arXiv subjects

Ram Band

Publications and source records attributed to Ram Band.

At least 19 recordsLinked to original sources

arXiv2026

Complete hierarchical structure of the spectral bands in the Kohmoto model

We study the Kohmoto model, a family of discrete Schr\"odinger operators with Sturmian potentials depending on a frequency and a coupling constant. We prove that, for all non-vanishing coupling constants, all spectral bands admit a hierarchical structure. This structure offers a variety of applications, including a detailed description of the Kohmoto butterfly and a central step towards the resolution of the dry ten Martini problem for Sturmian Hamiltonians, which we carry out in a subsequent work.

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arXiv2026

Spectral pollution in substitution systems

We study spectral properties of Schr\"odinger operators associated with substitution dynamical systems in higher dimensions. Focusing on periodic approximations generated by iterating substitutions on initial configurations, we analyze how structural defects influence the limiting spectral behavior. In contrast to the one-dimensional setting, we show that such approximations may exhibit significant spectral pollution, including changes in the essential spectrum and the Lebesgue measure.

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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.

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arXiv2025

From Quasiperiodicity to a Complete Coloring of the Kohmoto Butterfly

The spectra of the Kohmoto model give rise to a fractal phase diagram, known as the Kohmoto butterfly. The butterfly encapsulates the spectra of all periodic Kohmoto Hamiltonians, whose index invariants are sought after. Topological methods - such as Chern numbers - are ill defined due to the discontinuous potential, and hence fail to provide index invariants. This Letter overcomes that obstacle and provides a complete classification of the Kohmoto model indices. Our approach encodes the Kohmoto butterfly as a spectral tree graph, reflecting the quasiperiodic nature via the periodic spectra. This yields a complete coloring of the phase diagram and a new perspective on other spectral butterflies.

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arXiv2025

Spectral statistics of preferred orientation quantum graphs

We study the spectral statistics of quantum (metric) graphs whose vertices are equipped with preferred orientation vertex conditions. When comparing their spectral statistics to those predicted by suitable random matrix theory ensembles, one encounters some deviations. We point out these discrepancies and demonstrate that they occur in various graphs and even for Neumann-Kirchhoff vertex conditions, which was overlooked so far. Detailed explanations and computations are provided for this phenomena. To achieve this, we explore the combinatorics of periodic orbits, with a particular emphasis on counting Eulerian cycles.

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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.

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arXiv2024

A review of a work by Raymond: Sturmian Hamiltonians with a large coupling constant -- periodic approximations and gap labels

We present a review of the work L. Raymond from 1995. The review aims at making this work more accessible and offers adaptations of some statements and proofs. In addition, this review forms an applicable framework for the complete solution of the Dry Ten Martini Problem for Sturmian Hamiltonians as appears in the work arXiv:2402.16703 by R. Band, S. Beckus and R. Loewy. A Sturmian Hamiltonian is a one-dimensional Schr\"odinger operator whose potential is a Sturmian sequence multiplied by a coupling constant, $V\in\mathbb{R}$. The spectrum of such an operator is commonly approximated by the spectra of designated periodic operators. If $V>4$, then the spectral bands of the periodic operators exhibit a particular combinatorial structure. This structure provides a formula for the integrated density of states. Employing this, it is shown that if $V>4$, then all the gaps, as predicted by the gap labelling theorem, are there.

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arXiv2024

Spectral approximation for substitution systems

We study periodic approximations of aperiodic Schr\"odinger operators on lattices in Lie groups with dilation structure. The potentials arise through symbolic substitution systems that have been recently introduced in this setting. We characterize convergence of spectra of associated Schr\"odinger operators in the Hausdorff distance via properties of finite graphs. As a consequence, new examples of periodic approximations are obtained. We further prove that there are substitution systems that do not admit periodic approximations in higher dimensions, in contrast to the one-dimensional case. On the other hand, if the spectra converge, then we show that the rate of convergence is necessarily exponentially fast. These results are new even for substitutions over $\mathbb{Z}^d$.

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arXiv2024

The Dry Ten Martini Problem for Sturmian Hamiltonians

The dry ten Martini problem for Sturmian Hamiltonians is solved. Concretely, we prove that all the predicted spectral gaps "are there" for all the Schr\"odinger operators with Sturmian potentials and non-vanishing coupling constant. A key approach towards the solution is a representation of the spectrum as the boundary of an infinite tree. This tree is constructed using periodic approximations and encodes substantial spectral characteristics.

