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 ferm
ions. 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.
In this note we show that the weighted $L^{2}$-Sobolev estimates obtained by P. Charpentier, Y. Dupain & M. Mounkaila for the weighted Bergman projection of the Hilbert space $L^{2}left(Omega,dmu_{0}right)$ where $Omega$ is a smoothly bounded pseudoc
onvex domain of finite type in $mathbb{C}^{n}$ and $mu_{0}=left(-rho_{0}right)^{r}dlambda$, $lambda$ being the Lebesgue measure, $rinmathbb{Q}_{+}$ and $rho_{0}$ a special defining function of $Omega$, are still valid for the Bergman projection of $L^{2}left(Omega,dmuright)$ where $mu=left(-rhoright)^{r}dlambda$, $rho$ being any defining function of $Omega$. In fact a stronger directional Sobolev estimate is established. Moreover similar generalizations are obtained for weighted $L^{p}$-Sobolev and lipschitz estimates in the case of pseudoconvex domain of finite type in $mathbb{C}^{2}$ and for some convex domains of finite type.
We introduce the notion of extremal basis of tangent vector fields at a boundary point of finite type of a pseudo-convex domain in $mathbb{C}^n$. Then we define the class of geometrically separated domains at a boundary point, and give a description
of their complex geometry. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form, and domains for which the Levi form have comparable eigenvalues at a point. Moreover we show that these domains are localizable. Then we define the notion of adapted pluri-subharmonic function to these domains, and we give sufficient conditions for his existence. Then we show that all the sharp estimates for the Bergman ans Szego projections are valid in this case. Finally we apply these results to the examples to get global and local sharp estimates, improving, for examlple, a result of Fefferman, Kohn and Machedon on the Szego projection.
In this paper we investigate the regularity properties of weighted Bergman projections for smoothly bounded pseudo-convex domains of finite type in $mathbb{C}^{n}$. The main result is obtained for weights equal to a non negative rational power of the
absolute value of a special defining function $rho$ of the domain: we prove (weighted) Sobolev-$L^{p}$ and Lipchitz estimates for domains in $mathbb{C}^{2}$ (or, more generally, for domains having a Levi form of rank $geq n-2$ and for decoupled domains) and for convex domains. In particular, for these defining functions, we generalize results obtained by A. Bonami & S. Grellier and D. C. Chang & B. Q. Li. We also obtain a general (weighted) Sobolev-$L^{2}$ estimate.
In the late ten years, the resolution of the equation $barpartial u=f$ with sharp estimates has been intensively studied for convex domains of finite type by many authors. In this paper, we consider the case of lineally convex domains. As the method
used to obtain global estimates for a support function cannot be carried out in this case, we use a kernel that does not gives directly a solution of the $barpartial$-equation but only a representation formula which allows us to end the resolution of the equation using Kohns $L^2$ theory. As an application we give the characterization of the zero sets of the functions of the Nevanlinna class for lineally convex domains of finite type.
