Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên khớp nối các ngưỡng liên tục trong hai kích thước. | Copyright by ỈNCREST 1985 JJ. OPERATOR THEORY 24 1985 263-276 ON COUPLING CONSTANT THRESHOLDS IN TWO DIMENSIONS HELGE HOLDEN 1. INTRODUCTION Recently there has been an increased interest in low-energy phenomena and also in the. study of two-dimensional systems due to applications to surfaces in solid state physics. In this paper we study the low-energy behaviour of a two-dimensional two--body system. Consider namely the Schrodinger operator Hk - A ẢV on L2 RV where A is the Laplacian and V is a suitable short-range potential. The spectrum of Hk then consists of possibly some negative eigenvalues in addition to the continuous spectrum 0 co . Assume now that . is a negative eigenvalue of Hk such that z Ĩ 0 as Ấ ị Á . E Ả approaches the continuous spectrum. While perturbation of discrete eigenvalues for self-adjoint operators yield analytic expansions for the eigenvalue in terms of the perturbation parameter this is not necessarily the case when an eigenvalue approaches the continuous spectrum. An interesting question is therefore whether one can find convergent expansions in singular quantities like z z0 a or z 20 ln . z0 . This problem has recently been extensively studied by Klaus and Simon 8 in all dimensions V and they find convergent expansions for E J. in various functions of Ấ like the two mentioned above and others. The results do very much depend the dimension V. However there is one case in which Klaus and Simon 8 do not give a convergent expansion namely in the two dimensional case which corresponds to .s-waves when V is central and where z0 5Ể 0. 264 HELGE HOLDEN In this paper we show that z in this case has a convergent expansion given by . - Ễ c r-fz - .o 0 -2 where a exp a z . T G - Zo - 2exp -fl z - Zo -1 and a and c 00 are explicitly computed and given by 14 and 15 respectively provided ịd2A K .r 0. In one and two dimensions it is also possible to have z0 0. This case has been considered by Klaus 6 in one dimension and Simon 10 in both one .
