Construction of regular solutions of Schrödinger and Faddeev equations in the linear three-particle configuration limit

We study the six-dimensional Schrödinger and Faddeev equations for a three-particle system with central pairwise interactions more general than the Coulomb interactions. The regular general and particular physical solutions of such equations are represented by infinite series in integer powers of the distance from one of the particles to the center of mass of the other two particles and in some functions of the other three-particle coordinates. Constructing such functions in the angular bases formed by spherical and bispherical harmonics or by symmetrized Wigner D-functions reduces to solving simple algebraic recurrence relations. For the projections of physical solutions on the angular basis functions, we introduce the boundary conditions in the linear three-particle configuration limit.     

Point interactions in the problem of three quantum particles with internal structure

A system of three quantum particles with internal structure in which the two-body interactions are point interactions and are described in terms of two-channel Hamiltonians is considered. It is established that in the cases when the parameters of the model are such that the total Hamiltonian of the three-particle system is semibounded the Faddeev equations are Fredholm equations. Boundary conditions are formulated for the differential Faddeev equations whose solutions are the scattering wave functions.Joint Institute for Nuclear Research, Dubna. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 2, pp. 258–282, February, 1995.     

Three-Particle Problem with Pairwise Interactions Inversely Proportional to Squared Distance

The hyperharmonic method is used to investigate the three-particle Schrödinger and Faddeev equations with pairwise interactions inversely proportional to the squared distance. Exact solutions for such equations are constructed in the form of a product of the Bessel function depending on the hyperradius and a finite linear combination of the hyperharmonics. A criterion for the existence of such solutions is proved and analyzed.     

Faddeev differential equations as a spectral problem for a nonsymmetric operator

We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrödinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.We dedicate this paper to the memory of Stanislav Petrovitch Merkuriev, who left us three years ago.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 107, No. 3, pp. 513–528, June, 1996.     

Extension theory approach to scattering and annihilation in the\bar pd system

We consider the problems of three-particle scattering and annihilation in a system of three strongly interacting charged particles ( pn). We propose a model for the elastic scattering and the breakup process in the nucleon channel as well as for the annihilation into mesons. The mathematical foundation of the model is the extension theory of symmetrical operators. In the framework of this model, we construct the modified integral Faddeev equations with energy-dependent interactions taking the annihilation processes into account. These equations are uniquely resolvable for suitable classes of functions. On this basis, we deduce the corresponding differential Faddeev equations, construct asymptotic boundary conditions for wave function components, and formulate boundary problems for a system composed of nucleonic and mesonic channels. The results obtained are applied to scattering and annihilation processes in the three-particle system . Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 74–94, January, 1999.     

Spectral properties of Faddeev's equations

The spectral properties of the matrix operators corresponding to the three-particle Faddeev equations are investigated. It is shown that these operators have two types of invariant subspace. On the subspaces of the first type, the operators possess an eigenvalue spectrum identical to the spectrum of the three-particle Hamiltonian, while the eigenfunctions can be expressed in terms of solutions of the Schrödinger equation. On the subspaces of the second type, the operators are equivalent to the kinetic-energy operator of the system, and therefore their eigenfunctions do not correspond to the dynamics of the interacting particles.State University, St. Petersburg. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 323–336, March, 1995.     

Perturbation theory in the scattering problem for a three-particle system

We consider the scattering problem for a system of three nonrelativistic particles in the case of energies below the threshold of the system breakup into three free particles. We assume that the interaction potentials can be represented as a sum of two terms, one of which is a small perturbation. We develop a perturbation theory scheme for solving the scattering problem based on the three-particle Faddeev equations.     

Asymptotic behavior of the wave function of a system of several particles with pair interactions increasing at infinity

We construct an asymptotic representation of the wave functions of systems of two and three quantum particles with pair interactions increasing at infinity. We consider three-particle systems on the line and in the three-dimensional space. The eikonal and transport equations used to construct the asymptotic representation differ significantly from the corresponding equations in the case of decreasing potentials. We study the solution of the nonlinear eikonal equation in detail.     

The Faddeev equation and the location of the essential spectrum of a model multi-particle operator

In this paper we consider a model operator which acts in a three-particle cut subspace of the Fock space. We describe “two-particle” and “three-particle” branches of the essential spectrum and obtain an analog of the Faddeev equation for eigenfunctions of this operator.     

A stability result concerning the Shannon entropy

Summary. We are concerned with a well-known characterization of the Shannon entropy by Faddeev, suitably re-examined in the frame of Ulam--Hyers "stability" of functional equations.¶By use of some results about number theoretical functions, we give a sufficient condition that the solutions of a suitable system of countably many functional inequalities approximate the Shannon entropy uniformly.     

