Coupling of definitizable operators in Krein spaces

Indefinite Sturm–Liouville operators defined on $\mathbb{R}$ are often considered as a coupling of two semibounded symmetric operators defined on $\mathbb{R}^+$ and $\mathbb{R}^-$, respectively. In many situations, those two semibounded symmetric operators have in a special sense good properties like a Hilbert space self-adjoint extension. In this paper, we present an abstract approach to the coupling of two (definitizable) self-adjoint operators. We obtain a characterization for the definitizability and the regularity of the critical points. Finally we study a typical class of indefinite Sturm–Liouville problems on $\mathbb{R}$.