Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Study of the asymptotic eigenvalue distribution and trace formula of a second order operatordifferential equation | Aslanova Boundary Value Problems 2011 2011 7 http content 2011 1 7 0 Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Study of the asymptotic eigenvalue distribution and trace formula of a second order operatordifferential equation Nigar Mahar Aslanova1 2 Correspondence nigar. aslanova@ department of Differential Equation Institute of Mathematics and Mechanics-Azerbaijan National Academy of Science 9 F. Agayev Street Baku AZ1141 Azerbaijan Full list of author information is available at the end of the article SpringerOpen0 Abstract The purpose of writing this article is to show some spectral properties of the Bessel operator equation with spectral parameter-dependent boundary condition. This problem arises upon separation of variables in heat or wave equations when one of the boundary conditions contains partial derivative with respect to time. To illustrate the problem and the proof in detail as a first step the corresponding operator s discreteness of the spectrum is proved. Then the nature of the eigenvalue distribution is established. Finally based on these results a regularized trace formula for the eigenvalues is obtained. MSC 34B05 34G20 34L20 34L05 47A05 47A10. Keywords Hilbert space discrete spectrum regularized trace Introduction Let L2 L2 H 0 1 H where H is a separable Hilbert space with a scalar product and a norm J inside of it. By definition a scalar product in L2 is 1 Y Z l2 y y t z t dt - h yi Z1 h 0 1 0 where Y y t y1 Z z t z1 and y t z t e L2 H 0 1 for which L2 H 0 1 is a space of vector functions y t such that f0 y t II2 dt X. Now consider the equation w - -y t - p-y t Ay t q t y t ky t t e 0 1 v 1 2 y 1 - hy 1 ky 1 3 in L2 H 0 1 where A is a self-adjoint positive-definite operator in H which has a compact inverse operator. Further suppose the operator-valued function q t is weakly measurable and q t is bounded on 0 1 with the following properties 1. q t has a second-order weak derivative
