Application of the functional integration method to laser theory

An approach is proposed for the investigation on nonequilibrium processes, based on the functional integration method. A system of laser type is examined. Expressions for the density matrix of a two-level system in a classical field and for the effective action functional of the radiation field are obtained. The equation for the stationary point of the effective action characterizes the process of laser generation. In the equilibrium limit this equation describes the Bose condensation in Dicke-type models.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 131, pp. 148–165, 1983.In conclusion, the author expresses his gratitude to V. N. Popov for his interest in this work.     

Application of the method of functional integration to the theory of superradiation

The process of spontaneous decay of a concentrated atomic system in a radiation field is studied by reconstructing the standard functional integral for the evolution operator. An exact expression is found for the state of the system which arises as a result of this process. The present results generalize the results known for this model which are found by the method of the inverse problem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 161, pp. 146–154, 1987.     

Path integration method in the theory of model Hamiltonians

The path integtation method is used to obtain the asymptotic behavior of the ratio Z/Z₀ for a model of the BCS type (Z is the statistical sum of the model, Z₀ is the statistical sum of the free system) in the limit V → ∞. The asymptotic formulas for Z/Z₀ are different above and below the transition and are expressed in terms of infinite products. A rigorous proof is obtained for the formulas under the condition of fulfillment of a natural additional assumption.     

Introduction to the theory of noncommutative integration

We present a survey of the contemporary state of the theory of integration in von Neumann and JBW algebras.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki (Noveishie Dostizheniya), Vol. 27, pp. 167–190, 1985.     

Studying the properties of carbon nanotubes with the functional integration method

We obtain an exact, closed, self-consistent system of equations for describing nanotubes that takes electron and oscillation subsystems in the collective variables into account. Collective excitations in nanotubes are described by the quantum numbers n, m, and k, where n − 1 is the number of radial modes, m is the number of azimuthal modes, and k − 1 is the number of longitudinal modes of the wave function. The results obtained approximate the experimental data better than those obtained by the method of linear combinations of atomic orbitals. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 1, pp. 127–144, October, 2006.     

A functional approach to the theory of quantum crystals

An approach is developed for describing inhomogeneous periodic structures in a system of Fermi particles with a short-range pair potential in the formalism of a functional integral and a functional effective action. Instability of a homogeneous base state at large pressures and also instability of a weakly inhomogeneous bodycentered structure are demonstrated.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 145, pp. 46–61, 1985.     

Application of the finite-element method to improve the quasi-exact reinforcement theory of fibrous polymeric composites

In the paper, the WL quasi-exact reinforcement theory of fibrous polymeric composites is improved. An optimum compatibility condition related to the transverse shear problem for a unit cell, which brings solutions closest to reality, is derived. This condition is formulated in the form of a linear combination of maximum radial and circumferential displacements. Optimum coefficients of this combination are determined by comparing analytical and numerical solutions for a test specimen in the form of a rectangular thin plate, which is in a plane strain state and is subject to selected loading schemes. The analytic solutions are obtained for a homogenized material by using the WL reinforcement theory. The numerical solutions are found for an actual heterogeneous composite material by using the finite-element method, and they verify the WL reinforcement theory, in particular, the admissibility of Hills assumption. An analysis performed for two composite materials shows that the improved WL reinforcement theory gives adequate displacement fields.Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 79–92, January–Febrauary, 2005.