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Matthias Keller

Publications and source records attributed to Matthias Keller.

At least 37 records · Page 2Linked to original sources
arXiv2024

Large deviations in mean-field quantum spin systems

Continuous fields (or bundles) of $C^*$-algebras form an important ingredient for describing emergent phenomena, such as phase transitions and spontaneous symmetry breaking. In this work, we consider the continuous $C^*$-bundle generated by increasing symmetric tensor powers of the complex $\ell\times\ell$ matrices $M_\ell(\mathbb{C})$, which can be interpreted as abstract description of mean-field theories defining the macroscopic limit of infinite quantum systems. Within this framework we discuss the principle of large deviations for the local Gibbs state in the high temperature regime and characterize the limit of the ensuing logarithmic generating function.

math-ph↗
arXiv2024

The space of Hardy-weights for quasilinear operators on discrete graphs

We study Hardy inequalities for $p$-Schrödinger operators on general weighted graphs. Specifically, we prove a Maz'ya-type result, where we characterize the space of Hardy weights for $ p $-Schrödinger operators via a generalized capacity. The novel ingredient in the proof is the demonstration that the simplified energy of the $ p $-Schrödinger energy functional is compatible with certain normal contractions. As a consequence, we obtain a necessary integrability criterion for Hardy weights. Finally, using some tools of criticality theory, we investigate the existence of minimizers in the Hardy inequalities and discuss relations to Cheeger type estimates.

math.AP↗
arXiv2024

Gaussian upper bounds, volume doubling and Sobolev inequalities on graphs

We investigate the equivalence of Sobolev inequalities and the conjunction of Gaussian upper heat kernel bounds and volume doubling on large scales on graphs. For the normalizing measure, we obtain their equivalence up to constants by imposing comparability of small balls and the vertex degree at their centers. If arbitrary measures are considered, we incorporate a new local regularity condition. Furthermore, new correction functions for the Gaussian, doubling, and Sobolev dimension are introduced. For the Gaussian and doubling, the variable correction functions always tend to one at infinity. Moreover, the variable Sobolev dimension can be related to the doubling dimension and the vertex degree growth.

math.AP↗
arXiv2024

Recurrence and transience for non-Archimedean and directed graphs

We introduce the notion of recurrence and transience for graphs over non-Archimedean ordered field. To do so we relate these graphs to random walks of directed graphs over the reals. In particular, we give a characterization of the real directed graphs which can arise is such a way. As a main result, we give characterization for recurrence and transience in terms of a quantity related to the capacity.

math.CO↗
arXiv2023

Neumann semigroup, subgraph convergence, form uniqueness, stochastic completeness and the Feller property

We study heat kernel convergence of induced subgraphs with Neumann boundary conditions. We first establish convergence of the resulting semigroups to the Neumann semigroup in $\ell^2$. While convergence to the Neumann semigroup always holds, convergence to the Dirichlet semigroup in $\ell^2$ turns out to be equivalent to the coincidence of the Dirichlet and Neumann semigroups while convergence in $\ell^1$ is equivalent to stochastic completeness. We then investigate the Feller property for the Neumann semigroup via generalized solutions and give applications to graphs satisfying a condition on the edges as well as birth-death chains.

math.FA↗
arXiv2023

Capacity of infinite graphs over non-Archimedean ordered fields

In this article we study the notion of capacity of a vertex for infinite graphs over non-Archimedean fields. In contrast to graphs over the real field monotone limits do not need to exist. Thus, in our situation next to positive and null capacity there is a third case of divergent capacity. However, we show that either of these cases is independent of the choice of the vertex and is therefore a global property for connected graphs. The capacity is shown to connect the minimization of the energy, solutions of the Dirichlet problem and existence of a Green's function. We furthermore give sufficient criteria in form of a Nash-Williams test, study the relation to Hardy inequalities and discuss the existence of positive superharmonic functions. Finally, we investigate the analytic features of the transition operator in relation to the inverse of the Laplace operator.

math.AP↗
arXiv2023

A Fully Fiber-Integrated Ion Trap for Portable Optical Atomic Clocks

We present a novel, single-ion trap with integrated optical fibers directly embedded within the trap structure to deliver laser light as well as collect the ion's fluorescence. This eliminates the need for optical windows. We characterise the system's performance and measure signal-to-background ratios in the ion's fluorescence on the order of 50, which allows us to perform state readout with a fidelity over 99% in 600 $μ$s. We test the system's resilience to thermal variations in the range between 22°C and 53°C, and the system's vibration resilience at 34 Hz and 300 Hz and find no effect on its performance. The combination of compactness and robustness of our fiber-coupled trap makes it well suited for applications in, as well as outside, research laboratory environments and in particular for highly compact portable optical atomic clocks. While our system is designed for trapping $^{40}$Ca$^{+}$ ions the fundamental design principles can be applied to other ion species.

