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.     

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.     

Essential spectrum of a model operator associated with a three-particle system on a lattice

We consider a model operator H associated with the system of three particles interacting via nonlocal pair potentials on a ν-dimensional lattice. We identify channel operators and use their spectra to describe the position and structure of the essential spectrum of H. We obtain an analogue of the Faddeev equation for the eigenfunctions of H.     

The resolvent of a singularly perturbed elliptic operator

Let A and B be uniformly elliptic operators of orders 2m and 2n, respectively, m > n. We consider the Dirichlet problems for the equations (?^{2(m ? n)}A + B + λ²ⁿI)u_? = f and (B + λ²ⁿI)u = f in a bounded domain Ω in R^k with a smooth boundary ?Ω. The estimate is derived. This result extends the results of [7, 9, 10, 12, 14, 15, 18]by giving estimates up to the boundary, improving the rate of convergence in ?, using lower norms, and considering operators of higher order with variable coefficients. An application to a parabolic boundary value problem is given.     

The essential spectrum of the three-particle discrete operator corresponding to a system of three fermions on a lattice

We consider a family of three-particle discrete Shrödinger operators H _μ (K). These operators are associated with the Hamiltonian for a system of three identical particles (fermions) with pairwise two-particle interactions on neighboring junctions of the d-dimensional lattice Z ^d . We describe the location and the structure of the essential spectrum of the operator H _μ (K) for all values of the three-particle quasi-momentum K ∈ T ^d and the interaction energy μ > 0.     

An Ablowitz-Ladik system with a discrete potential: I. Extended resolvent

An Ablowitz-Ladik linear system with the potential taking the values 0 or 1 is considered. The extended resolvent of this system is constructed, and the singularities of this operator are analyzed in detail. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 20–33, April, 1999.     

On a nonconvolution Volterra resolvent

Under fairly weak assumptions, the solutions of the system of Volterra equations x(t) = ∝₀^ta(t, s) x(s) ds + f(t), t > 0, can be written in the form x(t) = f(t) + ∝₀^tr(t, s) f(s) ds, t > 0, where r is the resolvent of a, i.e., the solution of the equation r(t, s) = a(t, s) + ∝₀^ta(t, v) r(v, s)dv, 0 < s < t. Conditions on a are given which imply that the resolvent operator f ∝₀^tr(t, s) f(s) ds maps a weighted L¹ space continuously into another weighted L¹ space, and a weighted L^∞ space into another weighted L^∞ space. Our main theorem is used to study the asymptotic behavior of two differential delay equations.     

The resolvent of a periodic optical waveguide the continuous spectrum

We consider a two-dimensional optical waveguide with periodic media interface. We study the resolvent of the waveguide in a neighborhood of a purely imaginary point of the spectral parameter. We prove that the resolvent exists on the subspace of functions orthogonal ina certain sense to the singular functions of the continuous spectrum. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 13, 1992, pp. 79–89.     

On the minimal polynomial of a resolvent

We derive the connection between the minimal polynomial of an operator and the minimal polynomial of its resolvent. An analogous result is obtained for the minimal polynomials of the iterated resolvents.     

Wave operators in a three-particle model of a rigid kernel

For the three-particle scattering problem in the model of a rigid kernel one establishes the existence, the isometry, and the completeness of the wave operators. One proves the absolute continuity of the Hamiltonian system.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 45–49, 1986.