Complete Separation Of Variables Research Articles

Modeling of induced currents in a type-II superconducting cylinder of finite length in a transverse magnetic field

Induced currents in the bulk superconducting cylinder of finite length placed in a transverse magnetic field are calculated by the partial and complete separation of variables. The Fourier transform over the angular and axial coordinates reduces the problem to the solution of one-dimensional integral equations. The harmonic balance method is applied to account for the highly nonlinear constitutive relation between the voltage and current. The current and field distribution is compared with the results of FEA. The approximation of the current distribution by a set of cylindrical and disc layers is presented. The magnetic moment of induced currents is analyzed in the increasing applied field when the cylinder is magnetized up to the state of saturation.

Complete separation of variables in the geodesic Hamilton-Jacobi equation in four dimensions

We list all metrics of arbitrary signature in four dimensions which admit complete separation of variables in the Hamilton–Jacobi equation for geodesic Hamiltonians. There are only ten classes of separable metrics admitting Killing vector fields, indecomposable quadratic conservation laws, and coisotropic coordinates. Canonical separable metrics parameterized by several (up to twelve) arbitrary functions of single coordinates are written explicitly. The full set of independent conservation laws in involution for each canonical metrics is also found.

Three-body harmonic molecule

In this study, the quantum three-body harmonic system with finite rest length R and zero total angular momentum L = 0 is explored. It governs the near-equilibrium S-states eigenfunctions of three identical point particles interacting by means of any pairwise confining potential that entirely depends on the relative distances between particles. At R = 0, the system admits a complete separation of variables in Jacobi-coordinates, it is (maximally) superintegrable and exactly-solvable. The whole spectra of excited states is degenerate, and to analyze it a detailed comparison between two relevant Lie-algebraic representations of the corresponding reduced Hamiltonian is carried out. At R > 0, the problem is not even integrable nor exactly-solvable and the degeneration is partially removed. In this case, no exact solutions of the Schrödinger equation have been found so far whilst its classical counterpart turns out to be a chaotic system. For R > 0, accurate values for the total energy E of the lowest quantum states are obtained using the Lagrange-mesh method. Concrete explicit results with not less than eleven significant digits for the states are presented in the range a.u. In particular, it is shown that (I) the energy curve develops a global minimum as a function of the rest length R, and it tends asymptotically to a finite value at large R, and (II) the degenerate states split into sub-levels. For the ground state, perturbative (small-R) and two-parametric variational results (arbitrary R) are displayed as well. An extension of the model with applications in molecular physics is briefly discussed.

Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations

We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into sparse triangular form, where the diagonal blocks consist of spin-s scalar Regge-Wheeler equations (for spins s=0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.

Separation of variables in Hamilton–Jacobi and Klein–Gordon–Fock equations for a charged test particle in the stackel spaces of type (1.1)

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.

Integration of the Hamilton–Jacobi and Maxwell Equations in Diagonal Metrics of Stäckel Spaces

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.

Separation of variables in Hamilton–Jacobi equation for a charged test particle in the Stackel spaces of type (2.1)

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.

Wave-like spatially homogeneous models of Stäckel spacetimes (2.1) type in the scalar-tensor theory of gravity

Six exact solutions are obtained in the general scalar-tensor theory of gravity related to spatially homogeneous wave-like models of the Universe. Wave-like spacetime models allow the existence of privileged coordinate systems where the eikonal equation and the Hamilton–Jacobi equation of test particles can be integrated by the method of complete separation of variables with the separation of isotropic (wave) variables on which the space metric depends (non-ignored variables). An explicit form of the scalar field and two functions of the scalar field that are part of the general scalar-tensor theory of gravity are found. The explicit form of the eikonal function and the action function for test particles in the considered models is given. The obtained solutions are of type III according to the Bianchi classification and type N according to the Petrov classification. Wave-like spatially homogeneous spacetime models can describe primordial gravitational waves of the Universe.