Structure of Regular Solutions of the Three-Body Schrödinger Equation near the Pair Impact Point

We investigate the six-dimensional Schrödinger equation for a three-body system with central pair interactions of a more general form than Coulomb interactions. Regular general and special physical solutions of this equation are represented by infinite asymptotic series in integer powers of the distance between two particles and in the sought functions of the other three-body coordinates. Constructing such functions in angular bases composed of spherical and bispherical harmonics or symmetrized Wigner D-functions is reduced to solving simple recursive algebraic equations. For projections of physical solutions on the angular bases functions, we derive boundary conditions at the pair impact point.     

Extremal solutions to a system of n nonlinear differential equations and regularly varying functions

The strongly increasing and strongly decreasing solutions to a system of n nonlinear first order equations are here studied, under the assumption that both the coefficients and the nonlinearities are regularly varying functions. We establish conditions under which such solutions exist and are (all) regularly varying functions, we derive their index of regular variation and establish asymptotic representations. Several applications of the main results are given, involving n‐th order nonlinear differential equations, equations with a generalized ?‐Laplacian, and nonlinear partial differential systems.     

AnS-matrix regularization of the trace formula for a three-particle system

A regularized trace formula is derived which expresses the trace of the connected part of the resolvent for a three-particle system in terms of the scattering matrices for two- and three-particle systems. This relation allows us to express the third cluster integral for Boltzmann statistics in terms ofS -matrices.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 95–131, 1976.In conclusion, the author would like to express his deep gratitude to V. S. Buslaev and L. D. Faddeev for many fruitful discussions.     

Spurious solutions of three-dimensional Faddeev equations

We obtain explicit spurious solutions of three-dimensional Faddeev equations written in the total angular momentum and total space parity representation. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 2, pp. 227–242, August, 2006.     

Optimal method for truncating a chain of equations for two-time Green’s functions

We use an example of a chain of equations describing a system of Bose particles with pairwise interaction to develop a method for decoupling the chain at its second element. We obtain an approximation of the interacting-modes type, which results in a system of nonlinear equations for one-, two-, and three-particle functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 503–510, June, 2006.     

Method for diagonalizing pair correlation functions for a multicomponent liquid system

We consider the method for finding pair correlation functions for a multicomponent liquid system based on diagonalizing the initial system of integral equations for the pair correlation functions, which allows solving this system. We obtain asymptotic solutions for the correlation functions and analyze the introduced approximations from the physical standpoint. We interpret the obtained results physically.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 3, pp. 569–576, March, 2005.     

Three-particle Coulomb on-shell vertex functions. Behavior of the amplitude for simultaneous transfer of two charged particles in the vicinity of the nearest singularity in cos θ

The connection between the on-shell vertex function (OSVF) and the asymptotic normalized coefficient of the three-particle wave function of the bound system in the configuration space is found. The explicit form of the leading singular term of the OSVF and its Faddeev component for decay of the three-particle bound state into three charged particles is established. An expression is found for the leading singular term of the exact (in the model of four charged particles) amplitude of sub-Coulomb binary reactions in the vicinity of the nearest singularity.Institute of Nuclear Physics, Uzbek Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 3, pp. 448–462, March, 1993.     

Nonlinear Transformations of Evolution Families Generated by Diffusion Processes

We derive closed equations for some nonlinear transformations of solutions to forward and backward Kolmogorov equations (which are parabolic equations with respect to measures and functions). In particular, we obtain closed equations for the logarithmic derivatives of smooth diffusion measures and for the Radon–Nikodym derivatives of a pair of absolutely continuous diffusion measures. Similar results are obtained for the backward Kolmogorov equation. Bibliography: 14 titles.     

On the L2 improperly posed problems

This paper considers a problem of approximation of functions proposed by Bellman [1]. The results include a representation lemma for the solutions and an algorithm for computing such solutions. A sufficient condition for the convergence of the algorithm to the optimal solutions is shown to be related to the uniqueness of solutions of a pair of functional equations. To the author's knowledge, not much is known about these functional equations and these may prove to be of interest in future research. An example is included to illustrate the algorithm.     

Paths of random evolutions

We study the paths of general random evolution processes obtained by piecing together deterministic evolution functions according to the dictates of a regular step process. If the state space is metrizable we show that such processes are strong Markov; and they are even standard under a certain continuity condition on paths. We apply this result to solutions of stochastic delayed differential equations, and we make a connection between our processes and random evolutions associated with classes of semigroups.