tailieunhanh - Báo cáo toán học: "A complete treatment of low-energy scattering in one dimension "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một điều trị hoàn toàn của tán xạ năng lượng thấp trong một chiều. | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 3-31 A COMPLETE TREATMENT OF LOW-ENERGY SCATTERING IN ONE DIMENSION D. BOLLÉ F. GESZTESY and s. F. J. WILK 1. INTRODUCTION The purpose of this paper is to provide a systematic analysis of low-energy scattering on the entire real line taking into account explicitly the possibility of zero-energy resonances of the Hamiltonian. Such an analysis has been carried out very recently for three dimensions 1 2 13 . In one and two dimensions this problem is more involved due to the well-known additional difficulty that the free Green s function has a square root logarithmic singularity in the limit as the energy tends to zero. Different aspects of the one-dimensional scattering problem have received much attention in the past especially in connection with inverse scattering techniques which are used extensively in quantum mechanical problems cf. 14 16 34 and the references therein and quantum field theory cf. 17 46 for a review . More recently new rigorous results have appeared 3 7 15 19 20 26 30 33 36 38 39 41 47 . In particular one has studied the ground-state properties of one-dimensional Schrodinger operators with various potentials including long-range ones especially in the limit of weak coupling 7 26 27 38 39 41 . Also bounds for the number of bound states 26 27 36 as well as for the imaginary parts of resonances 20 have been obtained. Other results are concerned with the limit situation where some negative eigenvalues approach zero as the coupling constant approaches a critical value 29 30 . Furthermore scaling techniques have been applied to analyse in detail the limit of one-dimensional short-range interactions converging to point interactions 3 . We remark that the latter paper contains an extensive list of earlier one-dimensional results which are not explicitly mentioned here. Let us now give a short description of the results obtained in this paper. In Section 2 we study the occurrence and properties of .