arXiv subjects

Tal Malinovitch

Publications and source records attributed to Tal Malinovitch.

7 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

Spectral Multiplicity Bounds for Jacobi Operators on Star-Like Graphs

We study the spectral multiplicity of Jacobi operators on star-like graphs with $m$ branches. Recently, it was established that the multiplicity of the singular continuous spectrum is at most $m$. Building on these developments and using tools from the theory of generalized eigenfunction expansions, we improve this bound by showing that the singular continuous spectrum has multiplicity at most $m-1$. We also show that this bound is sharp, namely, we construct operators with purely singular continuous spectrum of multiplicity $m-1$.

math.SP↗

arXiv2024

Twisted Bilayer Graphene in Commensurate Angles

We study a 2D continuum model of electronic transport in twisted bilayer graphene (TBG) at commensurate angles. We use two honeycomb potentials with the symmetries of graphene, either sharing a common origin (AA stacking) or shifted by a half-lattice spacing (AB stacking), and twisted relative to each other. While the electronic properties of TBG are most commonly studied via the approximate Bistritzer-MacDonald (BM) model, our approach studies the exact continuum Schr\"{o}dinger operator without these approximations. Our results hold for a wide class of potentials in both stacking types. We describe the exact angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones at the vertices of the Brillouin zone for such angles. Additionally, we establish quantitative bounds showing that, for small potentials, the slope of the Dirac cones flattens at commensurate angles near incommensurate angles. This work is the first to rigorously establish the existence of Dirac cones for twisted bilayer graphene in the continuum setting, without the BM approximations.

math-ph↗

arXiv2024

What is Ballistic Transport?

In this article, we review some notions of ballistic transport from the mathematics and physics literature, describe their basic interrelations, and contrast them with other commonly studied notions of wave packet spread.

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↗

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↗