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Edward McDonald

Publications and source records attributed to Edward McDonald.

At least 19 recordsLinked to original sources
arXiv2026

Schur bounded patterns, submajorisation and operator Lipschitz functions

A Schur bounded pattern is a subset $S\subset \mathbb{N}^2$ such that Schur multiplication by every bounded function on $\mathbb{N}^2$ supported on $S$ defines a bounded linear operator in the norm of $\mathcal{B}(\ell_2(\mathbb{N})).$ Schur bounded patterns were characterised by Davidson-Donsig as being unions of row-bounded and column-bounded sets. We study the analogous question for sets $S$ such that element-wise multiplication by every bounded function on $S$ is bounded in ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals $\mathcal{L}_p$ with $0<p<1$ and the weak Schatten ideal $\mathcal{L}_{1,\infty}.$ Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns. This has implications for the functions which are Lipschitz in the norm of ideals that are not closed under submajorisation. In particular such functions must be differentiable and have derivative that is asymptotically constant at infinity.

math.FA↗
arXiv2026

Multiple operator integrals, pseudodifferential calculus, and asymptotic expansions

We push the definition of multiple operator integrals (MOIs) into the realm of unbounded operators, using the pseudodifferential calculus from the works of Connes and Moscovici, Higson, and Guillemin. This in particular provides a natural language for operator integrals in noncommutative geometry. For this purpose, we develop a functional calculus for these pseudodifferential operators. To illustrate the power of this framework, we provide a pertubative expansion of the spectral action for regular $s$-summable spectral triples $(\mathcal{A}, \mathcal{H}, D)$, and an asymptotic expansion of $\mathrm{Tr}(P e^{-t(D+V)^2})$ as $t \downarrow 0$, where $P$ and $V$ belong to the algebra generated by $\mathcal{A}$ and $D$, and $V$ is bounded and self-adjoint.

math.FA↗
arXiv2026

CuCrZr heat-sink irradiation performance reveals new challenges for thermonuclear fusion reactors

Commercial fusion energy requires materials that survive intense neutron bombardment whilst extracting extreme heat loads for conversion to electricity. The CuCrZr alloy, the leading heat-sink material for fusion reactors, derives its strength from a fine dispersion of nano-precipitates formed during prime-ageing heat-treatment. Whether this precipitation-hardening strategy can withstand fusion-relevant irradiation remains untested. Here we show, combining in situ transmission electron microscopy under heavy-ion irradiation and He implantation with thermodynamic and transmutation modelling, that the hardening precipitates dissolve under two opposing kinetic regimes: ballistic dissolution dominates at low temperatures, whilst dissolution and re-precipitation dominate at high temperatures. Although the accelerated dose rates inherent to ion irradiation shift the balance between ballistic mixing and thermal back-diffusion relative to reactor conditions, precipitate degradation at both kinetic extremes indicates that the prime-aged microstructure is unlikely to remain unaltered under prolonged neutron exposure. He bubbles and Kr-rich voids nucleate once vacancies become mobile, and transmutation over five service years irreversibly redirects the alloy chemistry towards Ni-Zr intermetallics. These three independent mechanisms converge to challenge the strategy on which CuCrZr performance depends, suggesting that the long-term performance of age-hardenable Cu-based heat-sink alloys in fusion reactors warrants further assessment. Our findings reveal a new materials challenge for fusion reactor design and commercialisation: the need for new Cu-based heat-sink alloys able to retain engineered strength whilst their chemistry is irreversibly rewritten - thermodynamically and ballistically - by the fusion neutron spectrum.

cond-mat.mtrl-sci↗
arXiv2026

Connes' trace theorem and the log-polyhomogeneous calculus for Carnot manifolds

The Wodzicki residue is the unique trace on the algebra of classical pseudodifferential operators on a closed manifold, and Connes in 1988 proved that it coincides with the Dixmier trace. There are also ``higher" residues, defined on the set of operators whose symbols that can be expressed as polynomials of a logarithm, introduced by Lesch, and these are related to other singular traces. A Carnot manifold is a manifold $M$ whose tangent bundle $TM$ is equipped with a nested family $H$ of sub-bundles $H_0\leq H_1 \leq \cdots \leq TM$ which defines a filtration of the Lie algebra of vector fields on $M.$ Associated to a Carnot manifold is a pseudodifferential calculus $Ψ_H(M),$ which measures sections of $H_k$ as having order $k.$ Recently, Dave-Haller and Couchet-Yuncken proposed definitions of a residue functional on the algebra of pseudodifferential operators adapted to a Carnot manifold. We prove that Connes' trace theorem holds in this setting. We also introduce an analogy of Lesch's log-polyhomogeneous calculus for Carnot manifolds, define the corresponding higher residues, and give their spectral description in terms of singular traces.

