TAILIEUCHUNG - Báo cáo nghiên cứu khoa học: "Schrödinger operators whose potentials have separated singularities "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhà khai thác Schrödinger có tiềm năng đã phân chia kỳ dị. | J. OPERATOR THEORY 1 1979 109-115 Copyright by INCREST 1979 SCHRỐDINGER OPERATORS WHOSE POTENTIALS HAVE SEPARATED SINGULARITIES JOHN D. MORGAN III I. INTRODUCTION This paper is mainly concerned with obtaining lower bounds to the bottom of the spectrum of a Schrodinger operator V2 V on R where the real-valued potential V has separated singularities. The main result is that if there is a possibly finite sequence of smooth real-valued functions on R to R with p 1 and sup I v p 2 4-00 Ỉ i such that for some real a and b II I P 1 2 pứ 2 a II V 2 4- b li ll2 for all ựr in z iV then there exists a constant c such that V ll2W a 2 e W for all ỷ inZ i V .That is given a uniform local estimate of the form we can derive an estimate of the form . We also present an operator version of this result given stronger differentiability assumptions on the p-s. These estimates are useful in the following physical situation. Consider a Hamiltonian H V2 4- V on R where the potential V Vỵ 4- V2. Suppose that Fl and Vz have separated singularities in the sense that there exists a real number K such that ĩíị ki x l K and x F2 x K are separated by a minimum distance R 0. Then it is natural to enquire whether the semiboundedness of both j V2 Vỵ and Hz V2 4- Vz implies that H V2 4- Vi Vz is semibounded. For example in R3 the Hamiltonian H pì V2 a r2 is semibounded by 0 if a 1 4 and unbounded below if a 1 4 one might then ask whether for ro O the Hamiltonian H a P V2 air P r r0 2 is semibounded if a p 1 4 but a p 1 4. no JOHN D. MORGAN III The basic technique is to partition the density distribution ợ 2 of a function ý in 2 V2 the form-domain of V2 as l p 2 I A1I2 l l2 where 1-6 supp IO n x I 72 x K suppdO n x l x K 0 and ựq and ự 2 are in Q V2 - Thus i zj and ự 2 will not see the singularities of v2 and Fj respectively. Then modulo some terms which are easily seen to be bounded the semiboundedness of Hỵ and H2 will imply the semiboundedness of H. The guiding .

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