Discrete minimisers are close to continuum minimisers for the interaction energy

Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy \(\Gamma \)-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural iscrete analogue containing all the discrete minimisers.     

Nonlinear mobility continuity equations and generalized displacement convexity

We consider the geometry of the space of Borel measures endowed with a distance that is defined by generalizing the dynamical formulation of the Wasserstein distance to concave, nonlinear mobilities. We investigate the energy landscape of internal, potential, and interaction energies. For the internal energy, we give an explicit sufficient condition for geodesic convexity which generalizes the condition of McCann. We take an eulerian approach that does not require global information on the geodesics. As by-product, we obtain existence, stability, and contraction results for the semigroup obtained by solving the homogeneous Neumann boundary value problem for a nonlinear diffusion equation in a convex bounded domain. For the potential energy and the interaction energy, we present a nonrigorous argument indicating that they are not displacement semiconvex.     

Non-autonomous second order Hamiltonian systems

We study the existence of periodic solutions for a second order non-autonomous dynamical system containing variable kinetic energy terms. Our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other. We study sublinear problems and the existence of non-constant solutions.     

Vacuum interaction of conic singularities

In the (tr log)-formalism, we consider the problem of the vacuum interaction of conic singularities in a D-dimensional (D ≥ 3) space–time. We show that the interaction energy regularized by dimensional regularization contains neither ultraviolet divergences nor divergences associated with the nonintegrable nature of the vacuum mean of the energy–momentum tensor operator. In the case of four space–time dimensions, the result coincides with those obtained previously in a local approach.     

Spontaneous fermion creation in the coulomb field and Aharonov-Bohm potential in 2+1 dimensions

We find exact solutions of the Dirac equation and the fermion energy spectrum in the Coulomb (vector and scalar) potential and Aharonov-Bohm potential in 2+1 dimensions taking the particle spin into account. We describe the fermion spin using the two-component Dirac equation with the additional (spin) parameter introduced by Hagen. We consider the effect of creation of fermion pairs from the vacuum by a strong Coulomb field in the Aharonov-Bohm potential in 2+1 dimensions. We obtain transcendental equations implicitly determining the electron energy spectrum near the boundary of the lower energy continuum and the critical charge. We numerically solve the equation for the critical charge. We show that for relatively weak magnetic flows, the critical charge decreases (compared with the case with no magnetic field) if the energy of interaction of the electron spin magnetic moment with the magnetic field is negative and increases if this energy is positive. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 250–262, February, 2009.     

Some problems of the theory of quantum statistical systems with an energy spectrum of the fractional-power type

We consider the problem of the effective interaction potential in a quantum many-particle system leading to the fractional-power dispersion law. We show that passing to fractional-order derivatives is equivalent to introducing a pair interparticle potential. We consider the case of a degenerate electron gas. Using the van der Waals equation, we study the equation of state for systems with a fractional-power spectrum. We obtain a relation between the van der Waals constant and the phenomenological parameter ??, the fractional-derivative order. We obtain a relation between energy, pressure, and volume for such systems: the coefficient of the thermal energy is a simple function of ??. We consider Bose??Einstein condensation in a system with a fractional-power spectrum. The critical condensation temperature for 1 < ?? < 2 is greater in the case under consideration than in the case of an ideal system, where ?? = 2.     

Dissipative Transport: Thermal Contacts and Tunnelling Junctions

The general theory of simple transport processes between quantum mechanical reservoirs is reviewed and extended. We focus on thermoelectric phenomena, involving exchange of energy and particles. The theory is illustrated on the example of two reservoirs of free fermions coupled through a local interaction. We construct a stationary state and determine energy and particle currents with the help of a convergent perturbation series. We explicitly calculate several interesting quantities to lowest order, such as the entropy production rate, the resistance, and the heat conductivity. Convergence of the perturbation series allows us to prove that they are strictly positive under suitable smallness and regularity assumptions on the interaction between the reservoirs. Communicated by Gian Michele Graf submitted 15/01/03, accepted: 25/02/03     

Multiple solutions for a Schrödinger–Bopp–Podolsky system with positive potentials

In this paper, we prove existence of solutions for a Schrödinger–Bopp–Podolsky system under positive potentials. We use the Ljusternick–Schnirelmann and Morse Theories to get multiple solutions with a priori given “interaction energy.”     

