The Cauchy Problem for the Wave Equation in the Schwarzschild Geometry

The Cauchy problem is considered for the scalar wave equation in the Schwarzschild geometry. We derive an integral spectral representation for the solution and prove pointwise decay in time.Comment: 33 page

Regularity of Bound States

We study regularity of bound states pertaining to embedded eigenvalues of a self-adjoint operator $H$, with respect to an auxiliary operator $A$ that is conjugate to $H$ in the sense of Mourre. We work within the framework of singular Mourre theory which enables us to deal with confined massless Pauli-Fierz models, our primary example, and many-body AC-Stark Hamiltonians. In the simpler context of regular Mourre theory our results boils down to an improvement of results obtained recently in \cite{CGH}.Comment: 70 page

Point interactions in acoustics: one dimensional models

A one dimensional system made up of a compressible fluid and several mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed for different settings of the oscillators array. The dynamical models are formulated in terms of singular perturbations of the decoupled dynamics of the acoustic field and the mechanical oscillators. Detailed spectral properties of the generators of the dynamics are given for each model we consider. In the case of a periodic array of mechanical oscillators it is shown that the energy spectrum presents a band structure.Comment: revised version, 30 pages, 2 figure

Quantum Singularities in Spacetimes with Spherical and Cylindrical Topological Defects

Exact solutions of Einstein equations with null Riemman-Christoffel curvature tensor everywhere, except on a hypersurface, are studied using quantum particles obeying the Klein-Gordon equation. We consider the particular cases when the curvature is represented by a Dirac delta function with support either on a sphere or on a cylinder (spherical and cylindrical shells). In particular, we analyze the necessity of extra boundary conditions on the shells.Comment: 7 page,1 fig., Revtex, J. Math. Phys, in pres

On the optical properties of carbon nanotubes--Part I. A general formula for the dynamical optical conductivity

This paper is the first one of a series of two articles in which we revisit the optical properties of single-walled carbon nanotubes (SWNT). Produced by rolling up a graphene sheet, SWNT owe their intriguing properties to their cylindrical quasi-one-dimensional (quasi-1D) structure (the ratio length/radius is experimentally of order of 10^3). We model SWNT by circular cylinders of small diameters on the surface of which the conduction electron gas is confined by the electric field generated by the fixed carbon ions. The pair-interaction potential considered is the 3D Coulomb potential restricted to the cylinder. To reflect the quasi-1D structure, we introduce a 1D effective many-body Hamiltonian which is the starting-point of our analysis. To investigate the optical properties, we consider a perturbation by a uniform time-dependent electric field modeling an incident light beam along the longitudinal direction. By using Kubo's method, we derive within the linear response theory an asymptotic expansion in the low-temperature regime for the dynamical optical conductivity at fixed density of particles. The leading term only involves the eigenvalues and associated eigenfunctions of the (unperturbed) 1D effective many-body Hamiltonian, and allows us to account for the sharp peaks observed in the optical absorption spectrum of SWNT.Comment: Comments: 24 pages. Revised version. Accepted for publication in J.M.

The influence of out-of-plane stress on a plane strain problem in rock mechanics

This paper analyses the stresses and displacements in a uniformly prestressed Mohr-Coulomb continuum, caused by the excavation of an infinitely long cylindrical cavity. It is shown that the solution to this axisymmetric problem passes through three stages as the pressure at the cavity wall is progressively reduced. In the first two stages it is possible to determine the stresses and displacements in the rθ-plane without consideration of the out-of plane stress . In the third stage it is shown that an inner plastic zone develops in whichzσzσ=σθ, so that the stress states lie on a singularity of the plastic yield surface. Using the correct flow rule for this situation, an analytic solution for the radial displacements is obtained. Numerical examples are given to demonstrate that a proper consideration of this third stage can have a significant effect on the cavity wall displacements

Notes on coherent backscattering from a random potential

We consider the quantum scattering from a random potential of strength $\lambda^{1/2}$ and with a support on the scale of the mean free path, which is of order $\lambda^{-1}$. On the basis of maximally crossed diagrams we provide a concise formula for the backscattering rate in terms of the Green's function for the kinetic Boltzmann equation. We briefly discuss the extension to wave scattering.Comment: 17 pages. 8 figure

Quantum singularities in (2+1) dimensional matter coupled black hole spacetimes

Quantum singularities considered in the 3D BTZ spacetime by Pitelli and Letelier (Phys. Rev. D77: 124030, 2008) is extended to charged BTZ and 3D Einstein-Maxwell-dilaton gravity spacetimes. The occurence of naked singularities in the Einstein-Maxwell extension of the BTZ spacetime both in linear and non-linear electrodynamics as well as in the Einstein-Maxwell-dilaton gravity spacetimes are analysed with the quantum test fields obeying the Klein-Gordon and Dirac equations. We show that with the inclusion of the matter fields; the conical geometry near r=0 is removed and restricted classes of solutions are admitted for the Klein-Gordon and Dirac equations. Hence, the classical central singularity at r=0 turns out to be quantum mechanically singular for quantum particles obeying Klein-Gordon equation but nonsingular for fermions obeying Dirac equation. Explicit calculations reveal that the occurrence of the timelike naked singularities in the considered spacetimes do not violate the cosmic censorship hypothesis as far as the Dirac fields are concerned. The role of horizons that clothes the singularity in the black hole cases is replaced by repulsive potential barrier against the propagation of Dirac fields.Comment: 13 pages, 1 figure. Final version, to appear in PR

Financial support for families with children: options for the new integrated child credit

This commentary discusses the rationale for directing financial support to families with children and assesses options for a new integrated child credit. It shows how the government intends to reform the existing system to separate out the 'adult' and 'child' components of financial support, and analyses various alternatives for how the integrated child credit could be structured to meet the costs of children in different sorts of households. It also assesses how the integrated child credit could respond to changes in income and family circumstances. In doing this, it examines the economics of financial support for children and the evidence on the 'cost' of children, and assesses both the objectives set by the government for an integrated child credit and other criteria that should be used to evaluate its eventual success