Boundary triplets, tensor products and point contacts to reservoirs

We consider symmetric operators of the form $S := A\otimes I_{\mathfrak T} + I_{\mathfrak H} \otimes T$where$A$issymmetricand$T = T^*$ is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet ${\mathrm{\Pi }}_S$ for $S^{\ast }$ preserving the tensor structure. The corresponding $\gamma$-field and Weyl function are expressed by means of the $\gamma$-field and Weyl function corresponding to the boundary triplet ${\mathrm{\Pi }}_A$ for $A^{\ast }$ and the spectral measure of $T$. Applications to 1-D Schr"odinger and Dirac operators are given. A model of electron transport through a quantum dot assisted by cavity photons is proposed. In this model the boundary operator is chosen to be the well-known Jaynes-Cumming operator which is regarded as the Hamiltonian of the quantum dot.