tailieunhanh - Báo cáo toán học: "Generalized Dirac-operators with several singularities "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Tổng quát Dirac khai thác một số kỳ dị. | J. OPERATOR THEORY 13 1985 171 188 Copyright by INCREST 1985 GENERALIZED DIRAC-OPERATORS WITH SEVERAL SINGULARITIES BERTEL KARNARSKI 1. INTRODUCTION AND NOTATIONS Jorgens generalizes in 6 the concept of Dirac-operators. But there he only allows singularities at infinity. For physical reasons . an electron possesses an anomalous magnetic moment 12 3 4 8 we have also to allow singularities localized at compact sets of measure zero 8 9 . In generalizing this concept one can also study multicenter operators of Jorgens-type see 9 12 for a physical nterpretation . Using a general decomposition principle we show in the following that the multicenter case can be reduced to the case of a single center. In the second and the third section of our paper we determine a formula for the deficiency indices and the essential spectral kernel. Further in all cases important for physical applications we also construct selfadjoint extensions and determine their essential spectrum. The techniques used are similar to those of Behncke 4 . Especially we generalize Theorem 1 in 4 in many ways. Finally we apply the results of the previous sections to Dirac-operators with several Coulomb-singularities. In particular we generalize the results of Landgren and Rejto 12 Arai 1 and Klaus 9 . At the end we also discuss the problem of determining a physically distinguished selfadjoint extension and generalize some results of Klaus 9 and Nenciu 13 . The notation will be mostly standard. Especially T denotes the closure of the operator T while T JTT and Ố T denote its domain kernel and range respectively. For a closed symmetric operator T we denote the deficiency indices the spectral kernel and the essential spectral kernel by def. T S T and S T respectively for a definition of these notions see 19 Chapter 8 . The spectrum and the essential spectrum of a selfadjoint operator T are denoted by ơ T and ơe T respectively. In the following all operators are defined in the Hilbert space 2 Rm Cp m peN 172