arXiv subjects

S. Charpentier

Publications and source records attributed to S. Charpentier.

6 recordsLinked to original sources

arXiv2024

Weighted holomorphic polynomial approximation

For $G$ an open set in $\mathbb{C}$ and $W$ a non-vanishing holomorphic function in $G$, in the late 1990's, Pritsker and Varga characterized pairs $(G,W)$ having the property that any $f$ holomorphic in $G$ can be locally uniformly approximated in $G$ by weighted holomorphic polynomials $\{W(z)^np_n(z)\}, \ deg(p_n)\leq n$. We further develop their theory in first proving a quantitative Bernstein-Walsh type theorem for certain pairs $(G,W)$. Then we consider the special case where $W(z)=1/(1+z)$ and $G$ is a loop of the lemniscate $\{z\in \mathbb{C}: |z(z+1)|=1/4\}$. We show the normalized measures associated to the zeros of the $n-th$ order Taylor polynomial about $0$ of the function $(1+z)^{-n}$ converge to the weighted equilibrium measure of $\overline G$ with weight $|W|$ as $n\to \infty$. This mimics the motivational case of Pritsker and Varga where $G$ is the inside of the Szego curve and $W(z)=e^{-z}$. Lastly, we initiate a study of weighted holomorphic polynomial approximation in $\mathbb{C}^n, \ n>1$.

math.CV↗

arXiv2023

An extremal problem for the Bergman kernel of orthogonal polynomials

Let $Γ\subset \mathbb C$ be a curve of class $C(2,α)$. For $z_{0}$ in the unbounded component of ${\mathbb C}\setminus Γ$, and for $n=1,2,...$, let $ν_n$ be a probability measure with supp$(ν_{n})\subset Γ$ which minimizes the Bergman function $B_{n}(ν,z):=\sum_{k=0}^{n}|q_{k}^ν(z)|^{2}$ at $z_{0}$ among all probability measures $ν$ on $Γ$ (here, $\{q_{0}^ν,\ldots,q_{n}^ν\}$ are an orthonormal basis in $L^2(ν)$ for the holomorphic polynomials of degree at most $n$). We show that $\{ν_{n}\}_n$ tends weak-* to $\hatδ_{z_{0}}$, the balayage of the point mass at $z_0$ onto $Γ$, by relating this to an optimization problem for probability measures on the unit circle. Our proof makes use of estimates for Faber polynomials associated to $Γ$.

math.CV↗

arXiv2018

Hypercyclic subsets

We completely characterize the finite dimensional subsets A of any separable Hilbert space for which the notion of A-hypercyclicity coincides with the notion of hypercyclicity, where an operator T on a topological vector space X is said to be A-hypercyclic if the set {T n x, n $\ge$ 0, x $\in$ A} is dense in X. We give a partial description for non necessarily finite dimensional subsets. We also characterize the finite dimensional subsets A of any separable Hilbert space H for which the somewhere density in H of {T n x, n $\ge$ 0, x $\in$ A} implies the hypercyclicity of T. We provide a partial description for infinite dimensional subsets. These improve results of Costakis and Peris, Bourdon and Feldman, and Charpentier, Ernst and Menet, and answer a number of related open questions.

math.FA↗

arXiv2016

Incipient Berezinskii-Kosterlitz-Thouless transition in two-dimensional coplanar Josephson junctions

Superconducting hybrid junctions are revealing a variety of novel effects. Some of them are due to the special layout of these devices, which often use a coplanar configuration with relatively large barrier channels and the possibility of hosting Pearl vortices. A Josephson junction with a quasi ideal two-dimensional barrier has been realized by growing graphene on SiC with Al electrodes. Chemical Vapor Deposition offers centimeter size monolayer areas where it is possible to realize a comparative analysis of different devices with nominally the same barrier. In samples with a graphene gap below 400 nm, we have found evidence of Josephson coherence in presence of an incipient Berezinskii-Kosterlitz-Thouless transition. When the magnetic field is cycled, a remarkable hysteretic collapse and revival of the Josephson supercurrent occurs. Similar hysteresis are found in granular systems and are usually justified within the Bean Critical State model (CSM). We show that the CSM, with appropriate account for the low dimensional geometry, can partly explain the odd features measured in these junctions.

cond-mat.supr-con↗

arXiv2016

Josephson coupling in junctions made of monolayer graphene on SiC

Graphene on silicon carbide (SiC) has proved to be highly successful in Hall conductance quantization for its homogeneity at the centimetre scale. Robust Josephson coupling has been measured in co-planar diffusive Al/monololayer graphene/Al junctions. Graphene on SiC substrates is a concrete candidate to provide scalability of hybrid Josephson graphene/superconductor devices, giving also promise of ballistic propagation.

cond-mat.supr-con↗

arXiv2014

Influence of Topological Edge States on the Properties of Al/Bi2Se3/Al Hybrid Josephson Devices

In superconductor-topological insulator-superconductor hybrid junctions, the barrier edge states are expected to be protected against backscattering, to generate unconventional proximity effects, and, possibly, to signal the presence of Majorana fermions. The standards of proximity modes for these types of structures have to be settled for a neat identification of possible new entities. Through a systematic and complete set of measurements of the Josephson properties we find evidence of ballistic transport in coplanar Al-Bi2Se3-Al junctions that we attribute to a coherent transport through the topological edge state. The shunting effect of the bulk only influences the normal transport. This behavior, which can be considered to some extent universal, is fairly independent of the specific features of superconducting electrodes. A comparative study of Shubnikov - de Haas oscillations and Scanning Tunneling Spectroscopy gave an experimental signature compatible with a two dimensional electron transport channel with a Dirac dispersion relation. A reduction of the size of the Bi2Se3 flakes to the nanoscale is an unavoidable step to drive Josephson junctions in the proper regime to detect possible distinctive features of Majorana fermions.

cond-mat.supr-con↗