math.FA↗
arXiv2026

Fujita exponents on quantum Euclidean spaces

We study the well-posedness of a non-linear heat equation with power nonlinearity with positive initial data on quantum Euclidean spaces. We prove a noncommutative analogue of the classical Fujita theorem by identifying the critical exponent separating finite-time blow-up from global existence for small initial data. Moreover, we establish a fundamental inequality in general semifinite von Neumann algebras that is of independent interest and plays a crucial role in the study of global existence and local well-posedness of solutions of nonlinear equations in noncommutative setting.

math.AP↗
arXiv2026

Weyl's laws and Connes' Trace Theorem for operator-valued pseudo-differential operators

We investigate the spectral asymptotic behavior of operator-valued classical pseudo-differential operators ($Ψ$DOs) for negative order with symbols taking values in a semifinite von Neumann algebran $\mathcal{M}$ equipped with a normal semifinite faithful trace. Within the framework of Connes' noncommutative geometry, we extend Connes' trace theorem to this operator-valued (type II) setting. Our main results are as follows: (i) a symbolic characterization of complex powers for operator-valued elliptic $Ψ$DOs, extending Seeley's classical construction; (ii) a trace formula for localized Riemann $ζ$-functions that links the spectral residues of operator-valued elliptic operators to their principal symbols, thereby providing an operator-valued extension of the Connes--Wodzicki residue; (iii) Weyl's law for right-compactly supported operator-valued classical $Ψ$DOs of arbitrary negative order, which yields a direct spectral proof of the noncommutative integral that bypasses the use of Dixmier traces; (iv) Weyl's law for operator-valued commutators of certain Fourier multipliers with multiplication operators.

math.OA↗
arXiv2026

Weyl asymptotic formulas in the nilpotent Lie group setting

The asymptotic properties of negative order pseudo-differential operators have been an important part of the spectral theory since H.Weyl's classical results. In this paper, we derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols; for these, our results generalize known results of Birman and Solomyak from 1977. Additionally, our work implies a version of Connes' integration formula for hypoelliptic operators on graded Lie groups. Our methods allow us to extend results from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is to adapt the singular value perturbation arguments of Birman and Solomyak to the setting of nilpotent Lie groups. The decomposing of graded Lie groups is inspired by Folland and Stein in their development of harmonic analysis on homogeneous groups.

math.FA↗
arXiv2025

Uniformity of Maximal Hypoellipticity on Graded Lie Groups: From Pointwise to Global

On graded Lie groups, we develop a mechanism that transfers the uniformity of maximal hypoellipcity from the frozen coefficients principal part of a differential operator to the full operator. Our approach brings the century-old "freeze-unfreeze" strategy into the hypoelliptic setting, and offers a transparent and flexible framework for lifting symbol-level hypoelliptic properties to global elliptic estimates, without relying on pseudodifferential calculus. In addition, we prove that symmetric operators of hypoelliptic type on a graded Lie group are self-adjoint.

math.AP↗
arXiv2024

Calderón's commutator on Stratified Lie groups

Motivated by the recent work of Gimperlein and Goffeng on Calderón's commutator on compact Heisenberg type manifolds and the related weak Schatten class estimates, we establish the characterisation of $L^p$ boundedness for Calderon's commutator on stratified Lie groups. We further study related weak Schatten class estimates for second order commutators on two step stratified Lie groups, which include the Heisenberg groups. This latter result is obtained using double operator integral techniques which are novel in this area.

math.FA↗
arXiv2024

A General Dixmier Trace Formula for the Density of States on Open Manifolds

We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of second order elliptic differential operators on manifolds of bounded geometry satisfying a certain geometric condition. This formula gives a new perspective on Roe's index on open manifolds.

math-ph↗
arXiv2023

Nonlinear partial differential equations on noncommutative Euclidean spaces

Noncommutative Euclidean spaces -- otherwise known as Moyal spaces or quantum Euclidean spaces -- are a standard example of a non-compact noncommutative geometry. Recent progress in the harmonic analysis of these spaces gives us the opportunity to highlight some of their peculiar features. For example, the theory of nonlinear partial differential equations has unexpected properties in this noncommutative setting. We develop elementary aspects of paradifferential calculus for noncommutative Euclidean spaces and give some applications to nonlinear evolution equations. We demonstrate how the analysis of some equations radically simplifies in the strictly noncommutative setting.

math.FA↗
arXiv2022

Dixmier Trace Formulas and Negative Eigenvalues of Schroedinger Operators on Curved Noncommutative Tori