physics.atom-ph↗
arXiv2022

Gaussian upper bounds for heat kernels on graphs with unbounded geometry

We prove large-time Gaussian upper bounds for continuous-time heat kernels of Laplacians on graphs with unbounded geometry. Our estimates hold for centers of large balls satisfying a Sobolev inequality and volume doubling. Distances are measured with respect to an intrinsic metric with finite distance balls and finite jump size. The Gaussian decay is given by Davies' function which is natural and sharp in the graph setting. Furthermore, we find a new polynomial correction term which does not blow up at zero. Although our main focus is unbounded Laplacians, the results are new even for the normalized Laplacian. In the case of unbounded vertex degree or degenerating measure, the estimates are affected by new error terms reflecting the unboundedness of the geometry.

math.AP↗
arXiv2022

Anchored heat kernel upper bounds on graphs with unbounded geometry and anti-trees

We derive Gaussian heat kernel bounds on graphs with respect to a fixed origin for large times under the assumption of a Sobolev inequality and volume doubling on large balls. The upper bound from our previous work [KR22] is affected by a new correction term measuring the distance to the origin. The main result is then applied to anti-trees with unbounded vertex degree, yielding Gaussian upper bounds for this class of graphs for the first time. In order to prove this, we show that isoperimetric estimates with respect to intrinsic metrics yield Sobolev inequalities. Finally, we prove that anti-trees are Ahlfors regular and that they satisfy an isoperimetric inequality of a larger dimension.

math.AP↗
arXiv2021

On $L^p$ Liouville theorems for Dirichlet forms

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.

math.FA↗
arXiv2021

Optimal Hardy weights on the Euclidean lattice

We investigate the large-distance asymptotics of optimal Hardy weights on $\mathbb Z^d$, $d\geq 3$, via the super solution construction. For the free discrete Laplacian, the Hardy weight asymptotic is the familiar $\frac{(d-2)^2}{4}|x|^{-2}$ as $|x|\to\infty$. We prove that the inverse-square behavior of the optimal Hardy weight is robust for general elliptic coefficients on $\mathbb Z^d$: (1) averages over large sectors have inverse-square scaling, (2), for ergodic coefficients, there is a pointwise inverse-square upper bound on moments, and (3), for i.i.d.\ coefficients, there is a matching inverse-square lower bound on moments. The results imply $|x|^{-4}$-scaling for Rellich weights on $\mathbb Z^d$. Analogous results are also new in the continuum setting. The proofs leverage Green's function estimates rooted in homogenization theory.

math.AP↗
arXiv2021

Asymptotic expansion of the annealed Green's function and its derivatives

We consider random elliptic equations in dimension $d\geq 3$ at small ellipticity contrast. We derive the large-distance asymptotic expansion of the annealed Green's function up to order $4$ in $d=3$ and up to order $d+2$ for $d\geq 4$. We also derive asymptotic expansions of its derivatives. The obtained precision lies far beyond what is established in prior results in stochastic homogenization theory. Our proof builds on a recent breakthrough in perturbative stochastic homogenization by Bourgain in a refined version shown by Kim and the second author, and on Fourier-analytic techniques of Uchiyama.

math.AP↗
arXiv2021

Eigenvalue asymptotics and unique continuation of eigenfunctions on planar graphs

We study planar graphs with large negative curvature outside of a finite set and the spectral theory of Schr{ö}dinger operators on these graphs. We obtain estimates on the first and second order term of the eigenvalue asymptotics. Moreover, we prove a unique continuation result for eigenfunctions and decay properties of general eigenfunctions. The proofs rely on a detailed analysis of the geometry which employs a Copy-and-Paste procedure based on the Gauß-Bonnet theorem.

math.CO↗
arXiv2020

Improving the Indistinguishability of Single Photons from an Ion-Cavity System

We investigate schemes for generating indistinguishable single photons, a key feature of quantum networks, from a trapped ion coupled to an optical cavity. Through selection of the initial state in a cavity-assisted Raman transition, we suppress the detrimental effect of spontaneous emission present in previously demonstrated schemes in similar systems. We measure a visibility of 72(2)% without correction for background counts in a Hong-Ou-Mandel interference measurement for the new scheme, with 51(2)% for a commonly-used scheme with similar parameters. Schemes such as the one demonstrated here have applications in distributed quantum computing and communications, where high fidelities are vital, and depend on the mutual indistinguishability of single photons.

quant-ph↗