arXiv subjects

Ram Band

Publications and source records attributed to Ram Band.

At least 37 records · Page 2Linked to original sources

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 Schrödinger 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.

math-ph↗

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.

math.SP↗

arXiv2018

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.

math.SP↗

arXiv2018

Nodal Statistics On Quantum Graphs

It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph's first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form --- it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.

math-ph↗

arXiv2017

Nonlinear Sturm Oscillation: from the interval to a star

The Sturm oscillation property, i.e. that the $n$-th eigenfunction of a Sturm-Liouville operator on an interval has $n -1$ zeros (nodes), has been well studied. This result is known to hold when the interval is replaced by a metric (quantum) tree graph. We prove that the solutions of the real stationary nonlinear Schrödinger equation on an interval satisfy a nonlinear version of the Sturm oscillation property. However, we show that unlike the linear theory, the nonlinear version of the Sturm oscillation breaks down already for a star graph. We point out conditions under which this violation can be assured.

math-ph↗

arXiv2017

Quantum graphs which optimize the spectral gap

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.

math-ph↗

arXiv2016

Courant-sharp eigenvalues of Neumann 2-rep-tiles

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In $\R^{2}$ the domains we consider are the isosceles right triangle and the rectangle with edge ratio $\sqrt{2}$ (also known as the A4 paper). In $\R^{n}$ the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\textbackslash{}unfolding. This structure affects the nodal set of the eigenfunctions, which in turn allows to derive necessary conditions for Courant-sharpness. In addition, the eigenvalues of these domains are arranged as a lattice which allows for a comparison between the nodal count and the spectral position. The Courant-sharpness of most eigenvalues is ruled out using those methods. In addition, this analysis allows to estimate the nodal deficiency - the difference between the spectral position and the nodal count.

math.SP↗

arXiv2015

Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs

We study the nodal count of the so-called bi-dendral graphs and show that it exhibits an anomaly: the nodal surplus is never equal to 0 or $β$, the first Betti number of the graph. According to the nodal-magnetic theorem, this means that bands of the magnetic spectrum (dispersion relation) of such graphs do not have maxima or minima at the "usual" symmetry points of the fundamental domain of the reciprocal space of magnetic parameters. In search of the missing extrema we prove a necessary condition for a smooth critical point to happen inside the reciprocal fundamental domain. Using this condition, we identify the extrema as the singularities in the dispersion relation of the maximal abelian cover of the graph (the honeycomb graph being an important example). In particular, our results show that the anomalous nodal count is an indication of the presence of the conical points in the dispersion relation of the maximal universal cover. Also, we discover that the conical points are present in the dispersion relation of graphs with much less symmetry than was required in previous investigations.

math-ph↗

arXiv2015

Universality and optimality of band-gaps in laminated media

We find that the frequency spectra of layered phononic and photonic composites admit a universal struc- ture, independent of the geometry of the periodic-cell and the specific physical properties. We show that this representation extends to highly deformable and multi-physical materials of tunable spectra. The structure is employed to establish universal properties of the corresponding band-gaps, and to rigorously determine their sta- tistical and optimal characteristics. Specifically, we investigate the density of the gaps, their maximal width and expected value. As a result, rules for tailoring the laminate according to desired spectra properties follow. Our representation further facilitates characterizing the tunability of the band-structure of soft and multi-physical materials.

cond-mat.mtrl-sci↗

arXiv2015

Topological Properties of Neumann Domains

A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction - those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry.

math.SP↗

arXiv2013

Universality of the momentum band density of periodic networks

The momentum spectrum of a periodic network (quantum graph) has a band-gap structure. We investigate the relative density of the bands or, equivalently, the probability that a randomly chosen momentum belongs to the spectrum of the periodic network. We show that this probability exhibits universal properties. More precisely, the probability to be in the spectrum does not depend on the edge lengths (as long as they are generic) and is also invariant within some classes of graph topologies.

math-ph↗

arXiv2013

The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree

Sturm's oscillation theorem states that the n-th eigenfunction of a Sturm-Liouville operator on the interval has n-1 zeros (nodes). This result was generalized for all metric tree graphs and an analogous theorem was proven for discrete tree graphs. We prove the converse theorems for both discrete and metric graphs. Namely, if for all n, the n-th eigenfunction of the graph has n-1 zeros then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues. In the course of the proof we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of 'discretized' versions of a metric graph and prove that their nodal counts are related to this of the metric graph.

math-ph↗

arXiv2012

Finite pseudo orbit expansions for spectral quantities of quantum graphs

We investigate spectral quantities of quantum graphs by expanding them as sums over pseudo orbits, sets of periodic orbits. Only a finite collection of pseudo orbits which are irreducible and where the total number of bonds is less than or equal to the number of bonds of the graph appear, analogous to a cut off at half the Heisenberg time. The calculation simplifies previous approaches to pseudo orbit expansions on graphs. We formulate coefficients of the characteristic polynomial and derive a secular equation in terms of the irreducible pseudo orbits. From the secular equation, whose roots provide the graph spectrum, the zeta function is derived using the argument principle. The spectral zeta function enables quantities, such as the spectral determinant and vacuum energy, to be obtained directly as finite expansions over the set of short irreducible pseudo orbits.

math-ph↗

arXiv2011

Note on the role of symmetry in scattering from isospectral graphs and drums

We discuss scattering from pairs of isospectral quantum graphs constructed using the method described in [1, 2]. It was shown in [3] that scattering matrices of such graphs have the same spectrum and polar structure, provided that infinite leads are attached in a way which preserves the symmetry of isospectral construction. In the current paper we compare this result with the conjecture put forward by Okada et al. [4] that the pole distribution of scattering matrices in the exterior of isospectral domains in R^2 are different.

math-ph↗

arXiv2011

Nodal domains of a non-separable problem - the right angled isosceles triangle

We study the nodal set of eigenfunctions of the Laplace operator on the right angled isosceles triangle. A local analysis of the nodal pattern provides an algorithm for computing the number of nodal domains for any eigenfunction. In addition, an exact recursive formula for the number of nodal domains is found to reproduce all existing data. Eventually we use the recursion formula to analyse a large sequence of nodal counts statistically. Our analysis shows that the distribution of nodal counts for this triangular shape has a much richer structure than the known cases of regular separable shapes or completely irregular shapes. Furthermore we demonstrate that the nodal count sequence contains information about the periodic orbits of the corresponding classical ray dynamics.

math-ph↗

arXiv2011

Isospectral graphs with identical nodal counts

According to a recent conjecture, isospectral objects have different nodal count sequences. We study generalized Laplacians on discrete graphs, and use them to construct the first non-trivial counter-examples to this conjecture. In addition, these examples demonstrate a surprising connection between isospectral discrete and quantum graphs.

math-ph↗

arXiv2011

On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions

Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains $ν_n$ of the $n$-th eigenfunction satisfies $n\ge ν_n$. Here, we provide a new interpretation for the Courant nodal deficiency $d_n = n-ν_n$ in the case of quantum graphs. It equals the Morse index --- at a critical point --- of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning --- it is the number of unstable directions in the vicinity of the critical point corresponding to the $n$-th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

math-ph↗

arXiv2010

Scattering from isospectral quantum graphs

Quantum graphs can be extended to scattering systems when they are connected by leads to infinity. It is shown that for certain extensions, the scattering matrices of isospectral graphs are conjugate to each other and their poles distributions are therefore identical. The scattering matrices are studied using a recently developed isospectral theory. At the same time, the scattering approach offers a new insight on the mentioned isospectral construction.

math-ph↗