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# A fully-connected graph with four vertices and sixteen directed bonds.

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For certain types of quantum graphs we show that the random-matrix form factor can be recovered to at least third order in the scaled time $\tau$ from periodic-orbit theory. We consider the contributions from pairs of periodic orbits represented by diagrams with up to two self-intersections connected by up to four arcs and explain why all other dia...

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... for a general class of graphs is a complicated and tedious task [16]. Fortunately, the calculation simplifies considerably for a special case described below. In this section we restrict our attention to fully-connected graphs with N vertices and B = N 2 directed bonds, including a loop at each of the vertices. An example with N = 4 is shown in Fig. 4. We assume that the vertex-scattering matrices ...

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## Citations

... In view of the Bohigas-Giannoni-Schmidt conjecture [10], this observation led to the investigation of quantum graphs as a model for quantum chaos. Additional results regarding convergence of spectral statistics to those of random matrix theory include [7,8,18,19,44], among others. ...
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We prove an analogue of the pointwise Weyl law for families of unitary matrices obtained from quantization of one-dimensional interval maps. This quantization for interval maps was introduced by Pako\'nski et al. [J. Phys. A: Math. Gen. 34 9303 (2001)] as a model for quantum chaos on graphs. Since we allow shrinking spectral windows in the pointwise Weyl law analogue, we obtain a strengthening of the quantum ergodic theorem for these models, and show in the semiclassical limit that a family of randomly perturbed quantizations has approximately Gaussian eigenvectors. We also examine further the specific case where the interval map is the doubling map.
... Kottos and Smilansky are considered the first to propose quantum graphs as a model of quantum chaos in 1999 [14], though Roth had discovered a trace formula using the heat kernel of a quantum graph as early as 1983 [18]. Berkolaiko, Schanz, and Whitney showed that the numbers and types of self-intersections in periodic orbits determine what term they contribute to in a series expansion of the form factor, specifically on quantum graphs [5,6], before general results were obtained. ...
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... It took a further fifteen years before Sieber and Sieber and Richter [42,43] obtained first order terms in the expansion for quantum billiards by considering figure eight orbits with a single self-intersection. Subsequently higher order terms were evaluated in quantum graphs by including orbits with multiple self-intersections by Berkolaiko, Schanz and Whitney [8,9] and the scheme was expanded to obtain all orders in the form factor expansion by Müller, Heusler, Braun, Haake, and Altland [36]. In this program, the contribution from certain classes of orbits is evaluated to obtain the relevant order in the expansion. ...
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We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example of a chaotic quantum system where a spectral statistic can be evaluated in terms of periodic orbits without taking the semiclassical limit, which is the limit of large graphs. The variance depends on the size of two classes of primitive pseudo orbits (sets of periodic orbits); pseudo orbits without self-intersections and those where all the self-intersections are 2-encounters at a single vertex. To show other pseudo orbits do not contribute we employ a parity argument for Lyndon word decompositions. For families of binary graphs with an increasing number of bonds, we show the periodic orbit formula approaches a universal constant independent of the coefficient of the polynomial. This constant is obtained by counting the total number of primitive pseudo orbits of a given length. To count periodic orbits and pseudo orbits we exploit further connections between orbits on binary graphs and Lyndon words.
... The study of statistical properties of quantum eigenfunctions and eigenvalues, particularly when the underlying classical system is chaotic, is part of the field of quantum chaos. The study of Schrödinger operators on one-dimensional networks is a prominent part of this field, going by the name of 'quantum graphs' [20,21,22,23,24,25,26,27,28,29,30,31,9,32,33] (the list of references is highly-incomplete but gives a flavour of the subject). In this article we report on constructions of scarred eigenfunctions of quantum graphs, that in some cases exhibit maximal delocalisation. ...
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We prove the existence of scarred eigenstates for star graphs with scattering matrices at the central vertex which are either a Fourier transform matrix, or a matrix that prohibits back-scattering. We prove the existence of scars that are half-delocalised on a single bond. Moreover we show that the scarred states we construct are maximal in the sense that it is impossible to have quantum eigenfunctions with a significantly lower entropy than our examples. These scarred eigenstates are on graphs that exhibit generic spectral statistics of random matrix type in the large graph limit, and, in contrast to other constructions, correspond to non-degenerate eigenvalues; they exist for almost all choices of lengths.
