TAILIEUCHUNG - Báo cáo toán học: Some norm bounds and quadratic form inequalities for Schroedinger operators

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số chỉ tiêu giới hạn và bất bình đẳng hình thức bậc hai cho các nhà khai thác Schroedinger. | J. OPERATOR THEORY 9 1983 147-162 Copyright by INCREST 1983 SOME NORM BOUNDS AND QUADRATIC FORM INEQUALITIES FOR SCHRỒDINGER OPERATORS E. B. DAVIES 1. INTRODUCTION We present a number of new norm bounds on some operators involving powers of the resolvent of H Hữ V on L2 RN where Hn A. We assume that Flies in the class Í of potentials for which 0 Ke L W R Q where Q is a closed set of zero Lebesgue measure depending on V. The operator H is then well-defined as a form sum with form domain Quad ff 2 C RN i2 and one may use the quadratic form version of the Trotter product formula due to Kato 4 p. 121 . This implies by the methods of 3 4 p. 179 or 8 p. 186 that the integral kernels of H -Ị- are non-negative and pointwise dominated by those of Ho z - for all p A 0. It is probable that many of the estimates we obtain could be extended to the case where r has a negative part r_ which is small enough would have to have form bound less than one with respect to 7 0 but we have not attempted such an extension. We start with an elementary result. Lemma 1. Ifv We then 11 0 v i -i- o w - K l v-w F 1 2 W I 1 2 CO 148 E. B. DAVIES Proof. 11 0 V I -1 - w l -1 - iiw. 4- y-i- l -1 - W Ơ4 4- 4- 1 -1 ll W0 K 1 - F l 1 2 - o w I -1 4- 1 1 2 - II K 4- 1 -V2 r - W IP -5-1 -U2ỊỊ. The result follows since K TO v 1 -V2 1 IIƠP4- l 1 2 4 4- wz4- 1 -V2 1. Corollary 2. Let H Ha 4- v and H - Ho V where v V 6 fS. Then Hn converges in the norm resolvent sense to H if either lim II r - P L 0 - co 1 r l 4- P l 4-x where lim IIXn IL - 0. M-iCO Proof The first statement is elementary. For the second we need only note that if I T nIL 4 1 2 then i V K Ị ------------- ---------11 4 21 2 x IL . I 1 4- P 1 2 1 1 2 . We next note that it is by no means the case that one always gets norm resolvent convergence when v - V. For example if N 1 and V x 14- cos 2n x and if Í V x if w n nW 12 otherwise then v converges locally uniformly and boundedly to V. Now f Hn is a compact operator for any continuous .

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• báo cáo của tạp chí Journal of Operator Theory
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học
• tài liệu báo cáo nghiện cứu khoa học