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Báo cáo nghiên cứu khoa học: "Maximal and minimal Schrödinger forms "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Schrödinger hình thức tối đa và tối thiểu. | J. OPERATOR THEORY 1 1979 37-47 Copyright by INCREST 1979 MAXIMAL AND MINIMAL SCHRODINGER FORMS BARRY SIMON 1. INTRODUCTION Let V 0 V e L oc a e L ioc. There are a priori two natural quadratic forms associated to i V a 2 V. Namely V p distributional gradient let 1-1 2 max pe 2l V WeL2 V1 2 peL2 and 1.2 hmJ p A Ẻ dj - MjW 2 p j i Define Amin to be the form closure of Amax restricted to Co X Co . Our main result in this note is that hmin Zzmax under the most general conditions considered above. Surprisingly this natural question appears to have been asked only recently by Kato 15 . One can phrase the extensively studied question of whether A V is essentially self-adjoint on Co in virtually identical terms namely let felloe and let cp e L21 - A p v p e L2 with Hm P A p v p. Let Hmin be the operator closure of Hmax restricted to cỏ . Since Hmax the operator equality is one of essential self-adjointness and is a necessary preliminary to the existence of a self-adjoint realization of A V. The form equality is only one of uniqueness since general theorems associate self-adjoint operators to any closed form 10 16 This is probably the reason that the question has only recently been raised. We also note that in finite volume problems Amin and ỉmax do differ one corresponds to Dirichlet boundary conditions the other to Neumann conditions see e.g. ref. 18 . 38 BARRY SIMON Our main tool in this note will be certain semigroup ideas which can be traced back to developments in constructive quantum field theory especially the P cp2 -Hamiltonian self-adjointness proof of Rosen 19 and Segal 22 abstracted in ref. 31 These ideas were useful in the study of Schrodinger operators 23 6 8 and in particular led to the proof of the fact that A V is essentially self-adjoint on Co if K 0 and V e L R e- 2 dx 23 They appear to have been abandoned as a self-adjointness tool in part because of the success of a different method of Kato 11 who in particular removed the global L RV e 2 dx restriction