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Maxwell’s vacuum equations are integrated for the case when the Hamilton–Jacobi equations for a charged test particle admit complete separation of variables of type (1.1). The results of classification of the Stackel metrics and the electromagnetic potentials of the external electromagnetic field carried out earlier are used.

Background. Stimulation of osteogenesis (SO) by biophysical methods has been widely used in practice to accelerate healing or stimulate the healing of fractures with non-unions, since the middle of the XIX century. SO can be carried out by direct current electrostimulation, or indirectly by low-intensity pulsed ultrasound, capacitive electrical coupling stimulation, and pulsed electromagnetic field stimulation. SO simulates natural physiological processes: in the case of electrical stimulation, it changes the electromagnetic potential of damaged cell tissues in a manner similar to normal healing processes, or in the case of low-intensity pulsed ultrasound, it produces weak mechanical effects on the fracture area. SO increases the expression of factors and signaling pathways responsible for tissue regeneration and bone mineralization and ultimately accelerates bone union.The purpose of this review was to present the most up-to-date data from laboratory and clinical studies of the effectiveness of SO.Material and Methods. The results of laboratory studies and the final results of metaanalyses for each of the four SO methods published from 1959 to 2020 in the PubMed, EMBASE, and eLibrary databases are reviewed.Conclusion. The use of SO effectively stimulates the healing of fractures with the correct location of the sensors, compliance with the intensity and time of exposure, as well as the timing of use for certain types of fractures. In case of non-union or delayed union of fractures, spondylodesis, arthrodesis, preference should be given to non-invasive methods of SO. Invasive direct current stimulation can be useful for non-union of long bones, spondylodesis with the risk of developing pseudoarthrosis.

A classification of metrics and electromagnetic potentials is carried out for the case when the Hamilton–Jacobi equation admits a complete separation of variables in a space with a complete set consisting of one vector and two tensor Killing fields. The classification is supplemented by solution of the vacuum Maxwell equations.

In this article, we prove that the function Ψ(t, x) belongs to the domain of the definition of the operator H. We discuss that the operator H and its powers is essentially self-adjo...

In this paper, we prove that the Feynman-Kac It^o formula of the Schrodinger operator with electromagnetic (t; x) in equation (1) in [8] is dierentiable of the variable t, and so establish that the innitely dierentiable in a region, therefore, investigate smoothness of this function.

We prove the existence of solutions to the nonlinear Schrodinger equation e 2 ( i ∇ + A ) 2 u + V ( y ) u − | u | p − 1 u = 0 in R 2 with a magnetic potential A = ( A 1 , A 2 ) . Here V represents the electric potential, the index p is greater than 1. Along some sequence { e n } tending to zero we exhibit complex-value solutions that concentrate along some closed curves.

In this paper, we present an approach to deal with the dynamics of the Dirac equation with time-dependent electromagnetic potentials using the fourth-order compact time-splitting method ($S_\text{4c}$). To this purpose, the time-ordering technique for time-dependent Hamiltonians is introduced, so that the influence of the time-dependence could be limited to certain steps which are easy to treat. Actually, in the case of the Dirac equation, it turns out that only those steps involving potentials need to be amended, and the scheme remains efficient, accurate, as well as easy to implement. Numerical examples in 1D and 2D are given to validate the scheme., 24pages, 8 figures

The relativistic quantum dynamics of a spin-0 scalar particle under the effects of the violation of Lorentz symmetry in the presence of a non-electromagnetic potential is analyzed. The central potential induced by the Lorentz symmetry violation is a linear electric and constant magnetic field and, analyze the effects on the eigenvalues and the wave function. We see there is a dependence of the linear charge density on the quantum numbers of the system

All equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that Hamilton-Jacobi equation and Klein-Gordon-Fock equation for a charged test particle can be integrated by the method of complete separation of variables are found. The separation is carried out using the complete sets of mutually-commuting integrals of motion of type (1.1). Whereby in a privileged coordinate system the given equations turn into parabolic type equations. Hence, these metrics can be used as models for describing plane gravitational waves., The article will be published in IJGMMP

We show that the presence of negative eigenvalues in the spectrum of the angular component of an electromagnetic Schrodinger Hamiltonian H generically produces a lack of the classical time-decay for the associated Schrodinger flow e^{-itH}. This is in contrast with the fact that dispersive estimates (Strichartz) still hold, in general, also in this case. We also observe an improvement of the decay for higher positive modes, showing that the time decay of the solution is due to the first nonzero term in the expansion of the initial datum as a series of eigenfunctions of a quantum harmonic oscillator with a singular potential. A completely analogous phenomenon is shown for the heat semigroup, as expected

The elastic differential cross section is calculated at low energies for the elements3He,20Ne,40Ar,14N,12C, and for208Pb using an effective 4D electromagnetic potential coming from the contribution of the massive Kaluza–Klein modes of the 5D vector field in a thick brane scenario. The length scale is adjusted in the potential to compare with known experimental data and to set bounds for the parameter of the model.

