Eigenvalue problem of Sturm-Liouville systems with separated boundary conditions

2015/04/08 by Xijun Hu, Hu, Xijun, Penghui Wang +1
Mathematics · #34B24 #34L15 #47E05 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP) #math.DS #math.FA #math.SP #msc:34B24 #msc:34L15 #msc:47E05

paper · pdf · doi:10.48550/arxiv.1504.01817

10 pages

arxiv created 2015/04/08 · arxiv updated 2015/04/09

Abstract

Let λ_j be the j-th eigenvalue of Sturm-Liouville systems with separated boundary conditions, we build up the Hill-type formula, which represent ∏_j(1-λ_j^-1) as a determinant of finite matrix. This is the first attack on such a formula under non-periodic type boundary conditions. Consequently, we get the Krein-type trace formula based on the Hill-type formula, which express ∑_j1\over λ_j^m as trace of finite matrices. The trace formula can be used to estimate the conjugate point alone a geodesic in Riemannian manifold and to get some infinite sum identities.

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