Casimir energy in noncompact lattice electrodynamics

We propose a new lattice method for calculating the Casimir energy for a U(1) gauge theory. Using this method, we analyze the standard problem of the Casimir interaction of two planar parallel plates with the boundary conditions induced by an additional Chern-Simons action localized on these boundary surfaces. From the physical standpoint, this boundary value problem models the interaction of two thin metal plates. The proposed method can be generalized to the case of more complicated surface shapes.     

Global weak solutions for the Vlasov–Poisson system with a point charge

We study the dynamics of three‐dimensional Vlasov‐Poisson system in the presence of a point charge with attractive interaction. Compared to the repulsive interaction,we cannot get a priori conversation law. Nevertheless,we are able to obtain bound of kinetic energy by introducing a Lyapunov functional. Combining this result with the concept of Diperna‐Lions flow, we establish global existence of weak solutions for this system. Copyright © 2014 John Wiley & Sons, Ltd.     

Point interaction of two distinguishable particles in an external field

We consider a point interaction model for two particles in an external field, which is similar to the Skornyakov-Ter-Martirosyan model and is treated using the theory of self-adjoint extensions of symmetric operators. The corresponding energy operator has an infinite set of bound states with the energy values receding to-∞. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 125, No. 1, pp. 74–90, October, 2000.     

Scattering of Dirac Particles by Electromagnetic Fields with Small Support in Two Dimensions and Effect from Scalar Potentials

We study the asymptotic behavior of scattering amplitudes for the scattering of Dirac particles in two dimensions when electromagnetic fields with small support shrink to point-like fields. The result is strongly affected by perturbations of scalar potentials and the asymptotic form changes discontinuously at half-integer fluxes of magnetic fields even for small perturbations. The analysis relies on the behavior at low energy of resolvents of magnetic Schrödinger operators with resonance at zero energy. The magnetic scattering of relativistic particles appears in the interaction of cosmic string with matter. We discuss this closely related subject as an application of the obtained results. Communicated by Bernard Helffersubmitted 05/05/03, accepted 31/07/03     

Global Jacobian and Γ-convergence in a two-dimensional Ginzburg-Landau model for boundary vortices

In the theory of 2D Ginzburg-Landau vortices, the Jacobian plays a crucial role for the detection of topological singularities. We introduce a related distributional quantity, called the global Jacobian that can detect both interior and boundary vortices for a 2D map u. We point out several features of the global Jacobian, in particular, we prove an important stability property. This property allows us to study boundary vortices in a 2D Ginzburg-Landau model arising in thin ferromagnetic films, where a weak anchoring boundary energy penalising the normal component of u at the boundary competes with the usual bulk potential energy. We prove an asymptotic expansion by Γ-convergence at the second order for this mixed boundary/interior energy in a regime where boundary vortices are preferred. More precisely, at the first order of the limiting expansion, the energy is quantised and determined by the number of boundary vortices detected by the global Jacobian, while the second order term in the limiting energy expansion accounts for the interaction between the boundary vortices.     

