Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

摘要

In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parametersboth in the equation and in one of the boundary conditions, are investigated. By using an operator-theoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtainasymptotic approximate formulae for eigenvaiues and normalized eigenfunctions. We modify sometechniques of[Fulton, C. T., Proc. Roy. Soc. Edin. 77 (A), 293-308 (1977)], [Walter, J., Math. Z.,133, 301-312 (1973)]and[Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second OrderDifferential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniqueswe obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.