By the large and small wave-function components approach we achieved the nonrela-tivistic limit of the Dirac equation in interaction with an electromagnetic potential in noncommutative phase-space, and we tested the effect of the phase-space noncommutativity on it, knowing that the nonrelativistic limit of the Dirac equation gives the Schr\"odinger-Pauli equation., Comment: 7 pages

We retrace an ab initio relativistic derivation of Maxwell's equations that was developed by Feynman in unpublished notes, clarifying the analogies and the differences with analogous treatments present in the literature. Unlike the latter, Feynman's approach stands out because it considers electromagnetic potentials as primary, reflecting his ideas about the quantum foundations of electromagnetism. Some considerations about the foundations of special relativity, which are naturally suggested by this approach, are given in appendix., 17 pages, no figures, submitted for publication

We consider a class of parabolic equations with critical electromagnetic potentials, for which we obtain a classification of local asymptotics, unique continuation results, and an integral representation formula for solutions., arXiv admin note: text overlap with arXiv:1002.3566

We can find all equivalence classes for electromagnetic potentials and space-time metrics of Stackel spaces, provided that the equations of motion of the classical charged test particles are integrated by the method of complete separation of variables in the Hamilton–Jacobi equation. Separation is carried out using the complete sets of mutually-commuting integrals of motion of type (2.1), whereby in a privileged coordinate system the Hamilton–Jacobi equation turns into a parabolic type equation.

We study the inverse problem of determining a magnetic Schrodinger operator in an unbounded closed waveguide from boundary measurements. We consider this problem with a general closed waveguide in the sense that we only require our unbounded domain to be contained into an infinite cylinder. In this context we prove the unique recovery of the magnetic field and the electric potential associated with general bounded and non-compactly supported electromagnetic potentials. By assuming that the electromagnetic potentials are known on the neighborhood of the boundary outside a compact set, we even prove the unique determination of the magnetic field and the electric potential from measurements restricted to a bounded subset of the infinite boundary. Finally, in the case of a waveguide taking the form of an infinite cylindrical domain, we prove the recovery of the magnetic field and the electric potential from partial data corresponding to restriction of Neumann boundary measurements to slightly more than half of the boundary. We establish all these results by mean of a new class of complex geometric optics solutions and of Carleman estimates suitably designed for our problem stated in an unbounded domain and with bounded electromagnetic potentials.

Nowadays, the dynamics of the erythrocyte membrane electromagnetic potential is not registered by the laboratory methods of practical medicine and is not used in the diagnostics. The value of the erythrocyte charge is a very thin and, which is more important, early diagnostic sign. Considering the method to detect and measure the charge of the erythrocyte membrane a visualization system for RBCs alternations between electrodes is proposed.

A fast evaluation of retarded electromagnetic potentials is presented for quantum calculations on a uniform Cartesian grid. The induced potentials are incorporated into the quantum Hamiltonian of the time-dependent density functional theory (TDDFT) in real space. Both Lorentz and Coulomb gauge-fixed Hamiltonians are implemented using a real-time propagation scheme. The efficiency of the resulting formalism and implementation enables the evaluation of retardation effects on long carbon chains.

We study the electromagnetic response of nanostructures to single-cycle THz excitation by using ultrafast electron diffraction. Although the nanostructures themselves are static, there exist complex sub-THz-cycle Bragg spot dynamics that relate via time-dependent Aharonov-Bohm-like phase shifts to the nanoscale electromagnetic potentials.

We prove Lp and smoothing estimates for the resolvent of magnetic Schrodinger operators. We allow electromagnetic potentials that are small perturbations of a smooth, but possibly unbounded background potential. As an application, we prove an estimate on the location of eigenvalues of magnetic Schrodinger and Pauli operators with complex electromagnetic potentials.

In his recent paper (2020 Eur. J. Phys. 41 045202), Davis makes the claim that potentials and fields are ill-defined in the conventional treatment of electromagnetism. He argues that ‘the usual treatment is ambiguous, with that ambiguity being reflected in the gauge transformation equations’. He then proposes an approach based on two operational versions of Helmholtz’s theorem and claims that his approach does not exhibit gauge freedom and allows a rigourous definition of electromagnetic potentials. Here I argue that Davis’s approach does not provide a more rigours definition of potentials than that provided by the standard approach. Apparently, Davis does not realize that when applying an operational version of Helmholtz’s theorem to Maxwell’s equations, he is not avoiding gauge invariance but tacitly applying it by choosing the particular gauge-condition related to this version of the theorem. The application of the instantaneous Helmholtz’s theorem to Maxwell’s equations is equivalent to the choice of the Coulomb-gauge condition, and the application of the retarded Helmholtz’s theorem to these equations is equivalent to the choice of the Lorentz-gauge condition.

The two commentary articles by van Hees and Heras both reject the conclusions of our paper. Our response here shows that they address neither the issues we raised nor the reasons for our conclusions. Rather, they merely reiterate the ‘conventional approach’ as arguments against them.