arXiv subjects

Gilad Sofer

Publications and source records attributed to Gilad Sofer.

7 recordsLinked to original sources

arXiv2026

Spectral properties of aperiodic metric and discrete graphs

In this thesis, we study the spectral properties of dynamically defined aperiodic metric and discrete graphs. Our goal is to determine to what extent spectral properties of discrete one-dimensional ergodic Schr\"odinger operators persist when the aperiodicity is manifested through the geometry rather than through a potential. The graphs considered here are inspired by one-dimensional aperiodic tilings, and are called tiling graphs and decorated $\mathbb{Z}$-graphs. For a large family of metric tiling graphs equipped with the standard Laplacian, we show that the spectrum is of zero Lebesgue measure, and is a generalized Cantor set up to a possible discrete set of energies. For decorated $\mathbb{Z}$-graphs, we further show that for a Baire-generic and Lebesgue almost-sure choice of the decoration edge lengths, the spectrum is a generalized Cantor set. We then study the integrated density of states (IDS) for metric and discrete decorated $\mathbb{Z}$-graphs. We prove a gap labelling theorem, which characterizes the set of possible values taken by the IDS inside spectral gaps. We show that the gap labels are contained in the Schwartzman group associated with the dynamical system generating the graph, up to a geometric scaling factor. Lastly, we consider the Dry Ten Martini Problem for discrete Sturmian decorated $\mathbb{Z}$-graphs, asking whether all possible values predicted by the gap labelling theorem are indeed attained by the IDS inside spectral gaps. We answer this question negatively, by identifying a large set of gap labels which are not attained due to jump discontinuities of the IDS. We then show that away from these jump discontinuities, the periodic approximants for Sturmian graphs display the same combinatorial structure as the standard Sturmian Hamiltonians, and use this to obtain an explicit characterization of the realized gap labels for Sturmian comb graphs.

math-ph↗

arXiv2026

Johnson-Schwartzman Gap Labelling for Metric and Discrete Decorated Graphs

We study Schr\"odinger operators on metric and discrete decorated graphs. The values taken by the integrated density of states (IDS) on spectral gaps are called gap labels. A natural question is which gap labels can occur. We answer this for graphs arising from uniquely ergodic one-dimensional dynamical systems by proving Johnson-Schwartzman gap-labelling theorems in both the metric and discrete settings. Our results extend Johnson-Schwartzman gap labelling beyond the standard one-dimensional setting. Unlike in one dimension, these graphs may contain cycles, which prevent the use of Sturm oscillation theory and require different spectral methods. We also analyze discontinuities of the IDS for certain graph families and show that not every admissible label corresponds to an open spectral gap. This reveals a mechanism of gap closing driven by graph geometry rather than by the underlying dynamics.

math.SP↗

arXiv2025

Spectral flow and Robin domains on metric graphs

This paper is devoted to the Neumann-Kirchhoff Laplacian on a finite metric graph. We prove an index theorem relating the nodal deficiency of an eigenfunction with (1) the Morse index of the Dirichlet-to-Neumann map, (2) its positive index and the first Betti number of the graph. We then generalize this result, replacing nodal points of an eigenfunction f with its Robin points (these are points with a prescribed value of f'/f, known as the Robin parameter, or delta coupling, or cotangent of Pr\"ufer angle). This provides the Robin count, a generalization of the nodal and Neumann counts of an eigenfunction. We relate the Robin count deficiency with the positive index of the Robin map (a generalization of the Dirichlet-to-Neumann map). In addition, we show that two of the relevant indices are independent of the Pr\"ufer angle. Our main tool is the spectral flow of the Laplacian with special families of boundary conditions. As an application of our results, we show that the spectral flow of these families is related to topological properties of the graph, such as its Betti number, the number of interaction vertices, and their positions with respect to the graph cycles.

math.SP↗

arXiv2022

Differences between Robin and Neumann eigenvalues on metric graphs

We consider the Laplacian on a metric graph, equipped with Robin ($\delta$-type) vertex condition at some of the graph vertices and Neumann-Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann-Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin-Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains. Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.

math-ph↗

arXiv2022

Spectral curves of quantum graphs with $\delta_s$ type vertex conditions

In this Thesis, we study the behavior of spectral curves of quantum graphs under certain families of vertex conditions, called the $\delta_s$ family, which we define in this work. We focus on studying two main quantities related to the spectral curves, known as the Robin-Neumann gap and the spectral flow. We show that these quantities hold information about the the spectral curves, the behavior of the corresponding eigenfunctions, and the geometry of the graph itself. For a specific subset of the $\delta_s$ family which is known as the $\delta$ family, we study the Robin-Neumann gap, which measures the total increase in the eigenvalues with respect to the perturbation parameter. We use this quantity to show that the growth of the spectral curves is uniformly bounded, and that on average it is linear, with proportionality factor determined by the geometry of the graph. For the general $\delta_s$ family of vertex conditions, we study a quantity known as the spectral flow, which counts the number of oriented intersections of the spectral curves with some given horizontal cross section. We use this quantity to prove an index theorem which connects between a generalized nodal deficiency of the eigenfunctions and the stability index of a generalized Dirichlet-to-Neumann map. We also show that the spectral flow holds information about the graph topology. Parts of the thesis are based on joint work with Ram Band, Marina Prokhorova, Holger Schanz, and Uzy Smilansky.

math-ph↗

arXiv2022

Time evolution and the Schr\"odinger equation on time dependent quantum graphs

The purpose of the present paper is to discuss the time dependent Schr\"odinger equation on a metric graph with time-dependent edge lengths, and the proper way to pose the problem so that the corresponding time evolution is unitary. We show that the well posedness of the Schr\"odinger equation can be guaranteed by replacing the standard Kirchhoff Laplacian with a magnetic Schr\"odinger operator with a harmonic potential. We then generalize the result to time dependent families of vertex conditions. We also apply the theory to show the existence of a geometric phase associated with a slowly changing quantum graph.

math-ph↗

arXiv2020

Three Classification Results In The Theory Of Weighted Hardy Spaces On The Ball

We present a natural family of Hilbert function spaces on the d-dimensional complex unit ball and classify which of them satisfy that subsets of the ball yield isometrically isomorphic subspaces if and only if there is an analytic automorphism of the ball taking one to the other. We also characterize pairs of weighted Hardy spaces on the unit disk which are isomorphic via a composition operator by a simple criterion on their respective sequences of weights.

math.FA↗