# Skrypnik W. I.

### On the post-Darwin approximation of the Maxwell – Lorentz equations of motion of point charges in the absence of neutrality

Ukr. Mat. Zh. - 2019. - 71, № 1. - pp. 117-128

The existence of holomorphic (in time) solutions of the nonrelativistic equations of motion of nonneutral systems of point charges that do not contain inverse powers of the velocity of light greater than three is proved by using the Cauchy theorem. The indicated equations contain time derivatives of the accelerations of charges.

### Mechanical systems with singular equilibria and the Coulomb dynamics of three charges

Ukr. Mat. Zh. - 2018. - 70, № 4. - pp. 519-533

We consider mechanical systems for which the matrices of second partial derivatives of the potential energies at equilibria have zero eigenvalues. It is assumed that their potential energies are holomorphic functions in these singular equilibrium states. For these systems, we prove the existence of proper bounded (for positive time) solutions of the Newton equation of motion convergent to the equilibria in the infinite-time limit. These results are applied to the Coulomb systems of three point charges with singular equilibrium in a line.

### Two-dimensional Coulomb dynamics of two and three equal negative charges in the field of two equal fixed positive charges

Ukr. Mat. Zh. - 2016. - 68, № 11. - pp. 1528-1539

Periodic and bounded for positive time solutions of the planar Coulomb equation of motion for two and three identical negative charges in the field of two equal fixed positive charges are found. The systems possess equilibrium configurations to which the found bounded solutions converge in the infinite time limit. The periodic solutions are obtained with the help of the Lyapunov center theorem.

### Coulomb dynamics near equilibrium of two equal negative charges in the field of fixed two equal positive charges

Ukr. Mat. Zh. - 2016. - 68, № 9. - pp. 1273-1285

Periodic and quasiperiodic solutions of the Coulomb equation of motion of two equal negative charges in the field of two fixed and equal positive charges are found with the help of the Lyapunov center theorem.

### Periodic and Bounded Solutions of the Coulomb Equation of Motion of Two and Three Point Charges with Equilibrium in the Line

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 679–693

Periodic and bounded solutions of the Coulomb equation of motion in the line are obtained for two and three identical negative point charges in the fields of two and three symmetrically located fixed point charges. The systems possess equilibrium configurations. The Lyapunov, Siegel, Moser, and Weinstein theorems are applied.

### On polymer expansions for generalized Gibbs lattice systems of oscillators with ternary interaction

Ukr. Mat. Zh. - 2013. - 65, № 5. - pp. 689–697

We propose a new short proof of the convergence of high-temperature polymer expansions in the thermodynamic limit of canonical correlation functions for classical and quantum Gibbs lattice systems of oscillators interacting via pair and ternary potentials and nonequilibrium stochastic systems of oscillators interacting via a pair potential with Gibbsian initial correlation functions.

### On Holomorphic Solutions of the Darwin Equations of Motion of Point Charges

Ukr. Mat. Zh. - 2013. - 65, № 4. - pp. 546-554

The existence of holomorphic (in time) solutions of the nonrelativistic Darwin equations of motion of point charges is proved with the help of the Cauchy theorem.

### On the holomorphic solutions of Hamiltonian equations of motion of point charges

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 270-280

The Maxwell - Lorenz system of an electromagnetic field interacting with charged particles (point charges) is considered in the Darwin approximation which is characterized by the Lagrangian and Hamiltonian of the particles both uncoupled with the field. The solution of the equation of motion of the particles with the approximated Darwin Hamiltonian is found on a finite time interval with the use of the Cauchy theorem. Components of this solution are represented as holomorphic functions of time.

### On lattice oscillator-type Kirkwood - Salsburg equation with attractive manybody potentials

Ukr. Mat. Zh. - 2010. - 62, № 12. - pp. 1687–1704

We consider a lattice oscillator-type Kirkwood–Salsburg (KS) equation with general one-body phase measurable space and many-body interaction potentials. For special choices of the measurable space, its solutions describe grand-canonical equilibrium states of lattice equilibrium classical and quantum linear oscillator systems. We prove the existence of the solution of the symmetrized KS equation for manybody interaction potentials which are either attractive (nonpositive) and finite-range or infinite-range and repulsive (positive). The proposed procedure of symmetrization of the KS equation is new and based on the superstability of many-body potentials.

