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Register a new userOn a spectral theorem of Weyl
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We give a geometric proof of a theorem of Weyl on the continuous part of the spectrum of Sturm-Liouville operators on the half-line with asymptotically constant coefficients. Earlier proofs due to Weyl and Kodaira depend on special features of Greens functions for linear ordinary differential operators; ours might offer better prospects for generalization to higher dimensions, as required for example in noncommutative harmonic analysis.
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We study bounded operators defined in terms of the regular representations of the $C^*$-algebra of an amenable, Hausdorff, second countable locally compact groupoid endowed with a continuous $2$-cocycle. We concentrate on spectral quantities associated to natural quotients of this twisted algebra, such as the essential spectrum, the essential numerical range, and Fredholm properties. We obtain decompositions for the regular representations associated to units of the groupoid belonging to a free locally closed orbit, in terms of spectral quantities attached to points (or orbits) in the boundary of this main orbit. As examples, we discuss various classes of magnetic pseudo-differential operators on nilpotent groups. We also prove localization and non-propagation properties associated to suitable parts of the essential spectrum. These are applied to twisted groupoids having a totally intransitive groupoid restriction at the boundary.
After recalling in detail some basic definitions on Hilbert C*-bimodules, Morita equivalence and imprimitivity, we discuss a spectral reconstruction theorem for imprimitivity Hilbert C*-bimodules over commutative unital C*-algebras and consider some of its applications in the theory of commutative full C*-categories.
Given two Hilbert spaces, $mathcal{H}$ and $mathcal{K}$, we introduce an abstract unitary operator $U$ on $mathcal{H}$ and its discriminant $T$ on $mathcal{K}$ induced by a coisometry from $mathcal{H}$ to $mathcal{K}$ and a unitary involution on $mathcal{H}$. In a particular case, these operators $U$ and $T$ become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between $U$ and $T$ via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpinski lattice, which is pure point and has a Cantor-like structure.
We prove a Tauberian theorem for singular values of noncommuting operators which allows us to prove exact asymptotic formulas in noncommutative geometry at a high degree of generality. We explain how, via the Birman--Schwinger principle, these asymptotics imply that a semiclassical Weyl law holds for many interesting noncommutative examples. In Connes notation for quantized calculus, we prove that for a wide class of $p$-summable spectral triples $(mathcal{A},H,D)$ and self-adjoint $V in mathcal{A}$, there holds [lim_{hdownarrow 0} h^pmathrm{Tr}(chi_{(-infty,0)}(h^2D^2+V)) = int V_-^{frac{p}{2}}|ds|^p.] where $int$ is Connes noncommutative integral.
In 1975, P.R. Chernoff used iterates of the Laplacian on $mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoffs theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.
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