arXiv subjects

Edward McDonald

Publications and source records attributed to Edward McDonald.

27 records · Page 2Linked to original sources

arXiv2021

Semiclassical Weyl law and exact spectral asymptotics in noncommutative geometry

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 \[\lim_{h\downarrow 0} h^p\mathrm{Tr}(χ_{(-\infty,0)}(h^2D^2+V)) = \int V_-^{\frac{p}{2}}|ds|^p.\] where $\int$ is Connes' noncommutative integral.

math.OA↗

arXiv2021

Estimates for Schur Multipliers and Double Operator Integrals -- A Wavelet Approach

We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $B^{\frac{1}{p}-\frac{1}{2}}_{p,p}(\mathbb R,L_2(\mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{\frac{1}{p}-\frac{1}{2}}_{\frac{2p}{2-p},p}(\mathbb R,L_\infty(\mathbb R))$, which extends Birman and Solomyak's result to symbols without compact domain.

math.CA↗

arXiv2021

Necessary and sufficient conditions for $n$-times Fréchet differentiability on ${\mathcal S}^p,$ $1 <p<\infty.$

Let $1<p<\infty$ and let $n\geq 1$. It is proved that a function $f:{\mathbb R}\to {\mathbb C}$ is $n$-times Fréchet differentiable on ${\mathcal S}^p$ at every self-adjoint operator if and only if $f$ is $n$-times differentiable, $f',f'',\ldots,f^{(n)}$ are bounded and $f^{(n)}$ is uniformly continuous.

math.FA↗

arXiv2020

A Dixmier trace formula for the density of states

A version of Connes trace formula allows to associate a measure on the essential spectrum of a Schrödinger operator with bounded potential. In solid state physics there is another celebrated measure associated with such operators --- the density of states. In this paper we demonstrate that these two measures coincide. We show how this equality can be used to give explicit formulae for the density of states in some circumstances.

math-ph↗

arXiv2019

Quantum differentiability on quantum tori

We provide a full characterisation of quantum differentiability (in the sense of Connes) on quantum tori. We also prove a quantum integration formula which differs substantially from the commutative case.

math.OA↗

arXiv2019

Quantum differentiability on noncommutative Euclidean spaces

We study the topic of quantum differentiability on quantum Euclidean $d$-dimensional spaces (otherwise known as Moyal $d$-spaces), and we find conditions that are necessary and sufficient for the singular values of the quantised differential to have decay $O(n^{-α})$ for $0 < α\leq \frac{1}{d}$. This result is substantially more difficult than the analogous problems for Euclidean space and for quantum $d$-tori.

math.OA↗

arXiv2019

Quantum Euclidean Spaces with Noncommutative Derivatives

Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semi-finite spectral triple with non-flat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a simplified local index formula (even case) that is similar to the commutative setting.

math.OA↗

arXiv2019

Noncommutative Geometry for Symmetric Non-Self-Adjoint Operators

We introduce the notion of a pre-spectral triple, which is a generalisation of a spectral triple $(\mathcal{A}, H, D)$ where $D$ is no longer required to be self-adjoint, but closed and symmetric. Despite having weaker assumptions, pre-spectral triples allow us to introduce noncompact noncommutative geometry with boundary. In particular, we derive the Hochschild character theorem in this setting. We give a detailed study of Dirac operators with Dirichlet boundary conditions on open subsets of $\mathbb{R}^d$, $d \geq 2$.

math.OA↗

arXiv2018

A $C^*$-algebraic approach to the principal symbol II

We introduce an abstract theory of the principal symbol mapping for pseudodifferential operators extending the results of a preceding paper and providing a simple algebraic approach to the theory of pseudodifferential operators in settings important in noncommutative geometry. We provide a variant of Connes' trace theorem which applies to certain noncommutative settings, with a minimum of technical preliminaries. Our approach allows us to consider in a operators with non-smooth symbols, and we demonstrate the power of our approach by extending Connes' trace theorem to operators with non-smooth symbols in three examples: the Lie group $\mathrm{SU}(2)$, noncommutative tori and Moyal planes.

math.OA↗