### Kirkwood–Salsburg equation for a quantum lattice system of oscillators with many-particle interaction potentials

Ukr. Mat. Zh. - 2009. - 61, № 5. - pp. 689-700

For a Gibbs system of one-dimensional quantum oscillators on a *d*-dimensional hypercubic lattice interacting via superstable pair and many-particle potentials of finite range, we prove the existence of a solution of the (lattice) Kirkwood–Salsburg equation for correlation functions depending on the Wiener paths. Some many-particle potentials may be nonpositive.

### On an unbounded order parameter in lattice equilibrium GKS-type oscillator systems

Ukr. Mat. Zh. - 2009. - 61, № 4. - pp. 538-547

Встановлено існування необмеженого параметра порядку (намагніченості) для широкого класу ґраткових гіббсівських (рівноважних) систем лінійних осциляторів, що взаємодіють завдяки сильному парному полiномiальному потенціалу близьких сусідів та іншим багаточастинковим потенціалам. Розглянуті системи характеризуються загальною поліноміальною близькодійовою потенціальною енергією, що породжує середні, які підкоряються двом нерівностям ГКШ.

### Solutions of the Kirkwood–Salsburg equation for a lattice classical system of one-dimensional oscillators with positive finite-range many-body interaction potentials

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1427–1433

For a system of classical one-dimensional oscillators on the *d*-dimensional hypercubic lattice interacting via pair superstable and many-body positive
finite-range potentials, the (lattice) Kirkwood–Salsburg equation is proposed for the first time and is solved.

### Solutions of the Kirkwood-Salsburg equation for particles with finite-range nonpairwise repulsion

Ukr. Mat. Zh. - 2008. - 60, № 8. - pp. 1138–1143

For a system of classical particles interacting via stable pairwise integrable and positive many-body (nonpairwise) finite-range potentials, we prove the existence of a solution of the symmetrized Kirkwood-Salsburg equation.

### Long-range order in quantum lattice systems of linear oscillators

Ukr. Mat. Zh. - 2006. - 58, № 10. - pp. 1407–1424

The existence of the ferromagnetic long-range order is proved for equilibrium quantum lattice systems of linear oscillators whose potential energy contains a strong ferromagnetic nearest-neighbor (nn) pair interaction term and a weak nonferromagnetic term under a special condition on a superstability bound. It is shown that the long-range order is possible if the mass of a quantum oscillator and the strength of the ferromagnetic nn interaction exceed special values. A generalized Peierls argument and a contour bound, proved with the help of a new superstability bound for correlation functions, are our main tools.

### On Gibbs quantum and classical particle systems with three-body forces

Ukr. Mat. Zh. - 2006. - 58, № 7. - pp. 976–996

For equilibrium quantum and classical systems of particles interacting via ternary and pair (nonpositive) infinite-range potentials, a low activity convergent cluster expansion for their grand canonical reduced density matrices and correlation functions is constructed in the thermodynamic limit.

### Long-range order in Gibbs lattice classical linear oscillator systems

Ukr. Mat. Zh. - 2006. - 58, № 3. - pp. 388–405

The existence of the ferromagnetic long-range order (lro) is proved for Gibbs classical lattice systems of linear oscillators interacting via a strong polynomial pair nearest neighbor (n-n) ferromagnetic potential and other (nonpair) potentials that are weak if they are not ferromagnetic. A generalized Peierls argument and two different contour bounds are our main tools.

### Long-range order in linear ferromagnetic oscillator systems. Strong pair quadratic n-n potential

Ukr. Mat. Zh. - 2004. - 56, № 6. - pp. 810–817

Long-range order is proved to exist for lattice linear oscillator systems with ferromagnetic potential energy containing a term with strong nearest-neighbor (n-n) quadratic pair potential. A contour bound and a generalized Peierls argument are used in the proof.

