Point Interaction Between Two Fermions and One Particle of a Different Nature

We consider a model of point interaction between two fermions and one particle of a different nature. The model is analogous to the Skornyakov–Ter-Martirosyan model. It is interpreted based on the self-adjoint extension theory for symmetric operators. We show that if the mass of the third particle is sufficiently smaller than the fermion mass, the corresponding energy operator has an infinite set of bound states with the energy values tending to –.     

Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

Summary In Part I of this work, numerical methods were derived for the solution of the equations of motion of a single particle subject to a central force which conserved exactly the energy and momenta. In the present work, the methodology of Part I is extended, in part, to motion of a system of particles in that the energy and linear momentum are conserved exactly. In addition, the angular momentum will be conserved to one more order of accuracy than in conventional methods. Exact conservation of the total angular momentum results only for the lowest order numerical approximation, which is equivalent to the discrete mechanics presented elsewhere.     

Computing the Particle Paths in an Open-Trap Sharp-Point Geometry

The article presents simulation results for the motion of electrons in two different regions: a high-current vacuum diode and an open trap. The entire system is immersed in an external magnetic field with sharp-point geometry. The electrons are continuously injected from a part of the cathode into the diode region, where they are accelerated to relativistic velocities. In this region, the self-consistent problem of particle motion in electromagnetic fields is solved by the large particle method. In the magnetic trap region, the particles experience only the external magnetic field, and their paths are computed by the test particle method. The simulation efficiency is improved by partitioning the problem into two: separate simulation of the high-voltage diode and integration of the particle paths in the trap. Calculations show that the bulk of the particles leave the trap through the side walls, moving along the magnetic forcelines, and only a small part of the particles undergo multiple reflections and remain longer inside the trap.     

Nonlinear Effects Induced by Vacuum Polarization by a Strong Electromagnetic Field in a Weak Static Electromagnetic Field

In the one-loop approximation of quantum electrodynamics, we study the Faraday rotation effect in the propagation of a weak linearly polarized wave through the field of a strong plane electromagnetic wave and also processes induced by a strong wave: the emission of a photon whose energy is an integer multiple of the strong-wave quantum and photon absorption with the emission of several photons identical (coherent) to the strong-wave quanta. The induced processes should occur in a vacuum in a superposition of external fields, namely, the field of a strong plane electromagnetic wave and a static spatially inhomogeneous electromagnetic field. We show that the induced photon emission in the field of a strong electromagnetic wave with circular polarization and in a static inhomogeneous field is equivalent to the merging of two strong-wave quanta with the production of one photon and that the induced photon absorption in the same combination of fields is equivalent to the decay (splitting) of a photon into two photons completely identical to the strong-wave quanta. All these effects are induced by the phenomenon of the vacuum polarization by a strong external time-dependent electromagnetic field. The probabilities of the induced processes are nontrivial nonlinear functions of the squared strength of the time-dependent field also depending on other parameters.     

Threshold Phenomena in Intense Electromagnetic Fields

The influence of intense electromagnetic fields on the formation and decay of quasistationary states of different quantum systems is investigated based on exact solutions of quantum equations for charged particle motion. The method allows examining systems where a spontaneous decay may occur as well as phenomena that occur only under the action of the field. Different values of the total magnetic moment of the system are taken into account in this consideration. A consistent use of the analytic continuation method allows obtaining nonlinear equations that determine complex energies in an external field. The asymptotic expansions for real and imaginary energy values under the action of weak and strong electromagnetic fields are investigated. The developed approach allows establishing the characteristic values for the length parameters that determine the formation of the processes in superstrong fields. We note that a significant decrease of distances in strong fields may lead to effects with a new characteristic length scale, characterizing a modified quantum electrodynamics (QED) formalism, namely, the QED with the fundamental mass formalism.     

Relativistically invariant representation of arbitrary bilinear combinations of Dirac spinors

It is shown that in the general case a bilinear combination of two arbitrary four-spinors with one of the 16 basic Dirac matrices cannot be expressed in terms of the momentum and spin four-vectors (Pauli-Lubaski vector). A universal general formula obtained in this paper also includes four-vectors corresponding to two three-dimensional unit vectors in the particle rest frame perpendicular to the direction of the particle spin. The transition to vanishing masses is considered. Special expressions that describe binary combinations of spinors in the case of elastic scattering for helicity and transversal amplitudes are given.Institute of High Energy Physics, Serpukhov. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 96, No. 1, pp. 3–22, July, 1993.     

A geometric Ginzburg–Landau problem

For surfaces embedded in a three-dimensional Euclidean space, consider a functional consisting of two terms: a version of the Willmore energy and an anisotropic area penalising the first component of the normal vector, the latter weighted with the factor ${1/\epsilon^2}$ . The asymptotic behaviour of such functionals as ${\epsilon}$ tends to 0 is studied in this paper. The results include a lower and an upper bound on the minimal energy subject to suitable constraints. Moreover, for embedded spheres, a compactness result is obtained under appropriate energy bounds.     

