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Báo cáo hóa học: " The equiconvergence of the eigenfunction expansion for a singular version of onedimensional Schrodinger operator with explosive factor"

Tham khảo tài liệu 'báo cáo hóa học: " the equiconvergence of the eigenfunction expansion for a singular version of onedimensional schrodinger operator with explosive factor"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | El-Raheem and Nasser Boundary Value Problems 2011 2011 45 http www.boundaryvalueproblems.eom content 2011 1 45 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access The equiconvergence of the eigenfunction expansion for a singular version of onedimensional Schrodinger operator with explosive factor Zaki FA El-Raheem1 and AH Nasser2 Correspondence zaki55@Alex-sci. edu.eg department of Mathematics Faculty of Education Alexandria University Alexandria Egypt Full list of author information is available at the end of the article Abstract This paper is devoted to prove the equiconvergence formula of the eigenfunction expansion for some version of Schrodinger operator with explosive factor. The analysis relies on asymptotic calculation and complex integration. The paper is of great interest for the community working in the area. 2000 Mathematics Subject Classification 34B05 43B24 43L10 47E05 Keywords Eigenfunctions Asymptotic formula Contour integration Equiconvergence 1 Introduction Consider the Dirichlet problem y q x y kp x y 0 x n 1.1 y 0 0 y n 0 1.2 where q x is a non-negative real function belonging to L1 0 n l is a spectral parameter and p x is of the form p x Í1 0 x a n 1 a x n. 1.3 In 1 the author studied the asymptotic formulas of the eigenvalues and eigenfunctions of problem 1.1 - 1.2 and proved that the eigenfunctions are orthogonal with weight function p x . In 2 the author also studied the eigenfunction expansion of the problem 1.1 - 1.2 . The calculation of the trace formula for the eigenvalues of the pro-blem 1.1 - 1.2 is to appear. We mention here the basic definition and results from 1 that are needed in the progress of this work. Let ộ x l x l be the solutions of the problem 1.1 - 1.2 with the boundary conditions Ộ 0 l 0 ộ 0 l 1 n l 0 n l 1 and let W l ộ x l x l - x l ộ x l be the Wronskian of the two linearly independent solutions ộ x l x l . It is known that W is independent Springer 2011 El-Raheem and Nasser licensee Springer. This is