tailieunhanh - Báo cáo toán học: "Theorems on almost everywhere convergence in von Neumann algebras"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hầu như ở khắp mọi nơi trên định lý hội tụ trong đại số von Neumann. | J. OPERATOR THEORY 6 1981 . 313-339 Copyright by INCREST 1981 POINT INTERACTIONS AS LIMITS OF SHORT RANGE INTERACTIONS SERGIO ALBEVERIO and RAPHAEL H0EGH-KROHN 1. INTRODUCTION In the physical literature in particular in nuclear physics and in solid state physics there has been a long standing interest in studying models of a quantum mechanical particle moving under the influence of fixed centers the force between the particle and the centers being a point one zero range of different strengths we speak for short of models with 5-potentials . For such models see . 1 6 . The mathematical description of the Hamiltonian for these models is by standard quadratic forms theory in the one dimensional case the ỗ-potential being in this d2 case relatively form-bounded with respect to-- see . 7 Ch. 8 . dx2 In the two and three-dimensional cases the definition of the Hamiltonian requires more subtle methods. The problem for the case of one center had been discussed in the physical literature starting with 9 see also 10 mathematical solutions were then provided by Berezin and Faddeev 11 by using Krein s method of self-adjoint extensions by Streit and ourselves using the method of Dirichlet forms 12 and by Nelson 34 and Fenstad and ourselves using non standard analysis 13 in the latter paper the many centers problem is also solved .1 The non standard analysis version exhibits the Hamiltonian as a smooth perturbation of A by a potential of infinitesimal support. An explicit computation of the resolvent was also given by these non standard methods. The detailed study of the resolvent and of the scattering was pursued further without non standard analytical methods in 14 15 where also applications to models of solid state physics were given see also 16 A natural question that arises is whether the so defined Hamiltonians for ổ-interactions can be approximated by Hamiltonians with smooth local potentials . by operators of the form A V A I x Xị f with some smooth .