A New Estimate on the Two-Dimensional Indirect Coulomb Energy

We prove a new lower bound on the indirect Coulomb energy in two-dimensional quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the Lieb–Solovej–Yngvason bound with a smaller constant, ${C = (4/3)^{3/2} \sqrt{5 \pi -1} \approx 5.90 < C_{\rm LSY} = 192 \sqrt{2 \pi} \approx 481.27}$ , which also involves an additive gradient energy term of the single particle density.     

Radiation Recoil Problem for Classical Charged Dust Grains

We consider several classical problems (Coulomb scattering, motion in a constant external electromagnetic field, Brownian motion) for the Sommerfeld model of an extended charged particle (dust grain). We find essential differences from the classical result in both quantitative aspects (a change of the angle in Coulomb scattering, decrease of the diffusion coefficient for a Brownian particle, deceleration of outspiraling of the dust grain in a cyclotron) and in qualitative aspects (occurrence of the tunneling effect through a potential barrier). These differences do not disappear as the size of the particle tends to zero. The equation of motion of an extended object therefore does not have a smooth limit transition to the Newton equation for a point.     

Phase analysis in the problem of scattering by a radial potential

Let (k,g) be the total scattering cross section of a three-dimensional quantum particle of energy K² by a radial potential. Under the assumption it is shown that in the domain one has the asymptotics where the coefficient is is expressed explicitly in terms of the Gamma function. For nonnegative potentials, the domain of validity of this asymptotic is even larger. For potentials with a strong positive singularity, it is established that as. Similar results are obtained for the forward scattering amplitude.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 147, pp. 155–178, 1985.     

The transport of mass, energy and momentum in physiologic systems

A generic equation is developed to define the flux, Φ, of mass, energy and momentum in physiologic systems. This equation is written in terms of the concentration, ρ, of the quantity to be transported, the force driving it, a corresponding kinematic transport coefficient, , and the path and mobility associated with the transport process. Driving forces are expressed in terms of the gradient of some intensive property, ø, of the system, such that:

, where n=0 for energy, for momentum and n=1 for mass; β is a nondimensional coefficient of proportionality. Values of φ and are tabulated for various scenarios typical of transport processes occuring in physiologic systems.     

Chebyshev Wavelet Procedure for Solving FLDEs

This paper studies the boundary behaviour at mechanical equilibrium at the ends of a finite interval of a class of systems of interacting particles with monotone decreasing repulsive force. This setting covers, for instance, pile-ups of dislocations, dislocation dipoles and dislocation walls. The main challenge in characterising the boundary behaviour is to control the nonlocal nature of the pairwise particle interactions. Using matched asymptotic expansions for the particle positions and rigorous development of an appropriate energy via \(\Gamma \)-convergence, we obtain the equilibrium equation solved by the boundary layer correction, associate an energy with an appropriate scaling to this correction, and provide decay rates into the bulk.     

On coupled Boltzmann transport equation related to radiation therapy

We consider a system of Boltzmann transport equations which models the charged particle evolution in media. The system is related to the dose calculation in radiation therapy. Although only one species of particles, say photons is invasing these particles mobilize other type of particles (electrons and positrons). Hence in realistic modelling of particle transport one needs a coupled system of three Boltzmann transport equations. The solution of this system must satisfy the inflow boundary condition. We show existence and uniqueness result of the solution applying generalized Lax-Milgram Theorem. In addition, we verify that (in the case of external therapy) under certain assumptions the “incoming flux to dose operator” D₁ is compact. Also the adjoint is analyzed. Finally we consider the inverse planning problem as an optimal control problem. Its solution can be used as an initial solution of the actual inverse planning problem.     

Existence,uniqueness and positivity of solutions for BGK models for mixtures

We consider kinetic models for a multi component gas mixture without chemical reactions. In the literature, one can find two types of BGK models in order to describe gas mixtures. One type has a sum of BGK type interaction terms in the relaxation operator, for example the model described by Klingenberg, Pirner and Puppo [20] which contains well-known models of physicists and engineers for example Hamel [16] and Gross and Krook [15] as special cases. The other type contains only one collision term on the right-hand side, for example the well-known model of Andries, Aoki and Perthame [1]. For each of these two models [20] and [1], we prove existence, uniqueness and positivity of solutions in the first part of the paper. In the second part, we use the first model [20] in order to determine an unknown function in the energy exchange of the macroscopic equations for gas mixtures described by Dellacherie [11].     

