On the spectrum of a multidimensional periodic magnetic Shrödinger operator with a singular electric potential

We prove absolute continuity of the spectrum of a periodic $n$-dimensional Schrödinger operator for $n\geqslant 4$. Certain conditions on the magnetic potential $A$ and the electric potential $V+\sum f_j\delta_{S_j}$ are supposed to be fulfilled. In particular, we can assume that the following conditions are satisfied. (1) The magnetic potential $A\colon{\mathbb{R}}^n\to{\mathbb{R}}^n$ either has an absolutely convergent Fourier series or belongs to the space $H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-1$, or to the space $C({\mathbb{R}}^n;{\mathbb{R}}^n)\cap H^q_{\mathrm{loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$, $2q>n-2$. (2) The function $V\colon{\mathbb{R}}^n\to\mathbb{R}$ belongs to Morrey space ${\mathfrak{L}}^{2,p}$, $p\in \big(\frac{n-1}{2},\frac{n}{2}\big]$, of periodic functions (with a given period lattice), and $$\lim\limits_{\tau\to+0}\sup\limits_{0<r\leqslant\tau}\sup\limits_{x\in{\mathbb{R}}^n}r^2\bigg(\big(v(B^n_r)\big)^{-1}\int_{B^n_r(x)}|{\mathcal{V}}(y)|^pdy\bigg)^{1/p}\leqslant C,$$ where $B^n_r(x)$ is a closed ball of radius $r>0$ centered at a point $x\in{\mathbb{R}}^n$, $B^n_r=B^n_r(0)$, $v(B^n_r)$ is volume of the ball $B^n_r$, $C=C(n,p;A)>0$. (3) $\delta_{S_j}$ are $\delta$-functions concentrated on (piecewise) $C^1$-smooth periodic hypersurfaces $S_j$, $f_j\in L^p_{\mathrm{loc}}(S_j)$, $j=1,\ldots,m$. Some additional geometric conditions are imposed on the hypersurfaces $S_j$, and these conditions determine the choice of numbers $p\geqslant n-1$. In particular, let hypersurfaces $S_j$ be $C^2$-smooth, the unit vector $e$ be arbitrarily taken from some dense set of the unit sphere $S^{n-1}$ dependent on the magnetic potential $A$, and the normal curvature of the hypersurfaces $S_j$ in the direction of the unit vector $e$ be nonzero at all points of tangency of the hypersurfaces $S_j$ and the lines $\{x_0+te\colon t\in\mathbb{R}\}$, $x_0\in{\mathbb{R}}^n$. Then we can choose the number $p>\frac{3n}{2}-3$, $n\geqslant 4$.

Special functions and Gauss–Thakur sums in higher rank and dimension

Anderson generating functions have received a growing attention in function field arithmetic in the last years. Despite their introduction by Anderson in the 1980s where they were at the heart of comparison isomorphisms, further important applications, e.g., to transcendence theory have only been discovered recently. The Anderson–Thakur special function interpolates L-values via Pellarin-type identities, and its values at algebraic elements recover Gauss–Thakur sums, as shown by Anglès and Pellarin. For Drinfeld–Hayes modules, generalizations of Anderson generating functions have been introduced by Green–Papanikolas and – under the name of “special functions” – by Anglès, Ngo Dac and Tavares Ribeiro. In this article, we provide a general construction of special functions attached to any Anderson A-module. We show direct links of the space of special functions to the period lattice, and to the Betti cohomology of the A-motive. We also undertake the study of Gauss–Thakur sums for Anderson A-modules, and show that the result of Anglès–Pellarin relating values of the special functions to Gauss–Thakur sums holds in this generality.

