Impossibility of simultaneous localization of wave packets in energy and in configuration space

We consider Schrödinger Hamiltonians H, the scalar and vector potential of which may be singular on a closed set of measure zero. Sufficient conditions are given implying that no state vector can have compact support both in the spectral representation of H and in configuration space. These conditions are verified if the scalar potential is locally not too singular as well as for certain strongly singular potentials.     

Exact solutions of Einstein's field equations for a massive point-particle with scalar point-charge

The exact static and spherically symmetric solution of Einstein's field equations for a massive point-particle with a scalar point-charge as source of a massless scalar field is derived in Schwarzschild coordinates. There exists no longer a Schwarzschild horizon. Only at the point-particle metric and scalar field are singular (naked singularity).     

Exact solutions of Einstein-conformal scalar equations

The massless scalar field which satisfies a conformally invariant equation is in some respects more interesting than the ordinary one. Unfortunately, few, if any, exact solutions of Einstein's equations for a conformal scalar stress-energy have appeared previously. Here we present a theorem by means of which one can generate two Einstein-conformal scalar solutions from a single Einstein-ordinary scalar solution (of which many are known). As an example we show how to obtain Weyl-like solutions with a conformal scalar field. We obtain and analyze in some detail two families of spherically symmetric static Einstein-conformal scalar solutions. We also exhibit a family of static spherically symmetric Einstein-Maxwell-conformal scalar solutions (parametrized by both electric and scalar charge), which have black-hole geometries but are not genuine black holes. Finally, we present all the Robertson-Walker cosmological models which contain both incoherent radiation and a homogeneous conformal scalar field. One class of these represents open universes which bounce and never pass through a singular state; they circumvent the “singularity theorems” by violating the energy condition.     

New Singular and Nonsingular Colliding Wave Solutions in Einstein-Maxwell-Scalar Theory

A technique is given to generate coupled scalar field solutions in colliding Einstein-Maxwell (EM) waves. By employing the Bell-Szekeres solution as seed and depending on the chosen scalar field, it is possible to construct nonsingular solutions. If the original EM solution is already singular, addition of scalar fields does not make the physics any better. In particular, scalar field solution that is transformable to spherical symmetry is plagued with singularities.     

Causal green function in relativistic quantum mechanics

The causal Green function or Feynman propagator for the free-field Klein-Gordon equation and related singular functions, defined as distributions, are related to the causal time-boundary data. Probability densities and amplitudes are defined in terms of the solutions of the Klein-Gordon equation for a complex scalar field interacting with an electromagnetic field. The convergence of the perturbation expansion of the solution of the Klein-Gordon equation for a charged scalar particle in an external field is shown for well-behaved electromagnetic potentials. Other relativistic wave equations are discussed briefly.     

Propagation of Scalar Fields in a Plane Symmetric Spacetime

The present article deals with solutions for a minimally coupled scalar field propagating in a static plane symmetric spacetime. The considered metric describes the curvature outside a massive infinity plate and exhibits an intrinsic naked singularity (a singular plane) that makes the accessible universe finite in extension. This solution can be interpreted as describing the spacetime of static domain walls. In this context, a first solution is given in terms of zero order Bessel functions of the first and second kind and presents a stationary pattern which is interpreted as a result of the reflection of the scalar waves at the singular plane. This is an evidence, at least for the massless scalar field, of an old interpretation given by Amundsen and Grøn regarding the behaviour of test particles near the singularity. A second solution is obtained in the limit of a weak gravitational field which is valid only far from the singularity. In this limit, it was possible to find out an analytic solution for the scalar field in terms of the Kummer and Tricomi confluent hypergeometric functions.     

Cosmological Dynamics of Scalar Field with Non-Minimal Kinetic Term

We investigate the dynamics of a scalar field in the framework of the scalar-tensor theory. A nontrivial behavior of the field in the vicinity of singular points of the kinetic term is observed. In particular, the singular points could serve as attractor for classical solutions.     

Event horizon and scalar potential

The introduction of a scalar potential with a more general schema than general Relativity eliminates the event horizon. Among possible solutions, the Schwarzschild one represents a singular case. A study of the geodesic properties of the matching with an approximated interior Solution are given. A new definition of the gravitational mass and function is deduced.     

