Basis properties of the eigenfunctions of two-interval Sturm–Liouville problems

Abstract

In this paper a Sturm–Liouville equation together with eigenparameter-dependent boundary-transmission conditions are considered on two disjoint intervals. We construct the resolvent operator and Green’s function and obtain asymptotic approximate formulas for eigenvalues and corresponding eigenfunctions. The obtained results are implemented to the investigation of the basis properties of the system of eigenfunctions in the Lebesgue space $$L_2$$ with new measures. In particular, we show that the eigenfunction expansion series regarding the convergence behaves in the same way as an ordinary Fourier series.