tailieunhanh - Báo cáo toán học: "On point interactions in one dimension "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Điểm trên các tương tác trong một chiều. | Copyright by INCREST 1984 J. OPERATOR THEORY 12 1984 101-126 ON POINT INTERACTIONS IN ONE DIMENSION s. ALBEVERIO F. GESZTESY R. H0EGH-KROHN and w. KIRSCH 1. INTRODUCTION The problem of studying the spectrum of a second order differential operator of the Schrodinger type A V with V a potential describing point interactions in the sense that V is supported by a discrete set arises in models for nuclear physics many-body theory solid state physics as well as in acoustics and optics see . 1 12 14 15 16 17 22 34 36 38 45 57 60 . The advantage of such interactions is that explicit computations are possible. Moreover it is possible to develop local short range interactions around point interactions a procedure which has been mathematically justified recently in the three-dimensional case by scaling techniques see 2 5 19 20 21 For further mathematical work on the definition and approximation of point interactions see also 6 8 10 11 13 14 17 18 26 27 28 35 49 50 59 61 62 . The approximation of the Schrodinger operators describing point interactions by scaled local potentials is closely connected with the study of the low energy limit of such Hamiltonians and in fact expansions around the zero energy limit have been obtained for physical quantities like energy eigenvalues and resonances as well as scattering amplitudes 3 5 19 20 21 . The purpose of this paper is to extend and continue the above results in the one dimensional case. In some cases we get stronger results and in all cases the extension is not immediate since the scaling behaviour is different in one and three dimensions. Section 2 contains in particular the proof of the norm resolvent convergence as e I 0 of Hamiltonians He N with scaled short range interactions Vj-. d2 1 - e s M dx2 j i V 8 in T2 R dx to point interactions tXj0x with centers at X . XN and strengths otj Ấ 0 ị ăxVj x . R 102 s. ALBEVERIO F. GESZTESY R. H0EGH-KROHN and w. KIRSCH The detailed behaviour of eigenvalues and resonances in the limit