On the spectrum of a Landau Hamiltonian with a periodic electric potential $V\in L^p_{\mathrm {loc}}(\mathbb{R}^2)$, $p>1$

We consider the two-dimensional Shrödinger operator $\widehat{H}_B+V$ with a homogeneous magnetic field $B\in {\mathbb R}$ and with an electric potential $V$ which belongs to the space $L^p_{\Lambda } ({\mathbb R}^2;{\mathbb R})$ of $\Lambda $-periodic real-valued functions from the space $L^p_{\mathrm {loc}} ({\mathbb R}^2)$, $p>1$. The magnetic field $B$ is supposed to have the rational flux $\eta =(2\pi )^{-1}Bv(K) \in {\mathbb Q}$ where $v(K)$ denotes the area of the elementary cell $K$ of the period lattice $\Lambda \subset {\mathbb R}^2$. Given $p>1$ and the period lattice $\Lambda $, we prove that in the Banach space $(L^p_{\Lambda } ({\mathbb R}^2;\mathbb R),\| \cdot \| _{L^p(K)})$ there exists a typical set $\mathcal O$ in the sense of Baire (which contains a dense $G_{\delta}$-set) such that the spectrum of the operator $\widehat H_B+V$ is absolutely continuous for any electric potential $V\in {\mathcal O}$ and for any homogeneous magnetic field $B$ with the rational flux $\eta \in {\mathbb Q}$.

The Origin of Long-Period Lattice Spacings Observed in Iron-Carbide Nanowires Encapsulated by Multiwall Carbon Nanotubes

Structures comprising single-crystal, iron-carbon-based nanowires encapsulated by multiwall carbon nanotubes self-organize on inert substrates exposed to the products of ferrocene pyrolysis at high temperature. The most commonly observed encapsulated phases are Fe3C, α-Fe, and γ-Fe. The observation of anomalously long-period lattice spacings in these nanowires has caused confusion since reflections from lattice spacings of ≥0.4 nm are kinematically forbidden for Fe3C, most of the rarely observed, less stable carbides, α-Fe, and γ-Fe. Through high-resolution electron microscopy, selective area electron diffraction, and electron energy loss spectroscopy we demonstrate that the observed long-period lattice spacings of 0.49, 0.66, and 0.44 nm correspond to reflections from the (100), (010), and (001) planes of orthorhombic Fe3C (space group Pnma). Observation of these forbidden reflections results from dynamic scattering of the incident beam as first observed in bulk Fe3C crystals. With small amounts of beam tilt these reflections can have significant intensities for crystals containing glide planes such as Fe3C with space groups Pnma or Pbmn.

Hausdorff dimension of the set of elliptic functions with critical values approaching infinity

Let Λ denote the Weierstrass function with a period lattice Λ. We consider escaping parameters in the family βΛ, i.e. the parameters β for which the orbits of all critical values of βΛ approach infinity under iteration. Unlike the exponential family, the functions considered here are ergodic and admit a non-atomic, σ-finite, ergodic, conservative and invariant measure μ absolutely continuous with respect to the Lebesgue measure. Under additional assumptions on Λ, we estimate the Hausdorff dimension of the set of escaping parameters in the family βΛ from below, and compare it with the Hausdorff dimension of the escaping set in the dynamical space, proving a similarity between the parameter plane and the dynamical space.

PARAMETERIZED DYNAMICS FOR THE WEIERSTRASS ELLIPTIC FUNCTION OVER SQUARE PERIOD LATTICES

We iterate the Weierstrass elliptic ℘ function in order to understand the dependence of the dynamics on the underlying period lattice L. We focus on square lattices and use the holomorphic dependence on the classical invariants (g2, g3) = (g2, 0) to show that in parameter space (g2-space) one sees both quadratic-like attracting orbit behavior and prepole dynamics. In the case of prepole parameters all critical orbits terminate at poles and the Julia set of ℘L is the entire sphere. We show that both the Mandelbrot-like dynamics and the prepole parameters accumulate on prepole parameters of lower order providing results on the dynamics occurring in parameter space "between Mandelbrot sets".

EXPLICIT GEOMETRY ON A FAMILY OF CURVES OF GENUS 3

An explicit geometrical study of the curves[formula here]is presented. These are non-singular curves of genus 3, defined over ℚ(a). By exploiting their symmetries, it is possible to determine most of their geometric invariants, such as their bitangent lines and their period lattice. An explicit description is given of the bijection induced by the Abel–Jacobi map between their bitangent lines and odd 2-torsion points on their jacobian. Finally, three elliptic quotients of these curves are constructed that provide a splitting of their jacobians. In the case of the curve [Cscr ]1±√2, which is isomorphic to the Fermat curve of degree 4, the computations yield a finer splitting of its jacobian than the classical one.