Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic   differential operators
  The classical concept of $Q$-functions associated to symmetric and selfadjoint operators due to M.G. Krein and H. Langer is extended in such a way that the Dirichlet-to-Neumann map in the theory of elliptic differential equations can be interpreted as a generalized $Q$-function. For couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit Krein type formulas for the difference of the resolvents and trace formulas in an $H^2$-framework are obtained. 
Daniel Alpay and Jussi Behrndt