This thesis consists of four papers dealing with generalizations of de Branges`s model theory of (cyclic) self-adjoint operators and applications to the extension theory of symmetric operators.

The main project of this PhD thesis, consisting of three papers, is concerned with constructing operator models for meromorphic functions of bounded type. Specifically, it is shown that these functions can be realized as Q-functions of partially fundamentally reducible relations on Krein spaces in a minimal way. The main result can be found in Paper III, while Papers I and II contain related results of a smaller scope.

The main argument of our construction in Paper 3 can be used to generalize the extension theory for symmetric operators with deficiency index (1,1). Specifically, we characterize all one dimensional extensions with non-empty resolvent set  of such an operator via a Krein-type resolvent formula and investigate their spectral properties. This is the content of Paper IV.