arXiv subjects

Eva-Maria Hekkelman

Publications and source records attributed to Eva-Maria Hekkelman.

6 recordsLinked to original sources

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

Power counting in the spectral action matrix model

We derive power counting formulas for ribbon graph amplitudes that were recently independently discovered in two contexts, namely as a generalization of the Kontsevich model, and as corresponding to a matrix model approach to the spectral action. The Feynman rules are based on divided difference functions of eigenvalues of an abstract Dirac operator. We obtain formulas for the order of divergence, depending on the spectral dimension $d$, the order of decay of the test function $f$ of the spectral action, and the graph properties. Several consequences are discussed, such as the fact that all graphs with maximal order of divergence (at a given loop order and number of external vertices) are planar. To derive our main results we establish two-sided bounds for divided differences, and in particular generalize Hunter's positivity theorem to a larger class of functions.

math-ph↗

arXiv2025

A noncommutative integral on spectrally truncated spectral triples, and a link with quantum ergodicity

We propose a simple approximation of the noncommutative integral in noncommutative geometry for the Connes--Van Suijlekom paradigm of spectrally truncated spectral triples. A close connection between this approximation and the field of quantum ergodicity and work by Widom in particular immediately provides a Szegő limit formula for noncommutative geometry. We then make a connection to the density of states. Finally, we propose a definition for the ergodicity of geodesic flow for compact spectral triples. This definition is known in quantum ergodicity as uniqueness of the vacuum state for $C^*$-dynamical systems, and for spectral triples where local Weyl laws hold this implies that the Dirac operator of the spectral triple is quantum ergodic. This brings to light a close connection between quantum ergodicity and Connes' integral formula.

math.OA↗

arXiv2025

Trace Formulas in Noncommutative Geometry

Trace formulas appear in many forms in noncommutative geometry (NCG). In the first part of this thesis, we obtain results for asymptotic expansions of trace formulas like heat trace expansions by adapting the theory of Multiple Operator Integration to NCG. More broadly, this construction provides a natural language for operator integrals in NCG, which systematises and simplifies operator integral arguments throughout the literature. Towards this end, we construct a functional calculus for abstract pseudodifferential operators and generalise Peller's construction of multiple operator integrals to this abstract pseudodifferential calculus. In the process, we obtain a noncommutative Taylor formula. In the second part of this thesis, we shift our attention to Dixmier trace formulas. First, we provide an approximation of the noncommutative integral for spectrally truncated spectral triples in the Connes--Van Suijlekom paradigm of operator system spectral triples. Our approximation has a close link to Quantum Ergodicity, which we will use to state an NCG analogue of the fundamental result that ergodic geodesic flow implies quantum ergodicity. Furthermore, we provide a Szegő limit theorem in NCG. Next, we provide a Dixmier trace formula for the density of states, a measure originating in solid state physics that can be associated with an operator on a geometric space. We first provide this formula in the setting of discrete metric spaces, and then in the setting of manifolds of bounded geometry. The latter leads to a Dixmier trace formula for Roe's index on open manifolds.

math.OA↗

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↗

arXiv2022

Truncated Geometry on the Circle

In this letter we prove that the pure state space on the $n \times n$ complex Toeplitz matrices converges in Gromov-Hausdorff sense to the state space on $C(S^1)$ as $n$ grows to infinity, if we equip these sets with the metrics defined by the Connes distance formula for their respective natural Dirac operators. A direct consequence of this fact is that the set of measures on $S^1$ with density functions $c \prod_{j=1}^n (1-\cos(t-θ_j))$ is dense in the set of all positive Borel measures on $S^1$ in the weak$^*$ topology.

math.QA↗