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Báo cáo toán học: "Applications of the Krein resolvent formula to the theory of self-adjoint extensions of positive symmetric operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các ứng dụng của công thức Krein chỉ nhưng thuốc làm tiêu độc vào lý thuyết tự liên hợp các phần mở rộng của các nhà khai thác tích cực đối xứng. | J. OPERATOR THEORY 10 1983 209-218 Copyright by INCREST 1983 APPLICATIONS OF THE KREIN RESOLVENT FORMULA TO THE THEORY OF SELF-ADJOINT EXTENSIONS OF POSITIVE SYMMETRIC OPERATORS G. NENCIU 1. INTRODUCTION Let T be the closure in L2 0 oo of d2 dx2 defined on Cỏ O oo . T has deficiency indices 1 1 and its self-adjoint extensions Ta are indexed by the boundary conditions 1-1 I a 0 a 6 - 00 00 dx The Friedrichs extension Tp corresponds to a oo more properly Speaking O 0 and at the formal level is obtained by taking the limit a - co in 1.1 . Indeed one can verify that 1.2 . lim II Ta I -1 Tp 1 _1 0. a- oo Again at the formal level one can see that Tp is also obtained by taking the limit a oo in 1.1 and indeed one can verify with some work that 1-3 lim Tx I -1 - TF 1 -1 0. a co Now for a 0 ơ Ta -- a2 u 0 oo so in the sense of 1.3 Tp is the limit of self-adjoint extensions of T which are not uniformly bounded from below. The same phenomenon has been observed recently in the study of regulari-sations of the one-dimensional Schrodinger operator 5 8 . The initial motivation of this paper was to see whether this phenomenon the fact that the Friedrichs extension of a positive symmetric operator is in some sense the limit of some sequences of self-adjoint extensions which are not bounded from below is a generic one or related to the structure of the above examples. Our results below prove the gene-icity of this phenomenon. Our main result contained in Theorem 2 below gives 210 G. NENCIU a parametrisation of classes of not bounded from below self-adjoint extensions which in some sense made precise in Theorem 3 are neighbourhoods of the Friedrichs extension. Theorem 4 concerns the finite deficiency indices. An application of Theorem 4 to the problem discussed at the beginning of the Introduction is given in 15 . Some basic results of the Krein-Birman theory are also recovered. After the first version of this paper was finished we became aware of the fact that a particular case of