... It turns out that the class coefficients V O N (π) depend only on the cycle type of the permutation τ defined above. They satisfy the recursion (10) ...
... The ribbon graphs used in the present article are not significantly different from the more traditional way the semiclassical diagrams are depicted in the physics literature (see, e.g. [10,47,49,69]). In the "physics" diagrams, the trajectories are shown as smooth with the reconnections (which distinguish trajectories γ from trajectories γ ) happening in the small encounter regions, see Fig. 5. Within the encounter region trajectories are drawn as intersecting. This has the unfortunate effect of confining a lot of information to a small part of the drawing. ...
... If r 1 belongs to the first cycle, it splits into two parts. Finally, if r 1 belongs to the mirror image of the first cycle, the product is Taking the generating function with respect to 1/N according to the middle expression in (28), we recover recursion (10), which completes the proof. ...
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To study electronic transport through chaotic quantum dots, there are two main theoretical approachs. One involves substituting the quantum system with a random scattering matrix and performing appropriate ensemble averaging. The other treats the transport in the semiclassical approximation and studies correlations among sets of classical trajectories. There are established evaluation procedures within the semiclassical evaluation that, for several linear and non-linear transport moments to which they were applied, have always resulted in the agreement with random matrix predictions. We prove that this agreement is universal: any semiclassical evaluation within the accepted procedures is equivalent to the evaluation within random matrix theory. The equivalence is shown by developing a combinatorial interpretation of the trajectory sets as ribbon graphs (maps) with certain properties and exhibiting systematic cancellations among their contributions. Remaining trajectory sets can be identified with primitive (palindromic) factorisations whose number gives the coefficients in the corresponding expansion of the moments of random matrices. The equivalence is proved for systems with and without time reversal symmetry.
... Furthermore, in [2] the spectral form factor and the level spacing distribution were investigated analytically and numerically and compared with the corresponding quantities of random matrix theory searching for "characteristic signals" of quantum chaos. Their analysis of the spectrum was continued in [6,7,8] by a sophisticated application and evaluation of the trace formula. Kostrykin and Schrader [9] presented a classification of all self adjoint characterizations of the Laplacian on compact graphs. ...
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We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of $\delta$-like potentials with strength $\kappa>0$ on the half line $\rz_{\geq0}$ and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval $I_n=[0,l_n]$, $n\in\nz$, to each edge with length $l_n=\frac{\pi}{n}$. We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each $\kappa>0$ and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by $\lambda_n=n^2$, $n\in\nz$, with multiplicities $d(n)$, where $d(n)$ denotes the divisor function. We thus can relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics $\mathcal{N}(\lambda)=\frac{\sqrt{\lambda}}{2}\ln\lambda +\Or(\sqrt{\lambda})$ for the eigenvalue counting function. Based on an exact trace formula, the Vorono\"i summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph, for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.
... The key interest in the theory of quantum graphs is to understand the statistics of the eigenvalues {λ n } ∞ n=0 ⊂ R + of ∆. A particular problem is to understand the distribution of differences {λ n − λ m } ∞ n,m=0 and, via a trace formula, this is related to the differences {l(γ) − l(γ )} for pairs of closed cycles in G [3], [4], [5], [12], [29]. Indeed, a important role is played by summations over pairs of closed cycles (γ, γ ) with l(γ) = l(γ ). ...
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In this note we show that the length spectrum for metric graphs exhibits a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesics (or closed cycles) with the same metric length. KeywordsMetric graph-Closed geodesic-Length spectrum-Thermodynamic formalism Mathematics Subject Classification (2000)05C22-05C38-37C30-37C35-37D35-81Q50
... Sieber [36] and Sieber and Richter [37] were the first to evaluate the second order contributions by considering orbits with one self-intersections. This was extended to higher orders for quantum graphs in [4,5] and recently for general contributions of orbit pairs with any number of self-intersections [30]. ...
