arXiv subjects

Jacob Shapiro

Publications and source records attributed to Jacob Shapiro.

52 records · Page 3Linked to original sources

arXiv2022

Semiclassical resolvent bounds for long range Lipschitz potentials

We give an elementary proof of weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) - E$ in dimension $n \neq 2$, where $h, \, E > 0$. The potential is real-valued, $V$ and $\partial_r V$ exhibit long range decay at infinity, and may grow like a sufficiently small negative power of $r$ as $r \to 0$. The resolvent norm grows exponentially in $h^{-1}$, but near infinity it grows linearly. When $V$ is compactly supported, we obtain linear growth if the resolvent is multiplied by weights supported outside a ball of radius $CE^{-1/2}$ for some $C > 0$. This $E$-dependence is sharp and answers a question of Datchev and Jin.

math.AP↗

arXiv2021

Incomplete Localization for Disordered Chiral Strips

We prove that a disordered analog of the Su-Schrieffer-Heeger model exhibits dynamical localization (i.e. the fractional moments condition) at all energies except possibly zero energy, which is singled out by chiral symmetry. Localization occurs at arbitrarily weak disorder, provided it is sufficiently random. If furthermore the hopping probability measures are properly tuned so that the zero energy Lyapunov spectrum does not contain zero, then the system exhibits localization also at that energy, which is of relevance for topological insulators. The method also applies to the usual Anderson model on the strip.

math-ph↗

arXiv2021

On the interaction of metric trapping and a boundary

By considering a two ended warped product manifold, we demonstrate a bifurcation that can occur when metric trapping interacts with a boundary. In this highly symmetric example, as the boundary passes through the trapped set, one goes from a nontrapping scenario where lossless local energy estimates are available for the wave equation to the case of stably trapped rays where all but a logarithmic amount of decay is lost.

math.AP↗

arXiv2020

Semiclassical resolvent bounds for weakly decaying potentials

In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator $-h^2 Δ+ V(x) : L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed to either decay at infinity or to obey a radial $α$-Hölder continuity condition, $0\leq α\leq 1$, with sufficient decay of the local radial $C^α$ norm toward infinity. Note, however, that in the Hölder case, the potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent bound is of the form $\exp \left(C h^{-1 - \frac{1 - α}{3 + α}} [(1-α) \log(h^{-1})+c]\right)$, while for $n = 1$ it is of the form $\exp(Ch^{-1})$. A new type of weight and phase function construction allows us to reduce the necessary decay even in the pure $L^\infty$ case.

math.AP↗

arXiv2019

Semiclassical Estimates for Scattering on the Real Line

We prove explicit semiclassical resolvent estimates for an integrable potential on the real line. The proof is a comparatively easy case of the spherical energies method, which has been used to prove similar theorems in higher dimensions and in more complicated geometric situations. The novelty in our results lies in the weakness of the assumptions on the potential.

math.AP↗

arXiv2019

The Topology of Mobility-Gapped Insulators

Studying deterministic operators, we define an appropriate topology on the space of mobility-gapped insulators such that topological invariants are continuous maps into discrete spaces, we prove that this is indeed the case for the integer quantum Hall effect, and lastly we show why our "insulator" condition makes sense from the point of view of the localization theory using the fractional moments method.

math-ph↗

arXiv2019

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

We prove the existence and asymptotic expansion of a large class of solutions to nonlinear Helmholtz equations of the form \begin{equation*} (Δ- λ^2) u = N[u], \end{equation*} where $Δ= -\sum_j \partial^2_j$ is the Laplacian on $\mathbb{R}^n$ with sign convention that it is positive as an operator, $λ$ is a positive real number, and $N[u]$ is a nonlinear operator that is a sum of monomials of degree $\geq p$ in $u$, $\overline{u}$ and their derivatives of order up to two, for some $p \geq 2$. Nonlinear Helmholtz eigenfunctions with $N[u]= \pm |u|^{p-1} u$ were first considered by Gutiérrez. Such equations are of interest in part because, for certain nonlinearities $N[u]$, they furnish standing waves for nonlinear evolution equations, that is, solutions that are time-harmonic. We show that, under the condition $(p-1)(n-1)/2 > 2$ and $k > (n-1)/2$, for every $f \in H^{k+2}(\mathbb{S}^{n-1})$ of sufficiently small norm, there is a nonlinear Helmholtz function taking the form \begin{equation*} u(r, ω) = r^{-(n-1)/2} \Big( e^{-iλr} f(ω) + e^{+iλr} g(ω) + O(r^{-ε}) \Big), \text{ as } r \to \infty, \quad ε> 0, \end{equation*} for some $g \in H^{k}(\mathbb{S}^{n-1})$. Moreover, we prove the result in the general setting of asymptotically conic manifolds.

math.AP↗

arXiv2019

Two-Dimensional Time-Reversal-Invariant Topological Insulators via Fredholm Theory

We study spinful non-interacting electrons moving in two-dimensional materials which exhibit a spectral gap about the Fermi energy as well as time-reversal invariance. Using Fredholm theory we revisit the (known) bulk topological invariant, define a new one for the edge, and show their equivalence (the bulk-edge correspondence) via homotopy.

