NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2017, 8 (2), P. 166–179
Coupling of definitizable operators in Krein spaces
V. Derkach – Department of Mathematics, Dragomanov National Pedagogical University, Pirogova 9, Kiev, 01601, Ukraine; Department of Mathematics, Vasyl Stus Donetsk National University, 600-Richchya Str 21, Vinnytsya, 21021, Ukraine; derkach.v@gmail.com
C. Trunk – Institut für Mathematik, Technische Universität Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany; carsten.trunk@tuilmenau.de
Indefinite Sturm–Liouville operators defined on R are often considered as a coupling of two semibounded symmetric operators defined on R+ and 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 R.
Keywords: self-adjoint extension, symmetric operator, Krein space, locally definitizable operator, coupling of operators, boundary triple, Weyl function, regular critical point.
DOI 10.17586/2220-8054-2017-8-2-166-179