Thermal self-action effects of acoustic beam in a vibrationally relaxing gas

Thermal self-action of acoustic beam in a molecular gas with excited internal degrees of molecules’ freedom, is studied. This kind of thermal self-action differs from that in a Newtonian fluid. Heating or cooling of a medium takes place due to transfer of internal vibrational energy. Equilibrium and non-equilibrium gases, which may be acoustically active, are considered. A beam in an acoustically active gas is self-focusing unlike a beam in a standard viscous gas. The self-action effects relating to wave beams containing shock fronts, are discussed. Stationary and non-stationary kinds of self-action are considered.     

Decay of Quasi-Stationary States of Nonrelativistic Quantum Systems in an Intense Electromagnetic Field

We investigate the influence of intense electromagnetic fields on the formation and decay of quasi-stationary states of different quantum systems. Nonlinear equations that determine complex energy values are obtained from the expressions for the probability of the processes in an external electromagnetic field using the analytic continuation method. We obtain asymptotic expansions describing the action of weak and strong electromagnetic fields. If the distances that determine the formation of the processes in superstrong fields decrease significantly, this may lead to effects whose characteristic length scale corresponds to the formalism of the modified QED, i.e., QED with fundamental mass.     

Geometric Dynamics

A kinematic differential system on a Riemann (or semi-Riemann) manifold induces a Lorentz-Udrite world-force law, i.e., any local group with one parameter (any local flow) on a Riemann (or semi-Riemann) manifold induces the dynamics of the given vector field or of an associated particle, which will be called geometric dynamics.The cases of Riemann-Jacobi or Riemann-Jacobi-Lagrange structures are imposed by the behavior of an external tensor field of type (1,1). The case of the Finsler-Jacobi structure appears if the initial metric is chosen such that the energy of the given vector field is constant (Sec. 1). At the end of Sec. 1 are formulated open problems regarding some extensions of geometric dynamics.Adequate structures on the tangent bundle describe the geometric dynamics in the Hamilton language (Sec. 2).Section 3 proves the existence of a Finsler-Jacobi structure induced by an almost contact metric structure.The theory is applied to electromagnetic dynamical systems (the starting point of our theory), offering new principles of unification of the gravitation and the electromagnetism. Also, here, one enounces open problems regarding the geometric dynamics induced by the electric intensity and magnetizing force (Sec. 4).From the geometrical point of view, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds ensuring that all trajectories of a given vector field are geodesics. Having T₁M²ⁿ⁺¹ in mind, the problem of creating a wider class of Riemannian manifolds, in which there exists a vector field such that (1) all trajectories of the vector field are geodesics; (2) the flow defined by is incompressible; (3) the condition which corresponds to the property that is the associate vector field of the contact structure is satisfied;was studied intensively by S. Sasaki. The results were not satisfactory, but Sasaki discovered (, , )-structures [10].AMS Subject Classification (1991): 70H35, 53C22, 58F25, 83C22     

Multiplicity of virtual levels at the lower edge of the continuous spectrum of a two-particle Hamiltonian on a lattice

We consider a system of two arbitrary quantum particles on a three-dimensional lattice with special dispersion functions (describing site-to-site particle transport), where the particles interact by a chosen attraction potential. We study how the number of eigenvalues of the family of the operators h(k) depends on the particle interaction energy and the total quasimomentum \(k \in \mathbb{T}^3\) (where \(\mathbb{T}^3\) is a three-dimensional torus). Depending on the particle interaction energy, we obtain conditions under which the left edge of the continuous spectrum is simultaneously a multiple virtual level and an eigenvalue of the operator h(0).     

Approximation of periodic solutions for a dissipative hyperbolic equation

This paper studies the numerical approximation of periodic solutions for an exponentially stable linear hyperbolic equation in the presence of a periodic external force $f$ . These approximations are obtained by combining a fixed point algorithm with the Galerkin method. It is known that the energy of the usual discrete models does not decay uniformly with respect to the mesh size. Our aim is to analyze this phenomenon’s consequences on the convergence of the approximation method and its error estimates. We prove that, under appropriate regularity assumptions on $f$ , the approximation method is always convergent. However, our error estimates show that the convergence’s properties are improved if a numerically vanishing viscosity is added to the system. The same is true if the nonhomogeneous term $f$ is monochromatic. To illustrate our theoretical results we present several numerical simulations with finite element approximations of the wave equation in one or two dimensional domains and with different forcing terms.     

Positive definiteness of the gravitational radiation intensity in the theory of gravity with a nonzero graviton mass

A version of the theory of gravity is considered where the graviton mass is nonzero, and the gravitation radiation flux from an arbitrary spatially-bounded source is positive definite. The relation between energy losses by emission and the work of the source is established. It is shown that the total work includes the part produced by the interaction of the source with the radiation field and the part produced by the self-action of the field. The total work proves to be positive definite. The general form of the spectrum-angular distribution is obtained, accounting for the spin and polarization states. For spherically symmetric sources, the states with zero spin and zero projection of spin two on the momentum contribute to the emission.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 2, pp. 329–343, May, 1996.Translated by V. I. Serdobol'skii.