Jacob Shapiro

activity

20162026

most citedThe Bulk-Edge Correspondence for Disordered Chiral Chains

51 citations · 146 across the 27 of their papers we have counts for

collaborators

32 papers

math-ph2026

The $Z_{2}$ -Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

Anna Mazhar, Jacob Shapiro

We define a relative $Z_{2}$ -index $N (ω_{1}, ω_{2})$ for locally-comparable, parity-invariant pure states on a unital $C^{*}$ -algebra. We prove that the index is well-def…

math-ph2026

Fractional Hall Conductance of Laughlin States from a Topological Index

Sven Bachmann, Severin Schraven, Jacob Shapiro +1

We provide a rigorous Laughlin-pump argument that establishes the fractional quantization of the Hall conductance directly from the Laughlin state as a topological index. Our proof…

math-ph2026

Euler Topological Metals in 1D

Yichen Hu, Jacob Shapiro

We give a rigorous one-dimensional formulation of Kane's transport proposal for probing the Euler characteristic of a Fermi sea. For a clean, translation-invariant continuum Hamilt…

math-ph2026

Instantons in a Double-Well are Poisson Distributed

Jacob Shapiro

We give a rigorous realization of the dilute instanton picture for a semiclassical Schrödinger operator with a symmetric double-well potential on $R^{n}$ . Using a localized…

math-ph2025

The index of a pair of pure states and the interacting integer quantum Hall effect

Sven Bachmann, Jacob Shapiro, Clément Tauber

We introduce the index $N (ω_{1}, ω_{2})$ of a pair of pure states on a unital C*-algebra, which is a generalization of the notion of the index of a pair of projections on a Hi…

math-ph2023

Chiral Random Band Matrices at Zero Energy

Jacob Shapiro

We present a special model of random band matrices where, at zero energy, the famous Fyodorov and Mirlin $N$ -conjecture can be established very simply.

math-ph2023

Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems

Jui-Hui Chung, Jacob Shapiro

We study non-interacting electrons in disordered one-dimensional materials which exhibit a spectral gap, in each of the ten Altland-Zirnbauer symmetry classes. We define an appropr…

math.AP2023

Semiclassical resolvent bounds for short range $L^{\infty}$ potentials with singularities at the origin

Jacob Shapiro

We consider, for $h, E > 0$ , resolvent estimates for the semiclassical Schrödinger operator $- h^{2} Δ + V - E$ . Near infinity, the potential takes the form $V = V_{L} + V_{S}$ , where $V_{L}$ …

math.AP2023

Semiclassical estimates for measure potentials on the real line

Andrés Larraín-Hubach, Jacob Shapiro

We prove an explicit weighted estimate for the semiclassical Schrödinger operator $P = - h^{2} \partial_{x}^{2} + V (x; h)$ on $L^{2} (R)$ , with $V (x; h)$ a finite signed measure, and…

math.AP2022

Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation

Kiril Datchev, Jacob Shapiro

We consider the initial-value problem for a one-dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Und…

math-ph2022★ 1 cited

Dynamical Localization for Random Band Matrices up to $W ≪ N^{1/4}$

Giorgio Cipolloni, Ron Peled, Jeffrey Schenker +1

We prove that a large class of $N \times N$ Gaussian random band matrices with band width $W$ exhibits dynamical Anderson localization at all energies when $W ≪ N^{1/4}$ . The proo…

math.AP2021

Semiclassical resolvent bounds for compactly supported radial potentials

Kiril Datchev, Jeffrey Galkowski, Jacob Shapiro

We employ separation of variables to prove weighted resolvent estimates for the semiclassical Schrödinger operator $- h^{2} Δ + V (∣ x ∣) - E$ in dimension $n \geq 2$ , where $h, E > 0$ ,…

math.PR2021★ 3 cited

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Michael Aizenman, Matan Harel, Ron Peled +1

For a family of integer-valued height functions defined over the faces of planar graphs, we establish a relation between the probability of connection by level sets and the spin-sp…

math-ph2021★ 12 cited

Fredholm Homotopies for Strongly-Disordered 2D Insulators

Alex Bols, Jeffrey Schenker, Jacob Shapiro

We study topological indices of Fermionic time-reversal invariant topological insulators in two dimensions, in the regime of strong Anderson localization. We devise a method to int…

math-ph2021★ 1 cited

Incomplete Localization for Disordered Chiral Strips

Jacob Shapiro

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 ene…

math-ph2021★ 9 cited

Is the continuum SSH model topological?

