tailieunhanh - Báo cáo toán học: ""Localized" self-adjointness of Schrödinger operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: "Địa phương hóa" tự adjointness của các nhà khai thác Schrödinger. | J. OPERATOR THEORY 1 1979 287-290 Copyright by INCREST 1979 LOCALIZED SELF-ADJOINTNESS OF SCHRODINGER OPERATORS H. BREZIS Let ei oc Rm be a real valued function. Assume A A q with Rm CS Rm satisfies 1 A Q . j I v p 2dx j ạ ợ 2dx 0 V p e Rm . Suppose in addition that for every x0 e R there exist q e áL Rm with q 1 near Ao 5 0 and c such that 2 A qq 5 A q q c 3 A qq is essentially self-adjoint on S R in L2 R . Our main result is the following Theorem 1. A is essentially self-adjoint in L2 R . Remark. Assumptions 2 and 3 hold for example if q- e Lfoc Rm where ĨĨĨ _ p 2 when m 3 p 2 when m 4 p - when m 5 see T. Kato 5 and 2 also M. Reed and B. Simon 6 Theorem . In this case the conclusion of Theorem 1 was obtained by Chernoff 2 Theorem under slightly more general assumptions using a completely different approach . Related results can be found in 3 4 7 8 9 . In the proof of Theorem 1 we shall use the following Lemma 1. Assume 1 . Let V E HfR be such that 7 r 2 6 Then 7 t 2 6 L lR and I Vf 2dx q y 2dx 0. 288 H. BREZIS Proof. The conclusion holds obviously if V e 1 Rm n ROT has compact support use smoothing by convolution . In the general case truncate f and multiply by cut-off functions. Lemma 2. Let Q 6 L oz Rm be a real valued function. Assume 4 A -r Q is essentially self-adjoint on Rm in 2 Rm 5 A Q Ỗ A Q -C for some Ỗ 0 and some c. Let V e L2 R be such that Av Qv e Then vsH Rm and 2 v 2 6 L Rm . Proof. Let B A Q c with D B SfRm . By 4 and 5 B is essentially self-adjoint and B 0. Consequently Nự 5 0 . Let w v 6 Q M2 e L fR with the Hilbert norm IHI2 ị I v K pdx r j Q w 2dx j H2dx. We deduce from Lemma 1 and 5 that if w 6 w then Q wI2 e L R and j Iv dx 2 v 2dx 5 vM2 Ỗ j Ổ W2 - c ị W2. Consequently if T is given in 7 -1 R there exists a unique weW such that 6 ị dx j Qwỹ dx ị C 1 wý dx T ý V ìỊj e w by Lax-Milgram . If ÍÍ e L2 R is such that Av Qv 6 we may choose in 6 T Av Qv C l p. It follows that V w e N I B and thus v w in particular V w. Proof of .