgason]—these results were generalized in the form of Helgason's conjecture, proved in [KKMOOT],—and of [Gaillard 1, 2], where harmonic forms on Hn were considered. The connection between invariant differential equations on a Riemannian semisimple symmetric space G/K and homological aspects of the representation theory of G, as exemplified in (8) below, does not seem to have been previously mentioned. This note is divided into three main parts respectively dedicated to the statement of the results, some reminders, and the proofs. I thank the referee for having suggested various improvements.

The Neumann operator is an operator on the boundary of a smooth manifold which maps the boundary value of a harmonic function to its normal derivative. The spectrum of the Neumann operator is studied on the curves bounding a family of Riemann surfaces. The Neumann operator is shown to be isospectral to a direct sum of symmetric Jacobi matrices, each acting on l2(ℤ). The Jacobi matrices are shown to be isospectral to generators of bilateral, linear birth-death processes. Using the connection between Jacobi matrices and continued fractions, it is shown that the eigenvalues of the Neumann operator must solve a certain equation involving hypergeometric functions. Study of the equation yields uniform bounds on the eigenvalues and also the asymptotics of the eigenvalues as the curves degenerate into a wedge of circles.

We study the asymptotic behavior of the Laplacian on functions when the underlying Riemannian metric is collapsed to a Carnot-Carathéodory metric. We obtain a uniform short time asymptotics for the trace of the heat kernel in the case when the limit Carnot-Carathéodory metric is almost Heisenberg, the limit of which is the result of Beal-Greiner-Stanton, and Stanton-Tartakoff.

The spectrum of the Laplacian or more generally of a Schrödinger operator on an open manifold may have possibly a complicated aspect. For example, a Cantor set in the real axis may appear as the spectrum even for an innocent looking potential on a standard Riemannian manifold (see J. Moser [10]). The fundamental result of the spectral theory of periodic Schrödinger operators, however, says that the picture of the spectrum of a Schrödinger operator on ℝn with a periodic potential is simple; indeed the spectrum consists of a series of closed intervals of the real axis without accumulation, separated in general by gaps outside the spectrum (see M. Reed and B. Simon [13] or M. M. Skriganov [15] for instance).

Si M est une variété riemannienne de dimension n, n ≧ 2, et de métrique g on s'intéresse au problème: trouver toutes les fonctions régulières F : M → IR qui sont valeurs propres ainsi que leur carré de l'opérateur de Laplace-Beltrami, Δ, associé à g.

This paper studies complex potentials on homogeneous bundles over a compact Lie group. It extends the previous work of V. Guillemin and A. Uribe on potentials isospectral to the zero potential. Then the notion of a universal potential is introduced, that is a potential which acts on sections by a group representation rather than as a scalar. Finally the inverse question of whether the spectral data of a complex potential on all bundles over S2 determines the potential is answered negatively.