Jacob Shapiro, Michael I. Weinstein

The discrete Hamiltonian of Su, Schrieffer and Heeger (SSH) is a well-known one-dimensional translation-invariant model in condensed matter physics. The model consists of two atoms…

math.AP2020

Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions

Jesse Gell-Redman, Andrew Hassell, Jacob Shapiro

We study the nonlinear Helmholtz equation $(Δ - λ^{2}) u = \pm ∣ u ∣^{p - 1} u$ on $R^{n}$ , $λ > 0$ , $p \in N$ odd, and more generally $(Δ_{g} + V - λ^{2}) u = N [u]$ , where $Δ_{g}$ …

math-ph2020

Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Jacob Shapiro, Michael I. Weinstein

Topological insulators (TIs) are a class of materials which are insulating in their bulk form yet, upon introduction of an a boundary or edge, e.g. by abruptly terminating the mate…

math.AP2020

Semiclassical resolvent bounds for long range Lipschitz potentials

Jeffrey Galkowski, Jacob Shapiro

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 potentia…

math.AP2020

On the interaction of metric trapping and a boundary

Kiril Datchev, Jason Metcalfe, Jacob Shapiro +1

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…

math-ph2020

Lower Bound on Quantum Tunneling for Strong Magnetic Fields

Charles L. Fefferman, Jacob Shapiro, Michael I. Weinstein

We consider a particle bound to a two-dimensional plane and a double well potential, subject to a perpendicular uniform magnetic field . The energy difference between the lowest tw…

math.AP2020

Semiclassical resolvent bounds for weakly decaying potentials

Jeffrey Galkowski, Jacob Shapiro

In this note, we prove weighted resolvent estimates for the semiclassical Schrödinger operator $- h^{2} Δ + V (x) : L^{2} (R^{n}) \to L^{2} (R^{n})$ , $n \neq = 2$ . The potential $…

math.AP2019★ 3 cited

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

Jesse Gell-Redman, Andrew Hassell, Jacob Shapiro +1

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*} wher…

math-ph2019★ 10 cited

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

Eli Fonseca, Jacob Shapiro, Ahmed Sheta +2

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 Fredho…

The Z2​-Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

Fractional Hall Conductance of Laughlin States from a Topological Index

Euler Topological Metals in 1D

Instantons in a Double-Well are Poisson Distributed

The index of a pair of pure states and the interacting integer quantum Hall effect

Chiral Random Band Matrices at Zero Energy

Topological Classification of Insulators: I. Non-interacting Spectrally-Gapped One-Dimensional Systems

Semiclassical resolvent bounds for short range L∞ potentials with singularities at the origin

Semiclassical estimates for measure potentials on the real line

Exponential time-decay for a one dimensional wave equation with coefficients of bounded variation

Dynamical Localization for Random Band Matrices up to W≪N1/4

Semiclassical resolvent bounds for compactly supported radial potentials

Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models

Fredholm Homotopies for Strongly-Disordered 2D Insulators

Incomplete Localization for Disordered Chiral Strips

Is the continuum SSH model topological?

Regularity of the Scattering Matrix for Nonlinear Helmholtz Eigenfunctions

Tight-Binding Reduction and Topological Equivalence in Strong Magnetic Fields

Semiclassical resolvent bounds for long range Lipschitz potentials

On the interaction of metric trapping and a boundary

Lower Bound on Quantum Tunneling for Strong Magnetic Fields

Semiclassical resolvent bounds for weakly decaying potentials

Existence and asymptotics of nonlinear Helmholtz eigenfunctions

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

The $Z_{2}$ -Index of a Pair of Pure States and the Topology of Interacting 1D Superconductors

Semiclassical resolvent bounds for short range $L^{\infty}$ potentials with singularities at the origin

Dynamical Localization for Random Band Matrices up to $W ≪ N^{1/4}$