arXiv subjects

Adam Black

Publications and source records attributed to Adam Black.

8 recordsLinked to original sources

arXiv2026

Directional Ballistic Transport in Quantum Waveguides

We study the transport properties of Schr\"odinger operators on $\mathbb{R}^d$ with potentials that are periodic in some directions and compactly supported in the others. Such systems are known to produce surface states that are weakly confined near the support of the potential. We show that a natural set of surface states exhibits directional ballistic transport, characterized by ballistic transport in the periodic directions and its absence in the others. To prove this, we develop a Floquet theory that captures the analytic variation of surface states. The main idea consists of reformulating the eigenvalue problem for surface states as a Fredholm problem via the Dirichlet-to-Neumann map.

math.SP↗

arXiv2025

Self-consistent equations and quantum diffusion for the Anderson model

We consider the Anderson tight-binding model on $\mathbb{Z}^d$, $d\geq 2$, with Gaussian noise and at low disorder $\lambda>0$. We derive a diffusive scaling limit for the entries of the resolvent $R(z)$ at imaginary part $\operatorname*{Im} z\sim\lambda^{2+\kappa_d}$, $\kappa_d>0$, with high probability. As consequences, we establish quantum diffusion (in a time-averaged sense) for the Schr\"{o}dinger propagator at the longest timescale known to date and improve the best available lower bounds on the localization length of eigenfunctions. Our results for $d=2$ are the first quantum diffusion results for the Anderson model on $\mathbb{Z}^2$. The proof avoids the use of diagrammatic expansions and instead proceeds by analyzing certain self-consistent equations for $R(z)$. This is facilitated by new estimates for $\|R(z)\|_{\ell^p\rightarrow \ell^q}$ that control the recollisions.

math-ph↗

arXiv2025

Tail bounds for the Dyson series of random Schr\"odinger equations

We study Schr\"odinger equations on $\mathbb{Z}^d$ and $\mathbb{R}^d$, $d\geq 2$ with random potentials of strength $\lambda$. Our main result gives tail bounds for the terms of the Dyson series that are effective at time scales on the order of $\lambda^{-2+\varepsilon}$. As corollaries, we obtain estimates on the frequency localization and spatial delocalization of approximate eigenfunctions in the spirit of works by Schlag-Shubin-Wolff and T. Chen. These estimates also apply to Floquet states associated to time-periodic potentials. Our proof is elementary in that we use neither sophisticated harmonic analysis nor diagrammatic arguments. Instead, we use only the noncommutative Khintchine inequality from random matrix theory combined with pointwise dispersive estimates for the free Schr\"odinger equation.

math-ph↗

arXiv2023

Directional Ballistic Transport for Partially Periodic Schr\"odinger Operators on $\mathbb{Z}^2$

We study the transport properties of Schr\"{o}dinger operators on $\mathbb{Z}^2$ with potentials that are periodic in one direction and compactly supported in the other. Such systems are known to produce surface states that are weakly confined near the support of the potential. We demonstrate that surface states exhibit what we describe as directional ballistic transport, characterized by a strong form of ballistic transport along the periodic direction and its absence in the compactly supported one. By showing that the scattering states exhibit ballistic transport, we obtain ballistic transport for a dense subset of all of $\ell^2(\mathbb {Z} ^2)$.

math.SP↗

arXiv2023

Pointwise decay for radial solutions of the Schr\"odinger equation with a repulsive Coulomb potential

We study the long-time behavior of solutions to the Schr\"odinger equation with a repulsive Coulomb potential on $\mathbb{R}^3$ for spherically symmetric initial data. Our approach involves computing the distorted Fourier transform of the action of the associated Hamiltonian $H=-\Delta+\frac{q}{|x|}$ on radial data $f$, which allows us to explicitly write the evolution $e^{itH}f$. A comprehensive analysis of the kernel is then used to establish that, for large times, $\|e^{i t H}f\|_{L^{\infty}} \leq C t^{-\frac{3}{2}}\|f\|_{L^1}$. Our analysis of the distorted Fourier transform is expected to have applications to other long-range repulsive problems.

math.AP↗

arXiv2022

Scattering for Schr\"{o}dinger operators with conical decay

We study the scattering properties of Schr\"{o}dinger operators with potentials that have short-range decay along a collection of rays in $\bbR^d$. This generalizes the classical setting of short-range scattering in which the potential is assumed to decay along \emph{all} rays. For these operators, we show that any state decomposes into an asymptotically free piece and a piece which may interact with the potential for long times. We give a microlocal characterization of the scattering states in terms of the dynamics and a corresponding description of their complement. We also show that in certain cases these characterizations can be purely spatial.

math-ph↗

arXiv2022

Scattering for Schr\"{o}dinger operators with potentials concentrated near a subspace

We study the scattering properties of Schr\"{o}dinger operators with bounded potentials concentrated near a subspace of $\mathbb{R}^d$. For such operators, we show the existence of scattering states and characterize their orthogonal complement as a set of surface states, which consists of states that are confined to the subspace (such as pure point states) and states that escape it at a sublinear rate, in a suitable sense. We provide examples of surface states for different systems including those that propagate along the subspace and those that escape the subspace arbitrarily slowly. Our proof uses a novel interpretation of the Enss method in order to obtain a dynamical characterisation of the orthogonal complement of the scattering states.

math-ph↗

arXiv2020

Three-dimensional cooling of an atom beam source for high-contrast atom interferometry

We present a compact, two-stage atomic beam source that produces a continuous, narrow, collimated and high-flux beam of rubidium atoms with sub-Doppler temperatures in three dimensions, which features very low emission of near-resonance fluorescence along the atomic trajectory. The atom beam source originates in a pushed two-dimensional magneto-optical trap (2D$^+$ MOT) feeding a slightly off-axis three-dimensional moving optical molasses stage that continuously cools and redirects the atom beam. The capture velocity of the moving optical molasses is deliberately chosen to be low, $\sim 3$ m/s, to reduce fluorescence, and the cooling light is detuned by several atomic linewidths from resonance to reduce the absorption cross-section of cooling-induced fluorescence. Near-resonance light from the 2D$^+$ MOT and the push beam does not propagate to the output atomic trajectory due to a 10 degree bend in the atomic trajectory. The atomic beam emitted from the two-stage source has a flux up to $1.6(3)\times 10^9\;\textrm{atoms/s}$, with an optimized temperature of $15.0(2)\;\mu$K. We employ continuous Raman-Ramsey interference measurements at the atom beam output to study the sources of decoherence in the presence of continuous cooling, and demonstrate that the atom beam source effectively preserves high fringe contrast even during cooling. This cold-atom beam source is appropriate for use in atom interferometers and clocks, where continuous operation eliminates dead time, the slow atom beam velocity (6 - 16 m/s) improves sensitivity, the narrow 3D velocity distribution improves fringe contrast, and the low reabsorption of scattered light mitigates decoherence caused by the continuous cooling process.

physics.atom-ph↗