Extensions, dilations and functional models of Dirac operators
The author constructs a boundary value space for the Dirac operator in the limit-circle case. That leads to a description of various classes of extension of the minimal operator (selfadjoint, maximal dissipative, etc.) in terms of boundary conditions at \(\pm \infty\). Two classes of maximal dissipative extensions with separated boundary conditions (at \(+\infty\) and \(-\infty\)) are studied in detail. For each of them, the author constructs a selfadjoint dilation and its incoming and outgoing spectral representations, which makes it possible to determine the scattering matrix within the scheme of Lax and Phillips. On the other hand, these constructions lead to functional models of the operators in question, the calculation of their characteristic functions, and completeness criteria for root vectors of maximal dissipative extensions. A similar program for the one-dimensional Schrödinger operator was implemented by the author and \textit{A.~Canoglu} in [Proc. R. Soc. Edinb., Sect. A, Math. 127, No.~6, 1113--1121 (1997; Zbl 0901.47001)].
scientific article; zbMATH DE number 2176369