A class of exact solutions for coupled electromagnetic and scalar fields for einstein-rosen metric. II

This paper is a continuation of the author's earlier work entitled “A class of exact solutions for coupled electromagnetic and scalar fields for Einstein-Rosen metric. I.” [Ann. Physics69 (1972), 473]. In this paper we have interpreted some of the solutions from different viewpoints with special reference to singular behavior. It has been observed that the presence of zero mass scalar meson fields does not affect the singular behavior as when the electromagnetic field only is present.     

Möbius invariant integrable lattice equations associated with KP and 2DTL hierarchies

The integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. They generate the corresponding continuous hierarchy of singular manifold equations, its Bäcklund transformations and different forms of superposition principles; their distinctive feature is invariance under the action of Möbius transformation. Geometric interpretation of these discrete equations is given.     

A scalar polynomial singularity without an event horizon

It is shown that the solution of the field equations for a static spherically symmetric scalar field has a scalar polynomial singularity and no event horizon. The solution does not develop from nonsingular data on any Cauchy surface. The possible existence of a universal scalar field, the conformal diagram and geodesies of the solution, and the energy and momentum of the field present are discussed.     

Approximation methods in the context of the bound-state Bethe-Salpeter equation

Numerical approximation schemes of the Wick-rotated scalar Bethe-Salpeter equation are discussed for general local potentials with special emphasis on mesh-point methods. Convergence properties are obtained by considering the analytic properties of the kernel. To this end, the four-dimensional partial wave equations are formulated in a new representation-independent way. The close relationship of variational and mesh-point methods is demonstrated and the difficulties which arise if singular potentials are introduced are discussed. For marginal singular potentials those difficulties are overcome in a new way by redefining the corresponding two-particle Green's function. Numerical examples for this case are given.     

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the $δ$-function. For the approximation of the $δ$-function, the direct projection method is used that was proposed in [6]. The $δ$-function is constructed in a consistent way to the derivative operator. Nonlinear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The $δ$-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singular source, however, the solution of the linear scalar wave equation with the non-stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.     

Exact Solitary Wave Solutions of the Coupled Nonlinear Scalar Field Equations

Because of the arbitrary property of the singular manifold, the usual singularity analysis can be extended to a different form. Using a nonstandard truncation approach, five special types of exact solution of the coupled nonlinear scalar field equations are obtained.     

Casimir effects near the big rip singularity in viscous cosmology

Analytical properties of the scalar expansion in the cosmic fluid are investigated, especially near the future singularity, when the fluid possesses a constant bulk viscosity ζ. In addition, we assume that there is a Casimir-induced term in the fluid’s energy-momentum tensor, in such a way that the Casimir contributions to the energy density and pressure are both proportional to 1/a ⁴, a being the scale factor. A series expansion is worked out for the scalar expansion under the condition that the Casimir influence is small. Close to the Big Rip singularity the Casimir term has however to fade away and we obtain the same singular behavior for the scalar expansion, the scale factor, and the energy density, as in the Casimir-free viscous case.     

The role of a scalar-field potential in the Einstein-Kartan cosmology

Flat space isotropic cosmological models including a nonminimally coupled scalar field with a nonlinear potential are studied within the Einstein-Kartan theory. The exact general solutions for various types of scalar fields are derived for arbitrary values of the coupling constant ξ. It is shown that both singular and nonsingular models are feasible. The spectral values of ξ and restriction on ξ are found for the foregoing solutions. The cosmological consequence of taking into account the scalar-field potential is discussed. __________ Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 67–71, June, 2006.     

BRST-BFV formalism for the generalized Schwinger model

We derive the Wess-Zumino scalar term of the generalized Schwinger model both in the singular and nonsingular cases by using BRST-BFV framework. The photon propagators are also computed in the extended Lorentz gauge.     

Electromagnetic waves in a plane wave background

Source-free Maxwell equations are equivalent to the scalar wave equation. Most of its solutions are singular at infinity. If the nail eigenvector of the background metric is an eigenvector of the Maxwell field, too, the e.m. field is not changed by the plane wave.     

Energy levels of a scalar particle in a static gravitational field close to the black hole limit

The bound-state energy levels of a scalar particle in the gravitational field of finite-sized objects with interiors described by the Florides and Schwarzschild metrics are found. For these metrics, bound states with zero energy (where the binding energy is equal to the rest mass of the scalar particle) only exist when a singularity occurs in the metric. Therefore, in contrast to the Coulomb case, no pairs are produced in the non-singular static metric. For the Florides metric the singularity occurs in the black hole limit, while for the Schwarzschild interior metric it corresponds to infinite pressure at the center. Moreover, the energy spectrum is shown to become quasi-continuous as the metric becomes singular.