The Laughlin liquid in an external potential

We study natural perturbations of the Laughlin state arising from the effects of trapping and disorder. These are N-particle wave functions that have the form of a product of Laughlin states and analytic functions of the N variables. We derive an upper bound to the ground state energy in a confining external potential, matching exactly a recently derived lower bound in the large N limit. Irrespective of the shape of the confining potential, this sharp upper bound can be achieved through a modification of the Laughlin function by suitably arranged quasi-holes.     

Two interacting particles in an external potential

The solution to the Schrödinger equation is analyzed for two particles interacting in an external potential. It is assumed that a shallow level exists for one of the particles in the external potential. It is shown that repulsion leads to a transfer of one of the particles to the continuous spectrum after attainment of the interaction threshold. For the shallow level, the threshold value of the interaction is independent of level depth.     

Response of an ideal gas in an external potential

We construct the exact solution for the time evolution of the local mass density in an ideal gas in which an external short-ranged separable potential is switched on at time t = 0. The redistribution of matter in strong potentials causing bound states or resonances is studied in great numerical detail.     

Quantum electrodynamic perturbation theory based on electrodynamics with an external field

A method is proposed for construction of mean values of quantum quantities in quantum optics and electrodynamics on the basis of exact solutions of the corresponding problems of electrodynamics with an external field. To illustrate the method the mean energy of a two-level atom in the Raby problem is calculated. It is shown that the mean value obtained with a specified accuracy coincides with the result of exact quantum electrodynamic calculation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 7–10, September, 1983.     

Calculation of an effective potential in an external gravitational field with torsion and phase transitions induced by external fields

We find a general expression for a one-loop effective potential of an arbitrary massless theory with accuracy up to terms linear in two-dimensional invariants. The potential is constructed from the metric and the torsion. We show that spontaneous symmetry breaking can occur in the form of a first-order phase transition induced jointly by the curvature and the torsion.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 3–9, March, 1987.The authors are grateful to I. V. Tyutin and E. S. Fradkin for discussion.     

The ideal Boson gas in an external scalar potential

We consider a new way of going to the infinite volume thermodynamic limit for a finite density quantum system and apply it to the case of an ideal Boson gas. We describe two procedures for calculating the particle density in the thermodynamic limit, one local and one global, and show that they give different values for the density. Further calculations show that this discrepancy is caused by lack of macroscopic translation invariance of the system, which is not apparent at the microscopic level. We calculate the limiting value of the expectation function of the Weyl operators both above and below the critical density for Bose-Einstein condensation, and show that the condensate has paradoxical properties of a similar type to those recently discovered for the rotating Boson gas.     

Pinning of a rough interface by an external potential

An analysis is given of the behavior of an interface between two phases in the presence of an external pinning potential in the solid-on-solid limit of the two-dimensional Ising model. It is found that the potential turns a rough interface into a smooth one, except in the case of a boundary potential, where a minimum potential strength is required. The connection with the roughening transition found by Abraham is discussed. The interface width is calculated as a function of the potential parameters in the limit of a weak pining potential.     

Four-cluster chimera state in non-locally coupled phase oscillator systems with an external potential

Dynamics of a one-dimensional array of non-locally coupled Kuramoto phase oscillators with an external potential is studied. A four-cluster chimera state is observed for the moderate strength of the external potential. Different from the clustered chimera states studied before, the instantaneous frequencies of the oscillators in a synchronized cluster are different in the presence of the external potential. As the strength of the external potential increases, a bifurcation from the two-cluster chimera state to the four-cluster chimera states can be found. These phenomena are well predicted analytically with the help of the Ott-Antonsen ansatz.     

Field theory

The possibility of determining the preferred stable trajectories of macroparticles when the initial conditions of their motion in a central field remain indeterminate is investigated. The concept of field-dynamical equilibrium is introduced, an equation is derived for it, and one of the possible physical interpretations of this state is submitted. General considerations are advanced with regard to the solution of the equation, and a similarity is noted between the resulting equation and the quantum-mechanical Schrödinger equation.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 46–52, October, 1971.I wish to gratefully acknowledge State Prize Laureate and member of the international Commission on Gravitation, Prof. D. D. Ivanenko, and the Director of the Astronomical Observatory of Kiev University, Prof. A. F. Bogorodskii, for affording me the opportunity on several occasions to present papers at seminars under their direction, as well as for their valuable consultations on individual aspects of the present study.     