In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on "curved" NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these estimates. First, we get $L_p$-versions of the usual formula for the trace of \psidos\ on NC tori, i.e., for combinations of \psidos\ with $L_p$-position operators. Next, we get $L_p$-versions of the analogues for NC tori Connes' trace theorem and Connes' integration formula. They give formulas for the NC integrals (a.k.a.\ Dixmier traces) of products of $L_p$-position operators with \psidos\ or powers of the Laplace-Beltrami operators. Moreover, by combining our Cwikel-type estimates with suitable versions of the Birman-Schwinger principle we get versions of the CLR and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators associated with powers of Laplace-Beltrami operators and $L_p$-potentials. As in the original Euclidean case the Lieb-Thirring inequalities imply a dual Sobolev inequality for orthonormal families. Finally, we discuss spectral asymptotics and semiclassical Weyl's laws for the our classes of operators on curved NC tori. This superseded a previous conjecture in our previous paper.

math.OA↗
arXiv2022

Endpoint weak Schatten class estimates and trace formula for commutators of Riesz transforms with multipliers on Heisenberg groups

Along the line of singular value estimates for commutators by Rochberg-Semmes, Lord-McDonald-Sukochev-Zanin and Fan-Lacey-Li, we establish the endpoint weak Schatten class estimate for commutators of Riesz transforms with multiplication operator $M_f$ on Heisenberg groups via homogeneous Sobolev norm of the symbol $f$. The new tool we exploit is the construction of a singular trace formula on Heisenberg groups, which, together with the use of double operator integrals, allows us to bypass the use of Fourier analysis and provides a solid foundation to investigate the singular values estimates for similar commutators in general stratified Lie groups.

math.FA↗
arXiv2022

Spectral estimates and asymptotics for stratified Lie groups

We study Cwikel-type estimates for the singular values and Schatten $\mathcal{L}_p$-norms of compositions of multiplication and convolution operators acting on stratified Lie groups. This enables us to obtain novel spectral asymptotic formulas for certain operators derived from sub-Laplacians.

math.FA↗
arXiv2021

Cwikel Estimates and Negative Eigenvalues of Schroedinger Operators on Noncommutative Tori

In this paper, we establish Cwikel-type estimates for noncommutative tori for any dimension~$n\geq 2$. We use them to derive Cwikel-Lieb-Rozenblum inequalities and and Lieb-Thirring inequalities for the number of negative eigenvalues of fractional Schroedinger operators on noncommutative tori in any dimension~$n\geq 2$. The latter leads to a Sobolev inequality for noncommutative tori. On the way we establish a "borderline version" of the abstract Birman-Schwinger principle for the number of negative eigenvalues of relatively compact form perturbations of a non-negative semi-bounded operator with isolated 0-eigenvalue.

math.OA↗
arXiv2021

Lipschitz estimates in quasi-Banach Schatten ideals

We study the class of functions $f$ on $\mathbb{R}$ satisfying a Lipschitz estimate in the Schatten ideal $\mathcal{L}_p$ for $0 < p \leq 1$. The corresponding problem with $p\geq 1$ has been extensively studied, but the quasi-Banach range $0 < p < 1$ is by comparison poorly understood. Using techniques from wavelet analysis, we prove that Lipschitz functions belonging to the homogeneous Besov class $\dot{B}^{\frac{1}{p}}_{\frac{p}{1-p},p}(\mathbb{R})$ obey the estimate $$ \|f(A)-f(B)\|_{p} \leq C_{p}(\|f'\|_{L_{\infty}(\mathbb{R})}+\|f\|_{\dot{B}^{\frac{1}{p}}_{\frac{p}{1-p},p}(\mathbb{R})})\|A-B\|_{p} $$ for all bounded self-adjoint operators $A$ and $B$ with $A-B\in \mathcal{L}_p$. In the case $p=1$, our methods recover and provide a new perspective on a result of Peller that $f \in \dot{B}^1_{\infty,1}$ is sufficient for a function to be Lipschitz in $\mathcal{L}_1$. We also provide related Hölder-type estimates, extending results of Aleksandrov and Peller. In addition, we prove the surprising fact that non-constant periodic functions on $\mathbb{R}$ are not Lipschitz in $\mathcal{L}_p$ for any $0 < p < 1$. This gives counterexamples to a 1991 conjecture of Peller that $f \in \dot{B}^{1/p}_{\infty,p}(\mathbb{R})$ is sufficient for $f$ to be Lipschitz in $\mathcal{L}_p$.

math.FA↗
arXiv2021

The density of states depends on the domain

In this short note we demonstrate that the definition of the density of states of a Schrödinger operator with bounded potential in general depends on the choice of the domain undergoing the thermodynamic limit.

math-ph↗