TAILIEUCHUNG - Báo cáo toán học: "On the Carleman resolvent estimate of Schroedinger type operator with arbitrary potential "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Về dự toán chỉ nhưng thuốc làm tiêu độc Carleman của người điều khiển loại Schroedinger với tiềm năng tùy ý. | Copyright by INCREST 1983 J. OPERATOR THEORY 9 1983 27 -52 ON THE CARLEMAN RESOLVENT ESTIMATE OF SCHRODINGER TYPE OPERATOR WITH ARBITRARY POTENTIAL YURI OROCHKO 1. INTRODUCTION Let a x aJk x xe R 1 m 3 be a real positive definite for each _Y e Rm m X m-matrix with the elements ajk x e C a Rm . We denote by v2 x p2 x the lowest respectively the largest eigenvalue of a x and we impose on a x the restriction v0 v x fi x u0 X e Rm with some constants 0 v0 ẬZ0. Consider an arbitrary self-adjoint extension A in L2 R of the minimal operator corresponding to the elliptic differential expression 1 s div a x grad V x with the real potential V x e C Rm having no restrictions at the infinity. Let R x y r be the kernel of the resolvent A it -1 T 0 Imr 0. It is known see . 3 Chapter 6 Section 2 that A it -1 is a Carleman operator that is the integral L x i R T OI2dp Rm is a locally bounded function of X. Certain problems of the spectral theory demand the information about the growth rate of ZT x at the infinity. The well-known result of this kind is the uniform boundedness of It x in Rm in the case of Schrodinger expression A V x with the potential bounded from below. It follows from Kovalenko-Semenov results 9 Theorem that the same property holds if we add to the semibounded function F x another real function V- x which is a relatively small perturbation of the Laplace operator. Let be the ball X y 5 in Rm xf S x being its characteristic function ll-llp stands for the norm in Lp Rm . The aim of the present paper is to 28 YURI OROCHKO prove the following estimate of IT x for expressions s with arbitrary behaviour of potential at the infinity. Theorem 1. Fix p 1 p - 1 M 3 . There exists a function c0 r 2 r 0 depending only on m p p0 0 lim i o r --- 0 such that F QO 2 I x Í 0- n4 h l r XG R K_ x -- min 0 V x for every ỗ G 0 4-17rpn . In particular this result delivers useful information on the existence of wave operators corresponding to the pair of real .

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