On improvable bounds for polaron systems in an external potential

A variational approach is proposed that allows one to obtain in a regular way a sequence of improvable upper bounds for the ground-state energy of various polaron models confined in an external electrostatic potential. The proposed approach can be used for an arbitrary electron–phonon interaction constant and allows generalization to the case of polaron-type systems in a constant external magnetic field.     

Asymptotes of the effective potential in an external gravitational field

Renormalization group equations are derived which permit study of the behavior of the quantum theory effective potential of a field in curved space-time. Within the framework of asymptotically free models the asymptotes of the potential are studied for the limit of a strong gravitational field, the limit of large scalar fields, and the composite limit. The conditions for stability of quantum field theory in an external gravitational field are investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 26–32, October, 1985.The authors are indebted to I. V. Tyutin for evaluating the study.     

Influence of an external field on the stochastic theory of polymerization

A stochastic model is proposed to take into account the effect of a weak external field associated to diffusion, a discrete Master Equation is written down for the spatial distribution of particles. Approximate solutions are found for the first two cumulants, and applied to an example of polymerization reactions.     

The propagator of the Calogero-Moser system in an external quadratic potential

We obtain the Hamilton operator of the Calogero-Moser quantum system in an external quadratic potential by conjugating the radial part for the action of SO(n) by conjugacy of the Hamilton operator of the quantum harmonic oscillator on the Euclidean vector space of real symmetric matrices. Then, with Mehler's formula, we derive the propagator of the problem. We also investigate some schemes to change the interaction constant. For two-particle systems, we obtain explicit formulae, whereas for many-particle systems, we reduce the computation of the propagator to finding a definite integral. We give also the short time approximation, the energy levels and the trace of the propagation operator.     

Effective field theory approach to light propagation in an external magnetic field

The recent PVLAS experiment observed rotation of polarization and ellipticity when a linearly polarized laser beam passes through a transverse magnetic field. The phenomenon cannot be explained in conventional QED. We attempt to accommodate the result by employing an effective theory for the electromagnetic field alone. No new particles with a mass of order the laser frequency or below are assumed. To quartic terms in the field strength, a parity-violating term appears besides the two ordinary terms. The rotation of polarization and ellipticity are computed for parity-asymmetric and -symmetric experimental set-ups. While rotation occurs in an ideal asymmetric case and has the same magnitude as ellipticity, it disappears in a symmetric set-up like PVLAS. This would mean that we have to appeal to some low-mass new particles with nontrivial interactions with photons to understand the PVLAS result. PACS 12.20.-m; 12.20.Fv; 42.25.Lc; 42.25.Ja     

Integral equation theory for fluids ordered by an external field: separable interactions

The structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories. A general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles. This approach is then illustrated for the case of two (off lattice) models: the ferromagnetic Heisenberg model and a simple liquid crystal model, for which the numerical solution of integral equations such as the Percus-Yevick, the hypernetted chain, and the reference hypernetted chain closure equations are presented and compared with Monte Carlo simulation results and the analytical solution of the mean spherical approximation. The zero-field case is also examined, and the spontaneous ordering is analyzed in detail, mainly in what concerns the appearance of infinite wavelength singularity in the Ornstein-Zernike equation and the relation with the one-body closure equations and the long range orientational ordering that occurs. In particular, it is shown that the Wertheim one-body closure equation appears as a sum rule compatible with the Ornstein-Zernike equation. The relation between the elastic constant and the long range tail of the pair correlation function is made explicit. In particular, the long range behavior of the various terms in the expansion of the pair correlation function is depicted. The numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations, as to which one would be the most accurate in all cases. The general trends in the thermodynamical and structural properties seem to indicate that the Percus-Yevick approximation is generally better in the strong ordering case, whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes.     

Particle densities for dilute hard-sphere Bose or Fermi gases in an external potential

We prove that if the diameter of a hard-sphere is much smaller than the size of an external potential, the s-wave pseudopotential reduces to the Huang-Yang s-wave pseudopotential. We obtain the first-order virial expansions of particle densities for dilute hard-sphere Bose or Fermi gases in an arbitrary external potential. In the absence of an external potential, the results reduce to the Huang-Yang-Luttinger and Lee-Yang virial expansions. In the quasi-classical limit, the results reduce to the results of the local density approximation.