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arXiv2023

Sturm-Hurwitz Theorem for quantum graphs

We prove upper and lower bounds for the number of zeroes of linear combinations of Schr\"odinger eigenfunctions on metric (quantum) graphs. These bounds are distinct from both the interval and manifolds. We complement these bounds by giving non-trivial examples for the lower bound as well as sharp examples for the upper bound. In particular, we show that even tree graphs differ from the interval with respect to the nodal count of linear combinations of eigenfunctions. This stands in distinction to previous results which show that all tree graphs have to same eigenfunction nodal count as the interval.

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arXiv2023

MFO Report: The dry ten Martini problem for Sturmian dynamical systems

This extended Oberwolfach report (to appear in the proceedings of the MFO Workshop 2335: Aspects of Aperiodic Order) announces the full solution to the Dry Ten Martini Problem for Sturmian Hamiltonians. Specifically, we show that all spectral gaps of Sturmian Hamiltonians (as predicted by the gap labeling theorem) are open for all nonzero couplings and all irrational rotations. We present here the proof strategy.

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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.

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arXiv2021

Universality of nodal count distribution in large metric graphs

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph's non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph's first Betti number $\beta$. We study the distribution of the nodal surplus values in the countably infinite set of the graph's eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing $\beta$. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.

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arXiv2020

Defining the spectral position of a Neumann domain

A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. a Morse--Smale complex). This partition is generated by gradient flow lines of the eigenfunction, which bound the so-called Neumann domains. We prove that the Neumann Laplacian defined on a Neumann domain is self-adjoint and has a purely discrete spectrum. In addition, we prove that the restriction of an eigenfunction to any one of its Neumann domains is an eigenfunction of the Neumann Laplacian. By comparison, similar statements about the Dirichlet Laplacian on a nodal domain of an eigenfunction are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.

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arXiv2019

Neumann Domains on Quantum Graphs

The Neumann points of an eigenfunction $f$ on a quantum (metric) graph are the interior zeros of $f'$. The Neumann domains of $f$ are the sub-graphs bounded by the Neumann points. Neumann points and Neumann domains are the counterparts of the well-studied nodal points and nodal domains. We prove some foundational results in this field: bounds on the number of Neumann points and properties of the probability distribution of this number. Two basic properties of Neumann domains are presented: the wavelength capacity and the spectral position. We state and prove bounds on those as well as key features of their probability distributions. To rigorously investigate those probabilities, we establish the notion of random variables for quantum graphs. In particular, we provide conditions for considering spectral functions of quantum graphs as random variables with respect to the natural density on $\mathbb{N}$.

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arXiv2019

On the nodal structure of nonlinear stationary waves on star graphs

We consider stationary waves on nonlinear quantum star graphs, i.e. solutions to the stationary (cubic) nonlinear Schr\"odinger equation on a metric star graph with Kirchhoff matching conditions at the centre. We prove the existence of solutions that vanish at the centre of the star and classify them according to the nodal structure on each edge (i.e. the number of nodal domains or nodal points that the solution has on each edge). We discuss the relevance of these solutions in more applied settings as starting points for numerical calculations of spectral curves and put our results into the wider context of nodal counting such as the classic Sturm oscillation theorem.

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arXiv2018

Neumann Domains on Graphs and Manifolds

The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. This paper reviews the subject, as appears in a few recent works and points out some open questions and conjectures. The paper concerns both manifolds and metric graphs and the exposition allows for a comparison between the results obtained for each of them.

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arXiv2017

Quantum Graphs via Exercises

Studying the spectral theory of Schroedinger operator on metric graphs (also known as quantum graphs) is advantageous on its own as well as to demonstrate key concepts of general spectral theory. There are some excellent references for this study such as a mathematically oriented book by Berkolaiko and Kuchment, a review with applications to theoretical physicsby Gnutzmann and Smilansky, and elementary lecture notes by Berkolaiko. Here, we provide a set of questions and exercises which can accompany the reading of these references or an elementary course on quantum graphs. The exercises are taken from courses on quantum graphs which were taught by the authors.

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