Eigenvalue problems for Sturm-Liouville equations with transmission conditions
For the Sturm-Liouville eigenvalue problem \(-u''+ q(x) u = \lambda u\) for \(x \in (-1, 1), x \not = 0\) subject to the eigenparameter-dependent boundary conditions \(u(-1) = 0, (\lambda - a) u'(1) + \lambda b u(1) = 0\) and the transmission conditions at \(x = 0\): \(u(-0) = h_1 u (+0), u' (-0) = h_2 u' (+0)\), where \(ab > 0, h_1 h_2 > 0\), it was established by the authors [Appl. Math. Comput. 157, 347--355 (2004; Zbl 1060.34007)], that all eigenvalues are real and the eigenfunctions corresponding to different eigenvalues are orthogonal in a Hilbert space. In this article, the asymptotic behavior of the eigenvalues and the corresponding eigenfunctions are given through a realization of the basic solutions which are solutions of the problem on \([-1, 0) \cup (0, 1]\) except that only one boundary condition is satisfied. A counterexample is given to show that \(ab > 0\) and \(h_1 h_2 > 0\) are necessary.
eigenparameter-depent boundary conditions
scientific article; zbMATH DE number 2147134
scientific article