Nodal domains on quantum graphs

We consider the real eigenfunctions of the Schr\"odinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds $B$. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations.Comment: 19 pages, uses IOP journal style file

Symplectic tracking using point magnets and a reference orbit made of circular arcs and straight lines

Symplectic tracking of beam particles using point magnets is achieved using a reference orbit made of circular arcs and straight lines that join smoothly with each other. For this choice of the reference orbit, results are given for the transfer functions, transfer matrices, and the transit times of the magnets and drift spaces. These results provide a symplectic integrator, and allow the linear orbit parameters to be computed by multiplying transfer matrices. It is shown that this integrator is a second-order integrator, and that the transfer functions can be derived from a hamiltonian.Comment: 16 pages, PDF fil

Nonlocal spectral properties of disordered alloys

A general method is proposed for calculating a fully k-dependent, continuous, and causal spectral function A(k,E) within the recently introduced nonlocal version of the coherent-potential approximation (NLCPA). The method involves the combination of both periodic and anti-periodic solutions to the associated cluster problem and also leads to correct bulk quantities for small cluster sizes. We illustrate the method by investigating the Fermi surface of a two-dimensional alloy. Dramatically, we find a smeared electronic topological transition not predicted by the conventional CPA.Comment: 17 pages, 5 figures, Submitted to: J. Phys.: Condens. Matter Editorial receipt 25 May 200

Proof that the Hydrogen-antihydrogen Molecule is Unstable

In the framework of nonrelativistic quantum mechanics we derive a necessary condition for four Coulomb charges $(m_{1}^+, m_{2}^-, m_{3}^+, m_{4}^-)$, where all masses are assumed finite, to form the stable system. The obtained stability condition is physical and is expressed through the required minimal ratio of Jacobi masses. In particular this provides the rigorous proof that the hydrogen-antihydrogen molecule is unstable. This is the first result of this sort for four particles.Comment: Submitted to Phys.Rev.Let

Effective capillary interaction of spherical particles at fluid interfaces

We present a detailed analysis of the effective force between two smooth spherical colloids floating at a fluid interface due to deformations of the interface. The results hold in general and are applicable independently of the source of the deformation provided the capillary deformations are small so that a superposition approximation for the deformations is valid. We conclude that an effective long--ranged attraction is possible if the net force on the system does not vanish. Otherwise, the interaction is short--ranged and cannot be computed reliably based on the superposition approximation. As an application, we consider the case of like--charged, smooth nanoparticles and electrostatically induced capillary deformation. The resulting long--ranged capillary attraction can be easily tuned by a relatively small external electrostatic field, but it cannot explain recent experimental observations of attraction if these experimental systems were indeed isolated.Comment: 23 page

Sub-leading contributions to the black hole entropy in the brick wall approach

[Abridged] We compute the canonical entropy of a quantum scalar field around static and spherically symmetric black holes through the brick wall approach at the higher orders (in fact, up to the sixth order in \hbar) in the WKB approximation. We explicitly show that the brick wall model generally predicts corrections to the Bekenstein-Hawking entropy in all spacetime dimensions. In four dimensions, we find that the corrections to the Bekenstein-Hawking entropy are of the form (A^n \log A), while, in six dimensions, the corrections behave as (A^m + A^n \log A), where A denotes the area of the black hole event horizon, and (m, n) < 1. We compare our results with the corrections to the Bekenstein-Hawking entropy that have been obtained through the other approaches in the literature, and discuss the implications.Comment: 21 pages, Revtex 4; Final verson - 22 pages, References added, Accepted in Phys. Rev.

Optimal Renormalization-Group Improvement of R(s) via the Method of Characteristics

We discuss the application of the method of characteristics to the renormalization-group equation for the perturbative QCD series within the electron-positron annihilation cross-section. We demonstrate how one such renormalization-group improvement of this series is equivalent to a closed-form summation of the first four towers of renormalization-group accessible logarithms to all orders of perturbation theory

Exact Solutions of a Model for Granular Avalanches

We present exact solutions of the non-linear {\sc bcre} model for granular avalanches without diffusion. We assume a generic sandpile profile consisting of two regions of constant but different slope. Our solution is constructed in terms of characteristic curves from which several novel predictions for experiments on avalanches are deduced: Analytical results are given for the shock condition, shock coordinates, universal quantities at the shock, slope relaxation at large times, velocities of the active region and of the sandpile profile.Comment: 7 pages, 2 figure

Gas-dynamic shock heating of post-flare loops due to retraction following localized, impulsive reconnection

We present a novel model in which shortening of a magnetic flux tube following localized, three-dimensional reconnection generates strong gas-dynamic shocks around its apex. The shortening releases magnetic energy by progressing away from the reconnection site at the Alfven speed. This launches inward flows along the field lines whose collision creates a pair of gas-dynamic shocks. The shocks raise both the mass density and temperature inside the newly shortened flux tube. Reconnecting field lines whose initial directions differ by more that 100 degrees can produce a concentrated knot of plasma hotter that 20 MK, consistent with observations. In spite of these high temperatures, the shocks convert less than 10% of the liberated magnetic energy into heat - the rest remains as kinetic energy of bulk motion. These gas-dynamic shocks arise only when the reconnection is impulsive and localized in all three dimensions; they are distinct from the slow magnetosonic shocks of the Petschek steady-state reconnection model

Fredholm's Minors of Arbitrary Order: Their Representations as a Determinant of Resolvents and in Terms of Free Fermions and an Explicit Formula for Their Functional Derivative

We study the Fredholm minors associated with a Fredholm equation of the second type. We present a couple of new linear recursion relations involving the $n$th and $n-1$th minors, whose solution is a representation of the $n$th minor as an $n\times n$ determinant of resolvents. The latter is given a simple interpretation in terms of a path integral over non-interacting fermions. We also provide an explicit formula for the functional derivative of a Fredholm minor of order $n$ with respect to the kernel. Our formula is a linear combination of the $n$th and the $n\pm 1$th minors.Comment: 17 pages, Latex, no figures connection to supplementary compound matrices mentioned, references added, typos correcte