tailieunhanh - Báo cáo toán học: "Spectral asymptotics for the "soft" selfadjoint extension of a symmetric elliptic differential operator "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Phổ asymptotics gia hạn "mềm" selfadjoint của một nhà điều hành khác biệt giữa đối xứng elip. | Copyright by INCREST 1985 J. OPERATOR THEORY 10 1983 9 - 20 SPECTRAL ASYMPTOTICS FOR THE SOFT SELF ADJOINT EXTENSION OF A SYMMETRIC ELLIPTIC DIFFERENTIAL OPERATOR GERD GRUBB 1. INTRODUCTION The purpose of this paper is to give an affirmative answer to a question raised by A. Alonso and B. Simon in 2 concerning the asymptotic behavior of the nonzero eigenvalues of the so-called soft extension AM or Krein extension or von Neumann extension of the minimal operator y4min associated with a strongly elliptic formally selfadjoint elliptic operator A of order 2m on a bounded smooth domain QcR . We show that the number N t AM of eigenvalues in 0 t satisfies 1-1 N t Am cAtnl2m O t n--9 2 for t - oo where CA is the same constant as for the Dirichlet problem and 0 max 1 2 . 2m 2m 4- n 1 for any e 0. Here 0 - 2m 2m 4 n 1 when 2m 1. The method consists of reducing the eigenvalue problem for AM to the eigenvalue problem for a certain compact operator s whose deviation from the inverse Ay of the Dirichlet realization Ay of A G-S Aỹ is an operator with the spectral behavior ij G O j-2m U-l Ị for j co. The eigenvalues of s are then estimated by application of a perturbation argument to a well known estimate for the eigenvalues of Ay. There are several ways to prove . A method is to use the very particular structure of G to derive from a result essentially due to M. s. Birman 5 by use of operator-theoretical arguments involving a theorem of Birman Koplienko 10 GERD GRUBB and Solomiak 6 . Another method is to view G as a kind of singular Green operator as introduced by L. Boutet de Monvel 8 for which the spectral asymptotics were studied systematically in G. Grubb 15 17 18 Since the second method involves more technical machinery than the first one we present the first method noting however that the second point of view should be of value in the treatment of more general problems. 2. BACKGROUND MATERIAL It was shown by M. G. Krein in 20 that the family of selfadjoint