### On Polymer Expansion for Gibbsian States of Nonequilibrium Systems of Interacting Brownian Oscillators

Ukr. Mat. Zh. - 2003. - 55, № 10. - pp. 1356-1377

The convergence of polymer cluster expansions for correlation functions of general Gibbs oscillator-type systems and related nonequilibrium systems of Brownian oscillators is established. The initial states for the latter are Gibbsian. It is proved that the sequence of the constructed correlation functions of the nonequilibrium system is a generalized solution of a diffusion BBGKY-type hierarchy.

### On the Evolution Operator of the Gradient Diffusion Hierarchy for Plane Rotators

Ukr. Mat. Zh. - 2001. - 53, № 12. - pp. 1664-1685

By using a high-temperature cluster expansion, we construct the evolution operator of the BBGKY-type gradient diffusion hierarchy for plane rotators that interact via a summable pair potential in a Banach space containing the Gibbs (stationary) correlation functions. We prove the convergence of this expansion for a sufficiently small time interval. As a result, we prove that weak solutions of the hierarchy exist in the same Banach space. If the initial correlation functions are locally perturbed Gibbs correlation functions, then these solutions are defined on an arbitrary time interval.

### On Polymer Expansions for Equilibrium Systems of Oscillators with Ternary Interaction

Ukr. Mat. Zh. - 2001. - 53, № 11. - pp. 1532-1544

For Gibbs lattice systems characterized by a measurable space at sites of a *d*-dimensional hypercubic lattice and potential energy with pair complex potential, we formulate conditions that guarantee the convergence of polymer (cluster) expansions. We establish that the Gibbs correlation functions and reduced density matrices of classical and quantum systems of linear oscillators with ternary interaction can be expressed in terms of correlation functions of these systems.

### Sine-gordon transformations in nonequilibrium systems of Brownian particles

Ukr. Mat. Zh. - 1997. - 49, № 10. - pp. 1404–1421

Finite volume grand canonical correlation functions of nonequilibrium systems of *d*-dimensional Brownian particles, interacting through a regular (long-range) pair potential with integrable first partial derivatives, are expressed in terms of the expectation values of a Gaussian random field. The initial correlation functions coincide with the Gibbs correlation functions corresponding to a more general pair long-range potential. Nonequilibrium Euclidean action is introduced, satisfying a thermodynamic stability property.

### Equilibrium in quantum systems of particles with magnetic interaction. Fermi and bose statistics

Ukr. Mat. Zh. - 1997. - 49, № 5. - pp. 691–698

Quantum systems of particles interacting via an effective electromagnetic potential with zero electrostatic component are considered (magnetic interaction). It is assumed that the *j* th component of the effective potential for *n* particles equals the partial derivative with respect to the coordinate of the *j*th particle of “magnetic potential energy” of *n* particles almost everywhere. The reduced density matrices for small values of the activity are computed in the thermodynamic limit for *d*-dimensional systems with short-range pair magnetic potentials and for one-dimensional systems with long-range pair magnetic interaction, which is an analog of the interaction of three-dimensional Chern-Simons electrodynamics (“magnetic potential energy” coincides with the one-dimensional Coulomb (electrostatic) potential energy).

### Equilibrium in quantum systems of particles with magnetic interaction

Ukr. Mat. Zh. - 1995. - 47, № 12. - pp. 1686–1691

We compute density matrices determining equilibrium states in the thermodynamic limit for a class of quantum systems. Interaction in these systems is described as a collective electromagnetic field whose scalar potential is equal to zero.

### BBGKY hierarchy and its evolution operator for one class of integrable systems

Ukr. Mat. Zh. - 1995. - 47, № 6. - pp. 853–858

We define the abstract BBGKY hierarchy and its formal evolution operator. The existence of the latter is established in a special Banach space for a system of charged particles with Chern-Simons interaction regularized for small and large distances. An analog of the high-temperature expansion of equilibrium statistical mechanics is applied.