tailieunhanh - Báo cáo toán học: "Các phương pháp Comutator và dự toán không gian Besov cho Schrà ¶ dinger các nhà khai thác"

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 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 . 11 for a discussion of thresholds and 2 R e oc R ị I x 2 J u x 2 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 . 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 .