On the dissipation due to wave ringing in nonelliptic elastic materials

Summary Initial-boundary value problems describing the mechanics of nonelliptic elastic materials give rise to solutions that involve phase boundaries, the motion of which can dissipate mechanical energy. We investigate whether this dissipation, acting alone, can drive such a system toward equilibrium. Moving phase boundaries are regarded as a localized dissipative mechanism, and we consider a model which specifically excludes dissipation away from a phase boundary (such as that due to viscoelastic damping). In the problem under consideration, wave packets reverberate between the fixed external boundary and a single internal phase boundary. The phase boundary remains stationary unless it is acted upon by one of these wave packets, and each such interaction dissipates a finite amount of energy while causing the initiating wave packet to split into a reflected wave packet and a transmitted wave packet. Consequently, the number of wave packets increases in a geometric fashion. Each individual interaction of a wave packet with the phase boundary is, in a certain sense, mechanically underdetermined, and we augment the mechanical theory with two alternative energy criteria, each of which determines a different interaction dynamics. These alternative energy criteria are motivated by considerations of maximizing the energy dissipation in the system. We treat a system that is perturbed out of an initial minimum energy equilibrium state by a disturbance at the external boundary. A framework is developed for treating the resulting wave reverberations and calculating the energy dissipation for large time. Numerical computation indicates that the total energy dissipated in both versions of the dynamical problem is that which is necessary to settle into a new energy-minimal equilibrium state. We then establish the same result analytically for a meaningful limit involving a vanishingly small dynamical perturbation.     

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 –.     

Electron in the Aharonov-Bohm potential and in the Coulomb field in 2+1 dimensions

We obtain exact solutions of the Dirac equation in 2+1 dimensions and the electron energy spectrum in the superposition of the Aharonov-Bohm and Coulomb potentials, which are used to study the Aharonov-Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov-Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov-Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 3, pp. 502–517, December, 2006. An erratum to this article is available at .     

Energy and coherence loss rates in a one-dimensional vibrational system interacting with a bath

We consider the problem of the dynamics of a Gaussian wave packet in a one-dimensional harmonic ocsillator interacting with a bath. This problem arises in many chemical and biochemical applications related to the dynamics of chemical reactions. We take the bath-oscillator interaction into account in the framework of the Redfield theory. We obtain closed expressions for Redfield-tensor elements, which allows finding the explicit time dependence of the average vibrational energy. We show that the energy loss rate is temperature-independent, is the same for all wave packets, and depends only on the spectral function of the bath. We determine the degree of coherence of the vibrational motion as the trace of the density-matrix projection on a coherently moving wave packet. We find an explicit expression for the initial coherence loss rate, which depends on the wave packet width and is directly proportional to the intensity of the interaction with the bath. The minimum coherence loss rate is observed for a “coherent” Gaussian wave packet whose width corresponds to the oscillator frequency. We calculate the limiting value of the degree of coherence for large times and show that it is independent of the structural characteristics of the bath and depends only on the parameters of the wave packet and on the temperature. It is possible that residual coherence can be preserved at low temperatures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 1, pp. 130–144, October, 2007.     

Gamow vectors for an unstable relativistic quantum field

We study a model of cubic interaction between two scalar fields with a scattering resonance. The resonance manifests as two poles of the analytic continuations of the Green function with respect to energy. The Gamow vectors associated to these resonances acquire meaning in suitable rigged Fock spaces. Finally, we discuss some properties of the S-matrix for unstable fields.     

The relativistic operator of interaction of two quasimolecular electrons as a third-order effect of quantum electrodynamics

We solve the problem of interaction two quasimolecular electrons located at an arbitrary separation near different atoms (nuclei). We consider third-order effects in quantum electrodynamics, which include the virtual photon exchange between electrons with emission (absorption) of a real photon. We obtain the general expression for matrix elements of the operator of the effective interaction energy of two quasimolecular electrons with the external radiation field, which allows calculating probabilities of inelastic processes with rearrangement at slow collisions of multicharge ions with relativistic atoms. We demonstrate that consistently taking the natural condition of the interaction symmetry with respect to the two electrons into account results in the appearance of additional terms in the operators of spin-orbit, spin-spin, and retarded interactions compared with the previously obtained expressions for these operators. We construct the operator of the dipole-dipole interaction of two neutral atoms located at an arbitrary separation.     

Partial identification of the potential from phase shifts

We consider the three-dimensional inverse scattering with fixed energy in the spherically symmetrical case. We give a characterization of the sequences of phase shifts for two potentials which can be different only in a ball of radius a. In other words we study how the large distance interaction influences the asymptotical behavior of the phase shifts. We also characterize the tail of the potential by the growth order of the scattering amplitude F(t) for large t.