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The article surveys quantization schemes for metric graphs with spin. Typically quantum graphs are defined with the Laplace or Schrodinger operator which describe particles whose intrinsic angular momentum (spin) is zero. However, in many applications, for example modeling an electron (which has spin-1/2) on a network of thin wires, it is necessary to consider operators which allow spin-orbit interaction. The article presents a review of quantization schemes for graphs with three such Hamiltonian operators, the Dirac, Pauli and Rashba Hamiltonians. Comparing results for the trace formula, spectral statistics and spin-orbit localization on quantum graphs with spin Hamiltonians.
... On the other hand, in the case when the Laplace operator produces an energyindependent S-matrix, the matrix S has some very special properties (see Lemma 2.1 and preceeding discussion). For example, this condition excludes such a popular choice of a scattering matrix, as the Fourier matrix [20,22]. In Section 2.3 we will discuss two differential operators on quantum graphs, one due to Carlson [23] and the other due to Bolte and Harrison [24], with spectrum actually described by equation (1.1) with arbitrary energy-independent matrix S. We will also explain how to obtain solutions of (1.1) with arbitrary S as one half of the spectrum of a self-adjoint Laplacian. ...
... The most work in spectral statistics on graphs has been done on nearestneighbor spacing distribution (mostly numeric, starting from [15]; but see also [28]), form factor ( [15,16,38,20,39,22,41] and others) and the two-point correlation function (see [29,42,40]). Most articles proceed to study properties of the statistical functions without considering whether the sought functions exist and in what sense. ...
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The purpose of this article is to address two questions on funda- mentals of quantum graphs. Quantum graphs are usually introduced either through the dierential operator acting on the functions dened on the edges of a graph or through directly specifying the scattering matrices at the vertices of the graphs. The rst question addressed in this article is the connection between these con- structions, mostly from the point of view of spectral statistics. Our answer to this question is, in most part, a review of the already available results. The second question we address is the equivalence of two types of spectral statistics of a graph. When spectral statistics of quantum graphs are discussed, the spectrum can refer to one of two things: the eigenvalue spectrum of the dierential operator or the eigenphases of the scattering matrix associated to the graph. In the second part of the article we announce and discuss new results explaining in which limit the two types of statistics will agree (complete proofs of the results will appear in (1)). In addition, we discuss the eect on the spectral statistics of the possible energy-dependence of the S-matrix of the graph.
... Such systems have recently been the subject of considerable interest [12]. In particular, quantum graphs have emerged as important toy models of quantum chaotic behaviour [13,14]: if one considers sequences of graphs with increasing numbers of edges then, under certain conditions, the quantum eigenvalue statistics converge to those of random matrix theory [13,14,15,16,17,18,19,20]. However, relatively little attention has been paid to their eigenfunction statistics. ...
... Such expansions are notoriously difficult to analyze rigorously as both T and the size of the graph increase. The starting point of any such analysis is the evaluation of the contribution from the diagonal terms, obtained by restricting the last sum in (16) to identical trajectories, τ 1 = τ 2 . It is usually assumed that the off-diagonal terms sum up to a subdominant contribution, when T and the size of the graph scale appropriately. ...
... On graphs it was explored, in particular, in [14,44]. It is difficult, however, to give an a priori estimate on the size of the off-diagonal contributions and the analysis is usually restricted to evaluating the contributions coming from specific classes of interacting trajectories [15,16,17,18]. ...
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We prove quantum ergodicity for a family of graphs that are obtained from ergodic one-dimensional maps of an interval using a procedure introduced by Pakonski et al (J. Phys. A, v. 34, 9303-9317 (2001)). As observables we take the L^2 functions on the interval. The proof is based on the periodic orbit expansion of a majorant of the quantum variance. Specifically, given a one-dimensional, Lebesgue-measure-preserving map of an interval, we consider an increasingly refined sequence of partitions of the interval. To this sequence we associate a sequence of graphs, whose directed edges correspond to elements of the partitions and on which the classical dynamics approximates the Perron-Frobenius operator corresponding to the map. We show that, except possibly for subsequences of density 0, the eigenstates of the quantum graphs equidistribute in the limit of large graphs. For a smaller class of observables we also show that the Egorov property, a correspondence between classical and quantum evolution in the semiclassical limit, holds for the quantum graphs in question.