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 12 1984 177 196 Copyright by INCREST 1984 SOME NORM BOUNDS AND QUADRATIC FORM INEQUALITIES FOR SCHRỒDỈNGER OPERATORS. II E. B. DAVIES 1. INTRODUCTION In this paper we discuss questions such as the boundedness of the operator V Hn L -1 when V is a potential on an open region Q R and Ho is minus the Laplacian with Dirichlet boundary conditions. Because we treat rather singular potentials V it is necessary to use quadratic form techniques systematically and to pay careful attention to domain questions. Although many of the results in 4 are superceded by the present paper we draw attention in Appendix 1 to a technical gap in f4 which makes several proofs here more devious than one might have thought necessary. The initial reason for the estimates in this paper was the desire to obtain a comparison theorem of the form m - Hữ k2 -nỉ CHIT1 - IVII where Vi are both strictly positive potentials which may have very strong local singularities. Such a bound is given in Section 4 and will be used in a subsequent paper to study double well Schrodinger operators. However in the course of the work we discovered a number of quite different applications of Theorem 6 our main result which we decided to include in this paper. In order to apply Theorem 6 effectively it is necessary to find suitable potentials X such that 0 X Ho where is minus the Laplacian in L2 i2 with Dirichlet boundary conditions. This is actually an entirely independent problem which we discuss in Section 5. An immediate consequence of these calculations is to give a new lower bound to the spectrum of Ho in terms of a quantity fl which we call the quasi-inradius of Q because of its similarity to the ordinary inradius known to be of relevance to this problem if N 2. This lower bound is given and discussed in Section 6. 12 - 2086 178 B. DAM1S We Start by assuming that Q is an open region in R v and impose no regularity properties on its boundary. We identify L- Q with the set of functions in RA

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