Nonlocality of Accelerated Systems

The conceptual basis for the nonlocality of accelerated systems is presented. The nonlocal theory of accelerated observers and its consequences are briefly described. Nonlocal field equations are developed for the case of the electrodynamics of linearly accelerated systems.Comment: LaTeX file, no figures, 9 pages, to appear in: "Black Holes, Gravitational Waves and Cosmology" (World Scientific, Singapore, 2003

The effect of short ray trajectories on the scattering statistics of wave chaotic systems

In many situations, the statistical properties of wave systems with chaotic classical limits are well-described by random matrix theory. However, applications of random matrix theory to scattering problems require introduction of system specific information into the statistical model, such as the introduction of the average scattering matrix in the Poisson kernel. Here it is shown that the average impedance matrix, which also characterizes the system-specific properties, can be expressed in terms of classical trajectories that travel between ports and thus can be calculated semiclassically. Theoretical results are compared with numerical solutions for a model wave-chaotic system

On the geometry of the domain of the solution of nonlinear Cauchy problem

We consider the Cauchy problem for a second order quasi-linear partial differential equation with an admissible parabolic degeneration such that the given functions described the initial conditions are defined on a closed interval. We study also a variant of the inverse problem of the Cauchy problem and prove that the considered inverse problem has a solution under certain regularity condition. We illustrate the Cauchy and the inverse problems in some interesting examples such that the families of the characteristic curves have either common envelopes or singular points. In these cases the definition domain of the solution of the differential equation contains a gap.Comment: accepted for publication in the book Lie groups, differential equations and geometry in Springer Unip

Sufficient conditions for the existence of bound states in a central potential

We show how a large class of sufficient conditions for the existence of bound states, in non-positive central potentials, can be constructed. These sufficient conditions yield upper limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears. These upper limits are significantly more stringent than hitherto known results.Comment: 7 page

Critical strength of attractive central potentials

We obtain several sequences of necessary and sufficient conditions for the existence of bound states applicable to attractive (purely negative) central potentials. These conditions yields several sequences of upper and lower limits on the critical value, $g_{\rm{c}}^{(\ell)}$, of the coupling constant (strength), $g$, of the potential, $V(r)=-g v(r)$, for which a first $\ell$-wave bound state appears, which converges to the exact critical value.Comment: 18 page

Nonlocal Gravity: Modified Poisson's Equation

The recent nonlocal generalization of Einstein's theory of gravitation reduces in the Newtonian regime to a nonlocal and nonlinear modification of Poisson's equation of Newtonian gravity. The nonlocally modified Poisson equation implies that nonlocality can simulate dark matter. Observational data regarding dark matter provide limited information about the functional form of the reciprocal kernel, from which the original nonlocal kernel of the theory must be determined. We study this inverse problem of nonlocal gravity in the linear domain, where the applicability of the Fourier transform method is critically examined and the conditions for the existence of the nonlocal kernel are discussed. This approach is illustrated via simple explicit examples for which the kernels are numerically evaluated. We then turn to a general discussion of the modified Poisson equation and present a formal solution of this equation via a successive approximation scheme. The treatment is specialized to the gravitational potential of a point mass, where in the linear regime we recover the Tohline-Kuhn approach to modified gravity.Comment: 27 pages, 4 figures; v2: minor improvements, accepted for publication in J. Math. Phy

Inversion of Gamow's Formula and Inverse Scattering

We present a pedagogical description of the inversion of Gamow's tunnelling formula and we compare it with the corresponding classical problem. We also discuss the issue of uniqueness in the solution and the result is compared with that obtained by the method of Gel'fand and Levitan. We hope that the article will be a valuable source to students who have studied classical mechanics and have some familiarity with quantum mechanics.Comment: LaTeX, 6 figurs in eps format. New abstract; notation in last equation has been correcte

Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices

We compute exact asymptotic results for the probability of the occurrence of large deviations of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the probability that all the eigenvalues of an (NxN) random matrix are positive (negative) decreases for large N as ~\exp[-\beta \theta(0) N^2] where the Dyson index \beta characterizes the ensemble and the exponent \theta(0)=(\ln 3)/4=0.274653... is universal. We compute the probability that the eigenvalues lie in the interval [\zeta_1,\zeta_2] which allows us to calculate the joint probability distribution of the minimum and the maximum eigenvalue. As a byproduct, we also obtain exactly the average density of states in Gaussian ensembles whose eigenvalues are restricted to lie in the interval [\zeta_1,\zeta_2], thus generalizing the celebrated Wigner semi-circle law to these restricted ensembles. It is found that the density of states generically exhibits an inverse square-root singularity at the location of the barriers. These results are confirmed by numerical simulations.Comment: 17 pages Revtex, 5 .eps figures include

The Index Distribution of Gaussian Random Matrices

We compute analytically, for large N, the probability distribution of the number of positive eigenvalues (the index N_{+}) of a random NxN matrix belonging to Gaussian orthogonal (\beta=1), unitary (\beta=2) or symplectic (\beta=4) ensembles. The distribution of the fraction of positive eigenvalues c=N_{+}/N scales, for large N, as Prob(c,N)\simeq\exp[-\beta N^2 \Phi(c)] where the rate function \Phi(c), symmetric around c=1/2 and universal (independent of $\beta$), is calculated exactly. The distribution has non-Gaussian tails, but even near its peak at c=1/2 it is not strictly Gaussian due to an unusual logarithmic singularity in the rate function.Comment: 4 pages Revtex, 4 .eps figures include

Controlled quantum evolutions and transitions

We study the nonstationary solutions of Fokker-Planck equations associated to either stationary or nonstationary quantum states. In particular we discuss the stationary states of quantum systems with singular velocity fields. We introduce a technique that allows to realize arbitrary evolutions ruled by these equations, to account for controlled quantum transitions. The method is illustrated by presenting the detailed treatment of the transition probabilities and of the controlling time-dependent potentials associated to the transitions between the stationary, the coherent, and the squeezed states of the harmonic oscillator. Possible extensions to anharmonic systems and mixed states are briefly discussed and assessed.Comment: 24 pages, 4 figure