Linear-time algorithms for scattering number and Hamilton-connectivity of interval graphs.

We prove that for all inline image an interval graph is inline image-Hamilton-connected if and only if its scattering number is at most k. This complements a previously known fact that an interval graph has a nonnegative scattering number if and only if it contains a Hamilton cycle, as well as a characterization of interval graphs with positive scattering numbers in terms of the minimum size of a path cover. We also give an inline image time algorithm for computing the scattering number of an interval graph with n vertices and m edges, which improves the previously best-known inline image time bound for solving this problem. As a consequence of our two results, the maximum k for which an interval graph is k-Hamilton-connected can be computed in inline image time

Periodic Orbit Theory and Spectral Statistics for Quantum Graphs

We quantize graphs (networks) which consist of a finite number of bonds and vertices. We show that the spectral statistics of fully connected graphs is well reproduced by random matrix theory. We also define a classical phase space for the graphs, where the dynamics is mixing and the periodic orbits proliferate exponentially. An exact trace formula for the quantum spectrum is developed in terms of the same periodic orbits, and it is used to investigate the origin of the connection between random matrix theory and the underlying chaotic classical dynamics. Being an exact theory, and due to its relative simplicity, it offers new insights into this problem which is at the fore-front of the research in Quantum Chaos and related fields.Comment: 37 pages, 20 figures, other comments, accepted for publication in the Annals of Physic

Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides the first explicit quantization of a classically chaotic system.Comment: 25 pages, 5 figure

An Exactly Solvable Model for Nonlinear Resonant Scattering

This work analyzes the effects of cubic nonlinearities on certain resonant scattering anomalies associated with the dissolution of an embedded eigenvalue of a linear scattering system. These sharp peak-dip anomalies in the frequency domain are often called Fano resonances. We study a simple model that incorporates the essential features of this kind of resonance. It features a linear scatterer attached to a transmission line with a point-mass defect and coupled to a nonlinear oscillator. We prove two power laws in the small coupling \to 0 and small nonlinearity \to 0 regime. The asymptotic relation ~ C^4 characterizes the emergence of a small frequency interval of triple harmonic solutions near the resonant frequency of the oscillator. As the nonlinearity grows or the coupling diminishes, this interval widens and, at the relation ~ C^2, merges with another evolving frequency interval of triple harmonic solutions that extends to infinity. Our model allows rigorous computation of stability in the small and limit. In the regime of triple harmonic solutions, those with largest and smallest response of the oscillator are linearly stable and the solution with intermediate response is unstable

Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs

Manifestly non-Hermitian quantum graphs with real spectra are introduced and shown tractable as a new class of phenomenological models with several appealing descriptive properties. For illustrative purposes, just equilateral star-graphs are considered here in detail, with non-Hermiticities introduced by interactions attached to the vertices. The facilitated feasibility of the analysis of their spectra is achieved via their systematic approximative Runge-Kutta-inspired reduction to star-shaped discrete lattices. The resulting bound-state spectra are found real in a discretization-independent interval of couplings. This conclusion is reinterpreted as the existence of a hidden Hermiticity of our models, i.e., as the standard and manifest Hermiticity of the underlying Hamiltonian in one of less usual, {\em ad hoc} representations ${\cal H}_j$ of the Hilbert space of states in which the inner product is local (at $j=0$) or increasingly nonlocal (at $j=1,2, ...$). Explicit examples of these (of course, Hamiltonian-dependent) hermitizing inner products are offered in closed form. In this way each initial quantum graph is assigned a menu of optional, non-equivalent standard probabilistic interpretations exhibiting a controlled, tunable nonlocality.Comment: 33 pp., 6 figure

Trace Formulae and Spectral Statistics for Discrete Laplacians on Regular Graphs (I)

Trace formulae for d-regular graphs are derived and used to express the spectral density in terms of the periodic walks on the graphs under consideration. The trace formulae depend on a parameter w which can be tuned continuously to assign different weights to different periodic orbit contributions. At the special value w=1, the only periodic orbits which contribute are the non back- scattering orbits, and the smooth part in the trace formula coincides with the Kesten-McKay expression. As w deviates from unity, non vanishing weights are assigned to the periodic walks with back-scatter, and the smooth part is modified in a consistent way. The trace formulae presented here are the tools to be used in the second paper in this sequence, for showing the connection between the spectral properties of d-regular graphs and the theory of random matrices.Comment: 22 pages, 3 figure