ﻻ يوجد ملخص باللغة العربية
In this paper we give a new and constructive approach to stationary scattering theory for pairs of self-adjoint operators $H_0$ and $H_1$ on a Hilbert space $mathcal H$ which satisfy the following conditions: (i) for any open bounded subset $Delta$ of $mathbb R,$ the operators $F E_Delta^{H_0}$ and $F E_Delta^{H_1}$ are Hilbert-Schmidt and (ii) $V = H_1- H_0$ is bounded and admits decomposition $V = F^*JF,$ where $F$ is a bounded operator with trivial kernel from $mathcal H$ to another Hilbert space $mathcal K$ and $J$ is a bounded self-adjoint operator on $mathcal K.$ An example of a pair of operators which satisfy these conditions is the Schrodinger operator $H_0 = -Delta + V_0$ acting on $L^2(mathbb R^ u),$ where $V_0$ is a potential of class $K_ u$ (see B.,Simon, {it Schrodinger semigroups,} Bull. AMS 7, 1982, 447--526) and $H_1 = H_0 + V_1,$ where $V_1 in L^infty(mathbb R^ u) cap L^1(mathbb R^ u).$ Among results of this paper is a new proof of existence and completeness of wave operators $W_pm(H_1,H_0)$ and a new constructive proof of stationary formula for the scattering matrix. This approach to scattering theory is based on explicit diagonalization of a self-adjoint operator $H$ on a sheaf of Hilbert spaces $EuScript S(H,F)$ associated with the pair $(H,F)$ and with subsequent construction and study of properties of wave matrices $w_pm(lambda; H_1,H_0)$ acting between fibers $mathfrak h_lambda(H_0,F)$ and $mathfrak h_lambda(H_1,F)$ of sheaves $EuScript S(H_0,F)$ and $EuScript S(H_1,F)$ respectively. The wave operators $W_pm(H_1,H_0)$ are then defined as direct integrals of wave matrices and are proved to coincide with classical time-dependent definition of wave operators.
We describe the spectral theory of the adjacency operator of a graph which is isomorphic to homogeneous trees at infinity. Using some combinatorics, we reduce the problem to a scattering problem for a finite rank perturbation of the adjacency operato
Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associ
In the recent literature there has been a resurgence of interest in the fourth-order field-theoretic model of Pais-Uhlenbeck cite {Pais-Uhlenbeck 50 a}, which has not had a good reception over the last half century due to the existence of {em ghosts}
A class of singular integral operators, encompassing two physically relevant cases arising in perturbative QCD and in classical fluid dynamics, is presented and analyzed. It is shown that three special values of the parameters allow for an exact eige
Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and also on the choice of the contravariant components, it was shown that a wide variety of third, fourth, fift