Shapovalov Wave-Like Spacetimes

A complete classification of space-time models is presented, which admit the privileged coordinate systems, where the Hamilton–Jacobi equation for a test particle is integrated by the method of complete separation of variables with separation of the isotropic (wave) variable, on which the metric depends (wave-like Shapovalov spaces). For all types of Shapovalov spaces, exact solutions of the Einstein equations with a cosmological constant in vacuum are found. Complete integrals are presented for the eikonal equation and the Hamilton–Jacobi equation of motion of test particles.

Hamilton–Jacobi Equation for a Charged Test Particle in the Stäckel Space of Type (2.0)

All electromagnetic potentials and space–time metrics of Stäckel spaces of type (2.0) in which the Hamilton–Jacobi equation for a charged test particle can be integrated by the method of complete separation of variables are found. Complete sets of motion integrals, as well as complete sets of killing vector and tensor fields, are constructed. The results can be used when studying solutions of field equations in the theory of gravity.

Spatially Homogeneous Conformally Stäckel Spaces of Type (3.1)

In this work, we have obtained all spatially homogeneous space-time models related to the intersection of the set of Stackel spaces of type (2.1) and the set of conformally Stackel spaces of type (3.1). These models allow a complete separation of variables both in the Hamilton-Jacobi equation for massive test particles and in the eikonal equation for radiation. The models obtained in this work relate to wave-like space-time models. For the found models, solutions of the Einstein equations with the cosmological constant and radiation are obtained. For the obtained solutions, the eikonal equation and the equations of motion of massive test particles in the Hamilton-Jacobi form have been integrated.

Spatially Homogeneous Models Stäckel Spaces of Type (2.1)

All classes of spatially homogeneous space-time models are found that allow the integration of the equations of motion of test particles and the eikonal equation by the method of complete separation of variables according to type (2.1). Four classes of model data are obtained. The resulting models can be applied in any modified metric theories of gravity. Two of the above models allow solutions of the Einstein equations with a cosmological constant and radiation. For the models of a spatially homogeneous Universe with a cosmological constant and radiation obtained in Einstein's theory of gravity, the Hamilton-Jacobi equations of motion of the test particles and the eikonal equation for radiation are integrated by the method of separation of variables.

Kinetic equation of concurrent nucleation and chemical aging of an ensemble of aqueous organic aerosols.

Using the formalism of classical nucleation theory, we derive a kinetic equation for the size and composition distribution of an ensemble of aqueous organic droplets evolving via nucleation and concomitant chemical aging. This distribution can be drastically affected by the enthalpy of heterogeneous chemical reactions and the depletion of organic trace gases absorbed by aerosols. A partial differential equation of second order for the temporal evolution of this distribution is obtained from the discrete equation of balance via Taylor series expansions. Once reduced to the canonical form of the multidimensional Fokker-Planck equation, this kinetic equation can be solved via the method of complete separation of variables. This kinetic equation opens a new direction in the development of the kinetic theory of first-order phase transitions, while its applications to the formation and evolution atmospheric organic aerosols via concurrent nucleation and chemical aging may drastically improve the accuracy of global climate models.

Momentum representation in the relativistic Jahn–Teller effect Gg [3/2] × (t2g + eg)

Relativistic multi-mode Jahn–Teller effect in cubic and octahedral molecular systems is studied in a momentum representation. In the case under consideration, the momentum representation is equivalent to the five-dimensional Fourier transform the use of which allows to reduce the order of vibronic Schrödinger equation for nuclear amplitudes thus simplifying the analytical treatment. The basic result of this work is a quantum calculation of the non-adiabatic transition probability in the multi-mode vibronic system, which does not allow complete separation of variables.

Models of Generalized Scalar-Tensor Gravitation Theories with Radiation Allowing the Separation of Variables in the Eikonal Equation

Exact solutions are obtained for models of general scalar-tensor gravitational theories admitting the existence of privileged coordinate systems in which the eikonal equation can be integrated by the method of complete separation of variables of (2.1) type. Characteristics of radiation and explicit forms of functions of the scalar field entering into the field equations of generalized scalar-tensor gravitational theory are obtained.