math-ph↗

arXiv2018

Strongly Disordered Floquet Topological Systems

We study the strong disorder regime of Floquet topological systems in dimension two, that describe independent electrons on a lattice subject to a periodic driving. In the spectrum of the Floquet propagator we assume the existence of an interval in which all states are localized--a mobility gap. First we generalize the relative construction from spectral to mobility gap, define a bulk index for an infinite sample and an edge index for the half-infinite one and prove the bulk-edge correspondence. Second, we consider completely localized systems where the mobility gap is the whole circle, and define alternative bulk and edge indices that circumvent the relative construction and match with quantized magnetization and pumping observables from the physics literature. Finally, we show that any system with a mobility gap can be reduced to a completely localized one. All the indices defined throughout are equal.

math-ph↗

arXiv2018

Mining the GPIES database

The Gemini Planet Imager Exoplanet Survey (GPIES) is a direct imaging campaign designed to search for young, self-luminous, giant exoplanets. To date, GPIES has observed nearly 500 targets, and generated over 30,000 individual exposures using its integral field spectrograph (IFS) instrument. The GPIES team has developed a campaign data system with a database incorporating all of the metadata for all individual raw data products, including environmental conditions and instrument performance metrics. The same database also indexes metadata associated with multiple levels of reduced data products, including contrast measures for individual images and combined image sequences, which serve as the primary metric of performance for the final science products. The database is also used to track telemetry products from the adaptive optics subsystem, and associate these with corresponding IFS data. Here, we discuss several data exploration and visualization projects enabled by the GPIES database. Of particular interest are any correlations between instrument performance and environmental or operating conditions. We show single and multiple-parameter fits of single-image and observing sequence contrast as functions of various seeing measures, and discuss automated outlier rejection and other fitting concerns. Supervised learning techniques are employed in order to partition the space of raw (single image) to final (full sequence) contrast in order to better predict the value of the final data set from the first few completed observations. Finally, we discuss the particular features of the database design that aid in performing these analyses, and suggest potential future upgrades and refinements.

astro-ph.IM↗

arXiv2018

Local energy decay for Lipschitz wavespeeds

We prove a logarithmic local energy decay rate for the wave equation with a wavespeed that is a compactly supported Lipschitz perturbation of unity. The key is to establish suitable resolvent estimates at high and low energy for the meromorphic continuation of the cutoff resolvent. The decay rate is the same as that proved by Burq for a smooth perturbation of the Laplacian outside an obstacle.

math.AP↗

arXiv2018

Semiclassical resolvent bound for compactly supported $L^\infty$ potentials

We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension $n \ge 1$. We require the potential belong to $L^\infty(\mathbb{R}^n)$ and have compact support, but do not require that it have derivatives in $L^\infty(\mathbb{R}^n)$. The weighted resolvent norm is bounded by $e^{Ch^{-4/3}\log(h^{-1})}$, where $h$ is the semiclassical parameter.

math.AP↗

arXiv2018

The Bulk-Edge Correspondence for Disordered Chiral Chains

We study one-dimensional insulators obeying a chiral symmetry in the single-particle picture. The Fermi level is assumed to lie in a mobility gap. Topological indices are defined for infinite (bulk) or half-infinite (edge) systems, and it is shown that for a given Hamiltonian with nearest neighbor hopping the two indices are equal. We also give a new formulation of the index in terms of the Lyapunov exponents of the zero energy Schrödinger equation, which illustrates the conditions for a topological phase transition occurring in the mobility gap regime.

math-ph↗

arXiv2018

Automated data processing architecture for the Gemini Planet Imager Exoplanet Survey

The Gemini Planet Imager Exoplanet Survey (GPIES) is a multi-year direct imaging survey of 600 stars to discover and characterize young Jovian exoplanets and their environments. We have developed an automated data architecture to process and index all data related to the survey uniformly. An automated and flexible data processing framework, which we term the Data Cruncher, combines multiple data reduction pipelines together to process all spectroscopic, polarimetric, and calibration data taken with GPIES. With no human intervention, fully reduced and calibrated data products are available less than an hour after the data are taken to expedite follow-up on potential objects of interest. The Data Cruncher can run on a supercomputer to reprocess all GPIES data in a single day as improvements are made to our data reduction pipelines. A backend MySQL database indexes all files, which are synced to the cloud, and a front-end web server allows for easy browsing of all files associated with GPIES. To help observers, quicklook displays show reduced data as they are processed in real-time, and chatbots on Slack post observing information as well as reduced data products. Together, the GPIES automated data processing architecture reduces our workload, provides real-time data reduction, optimizes our observing strategy, and maintains a homogeneously reduced dataset to study planet occurrence and instrument performance.

astro-ph.IM↗

arXiv2017

The Bulk-Edge Correspondence in Three Simple Cases

We present examples in three symmetry classes of topological insulators in one or two dimensions where the proof of the bulk-edge correspondence is particularly simple. This serves to illustrate the mechanism behind the bulk-edge principle without the overhead of the more general proofs which are available. We also give a new formula for the \mathbb{Z}_{2}-index of our time-reversal invariant systems inspired by Moore and Balents.

math-ph↗

arXiv2017

Semiclassical resolvent bounds in dimension two

We give an elementary proof of weighted resolvent bounds for semiclassical Schrödinger operators in dimension two. We require the potential function to be Lipschitz with long range decay. The resolvent norm grows exponentially in the inverse semiclassical parameter, but near infinity it grows linearly. Our result covers the missing case from the work of Datchev.

math.AP↗