Đang chuẩn bị liên kết để tải về tài liệu:
Báo cáo toán học: "Các phương pháp Comutator và dự toán không gian Besov cho Schrà ¶ dinger các nhà khai thác"
Đang chuẩn bị nút TẢI XUỐNG, xin hãy chờ
Tải xuống
Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các phương pháp Comutator và dự toán không gian Besov cho Schrà ¶ dinger các nhà khai thác | J. OPERATOR THEORY 14 1985 181-188 Copyright by INCREST 1985 COMMUTATOR METHODS AND BESOV SPACE ESTIMATES FOR SCHRODINGER OPERATORS ARNE JENSEN and PETER PERRY I. INTRODUCTION In this note we show how Mourre s commutator methods 7 8 9 can be used to prove resolvent estimates of the optimal type introduced by Agmon and Hormander 3 for a large class of Schrodinger operators. These estimates are sharp in a sense made precise below in the case of rV-body Schrodinger operators see below they are new. To state the class of operators we will study let A be the Laplace operator on R and let n be projections onto subspaces .T - of R . If A - is the Laplacian on X and L .T - R is a measurable function such that the operator L A 1 -1 is compact on 2 iZ the differential operator generalized 7V-body Schrodinger operator M p - A ỵ r . 77 . i 1 is essentially self-adjoint on C R . Letting H denote its self-adjoint extension we want to study the behaviour of R z H z -1 Im z 0 as z approaches points A in the continuous spectrum of H. For the class of Schrodinger operators considered in Theorem 1.1 below it is known cf. II and references therein that R z is bounded as a map from 2 R to is R for any .S 1 2 with bound uniform in z with Imz o and Reze R 5 . Here 6 H is a closed countable set 7 11 consisting of eigenvalues and thresholds of 7 see e.g. 11 for a discussion of thresholds and 2 R e oc R ị I x 2 J u x 2 d.v ooị with the obvious norm. It is known moreover that the resolvent has Holder continuous boundary values RỌ. Í0 for A ệ 0 H as maps from R to 1 R .V 1 2. These boundary values are basic objects in the stationary scattering theory and the theory of eigenfunction expansions for H. 182 ARNE JENSEN and PETER PERRY For the case of two-body Schrodinger operators i.e. M 1 and I71 identity Agmon and Hormander 3 introduced an optimal framework in which to study boundary values of R z . They defined the space B R and its dualB R as follows. Let Rj 2j for j 0 1 .and let Qj x 6 R .