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

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

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

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

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

The 3D version of the finite element program FESTER

In this report, a detailed description of the 3-D version finite element pro-gram FESTER is given. This includes: 1. A brief introduction to the package FESTER; 2. Preparing an input data file for the 3D version of FESTER; 3. Principal stress and stress invariant analyses; 4. 2D joint element (surface contact) characterisation and its mathematical formulation; 5. Formulations of the 3D stress-strain analyses for both isotropic and anisotropic materials, plane of weakness and cracking criteria; 6. 3D brick elements, infinity elements and their corresponding shape and mapping functions; 7. Large-displacement formulations; 8. Modifications to the subroutines INVAR, JNTB, TMAT, MOD2 etc; 9. Numerical examples; and 10. Conclusions

Decoherence rates for Galilean covariant dynamics

We introduce a measure of decoherence for a class of density operators. For Gaussian density operators in dimension one it coincides with an index used by Morikawa (1990). Spatial decoherence rates are derived for three large classes of the Galilean covariant quantum semigroups introduced by Holevo. We also characterize the relaxation to a Gaussian state for these dynamics and give a theorem for the convergence of the Wigner function to the probability distribution of the classical analog of the process.Comment: 23 page

Exterior complex scaling as a perfect absorber in time-dependent problems

It is shown that exterior complex scaling provides for complete absorption of outgoing flux in numerical solutions of the time-dependent Schr\"odinger equation with strong infrared fields. This is demonstrated by computing high harmonic spectra and wave-function overlaps with the exact solution for a one-dimensional model system and by three-dimensional calculations for the H atom and a Ne atom model. We lay out the key ingredients for correct implementation and identify criteria for efficient discretization

Perturbation Theory of Schr\"odinger Operators in Infinitely Many Coupling Parameters

In this paper we study the behavior of Hamilton operators and their spectra which depend on infinitely many coupling parameters or, more generally, parameters taking values in some Banach space. One of the physical models which motivate this framework is a quantum particle moving in a more or less disordered medium. One may however also envisage other scenarios where operators are allowed to depend on interaction terms in a manner we are going to discuss below. The central idea is to vary the occurring infinitely many perturbing potentials independently. As a side aspect this then leads naturally to the analysis of a couple of interesting questions of a more or less purely mathematical flavor which belong to the field of infinite dimensional holomorphy or holomorphy in Banach spaces. In this general setting we study in particular the stability of selfadjointness of the operators under discussion and the analyticity of eigenvalues under the condition that the perturbing potentials belong to certain classes.Comment: 25 pages, Late

Generalized Wannier Functions

We consider single particle Schrodinger operators with a gap in the en ergy spectrum. We construct a complete, orthonormal basis function set for the inv ariant space corresponding to the spectrum below the spectral gap, which are exponentially localized a round a set of closed surfaces of monotonically increasing sizes. Estimates on the exponential dec ay rate and a discussion of the geometry of these surfaces is included