Symmetry operators and separation of variables in the (2 + 1)-dimensional Dirac equation with external electromagnetic field

We obtain and analyze equations determining first-order differential symmetry operators with matrix coefficients for the Dirac equation with an external electromagnetic potential in a [Formula: see text]-dimensional Riemann (curved) spacetime. Nonequivalent complete sets of mutually commuting symmetry operators are classified in a [Formula: see text]-dimensional Minkowski (flat) space. For each of the sets, we carry out a complete separation of variables in the Dirac equation and find a corresponding electromagnetic potential permitting separation of variables.

The spacetime models with dust matter that admit separation of variables in Hamilton–Jacobi equations of a test particle

The characteristics of dust matter in spacetime models, admitting the existence of privilege coordinate systems are given, where the single-particle Hamilton–Jacobi equation can be integrated by the method of complete separation of variables. The resulting functional form of the 4-velocity field and energy density of matter for all types of spaces under consideration is presented.

Factorization of Dirac equation in two space dimensions

We present a systematic approach for the separation of variables for the two-dimensional (2D) Dirac equation in polar coordinates. The three vector potential, which couple to the Dirac spinor via minimal coupling, along with the scalar potential are chosen to have angular dependence which emanate the Dirac equation to complete separation of variables. Exact solutions are obtained for a class of solvable potentials along with their relativistic spinor wavefunctions. Particular attention is paid to the situation where the potentials are confined to a quantum dot region and are of scalar, vector and pseudo-scalar type. The study of a single charged impurity embedded in a 2D Dirac equation in the presence of a uniform magnetic field was treated as a particular case of our general study.

Separability and hidden symmetries of Kerr-Taub-NUT spacetime in Kaluza-Klein theory

The Kerr-Taub-NUT spacetime in the Kaluza-Klein theory represents a localized stationary and axisymmetric object in four dimensions from the Kaluza-Klein viewpoint. That is, it harbors companion electromagnetic and dilaton fields, thereby showing up the signature of the extra fifth dimension. We explore the separability structure of this spacetime and show that the Hamilton-Jacobi equation for geodesics admits complete separation of variables only for massless geodesics. This implies the existence of hidden symmetries in the spacetime, which are generated by a conformal Killing tensor. Using a simple trick built up on a conformally related metric (an “effective” metric) with the Killing tensor, we construct an explicit expression for the conformal Killing tensor.

Hidden symmetries and geodesics of Kerr spacetime in Kaluza-Klein theory

The Kerr spacetime in Kaluza-Klein theory describes a rotating black hole in four dimensions from the Kaluza-Klein point of view and involves the signature of an extra dimension that shows up through the appearance of the electric and dilaton charges. In this paper, we study the separability properties of the Hamilton-Jacobi equation for geodesics and the associated hidden symmetries in the spacetime of the Kerr-Kaluza-Klein black hole. We show that the complete separation of variables occurs only for massless geodesics, implying the existence of hidden symmetries generated by a second rank conformal Killing tensor. Employing a simple procedure built up on an "effective" metric, which is conformally related to the original spacetime metric and admits a complete separability structure, we construct the explicit expression for the conformal Killing tensor. Next, we study the properties of the geodesic motion in the equatorial plane, focusing on the cases of static and rotating Kaluza-Klein black holes separately. In both cases, we obtain the defining equations for the boundaries of the regions of existence, boundedness and stability of the circular orbits as well as the analytical formulas for the orbital frequency, the radial and vertical epicyclic frequencies of the geodesic motion. Performing a detailed numerical analysis of these equations and frequencies, we show that the physical effect of the extra dimension amounts to the significant enlarging of the regions of existence, boundedness and stability towards the event horizon, regardless of the classes of orbits.