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 is symmetric and \(T = T^*\) is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet \(\Pi _S\) for \(S^*\) 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 \(\Pi _A\) for \(A^*\) and the spectral measure of T. An application to 1-D Schrodinger operators is 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–Cummings operator which is regarded as the Hamiltonian of the quantum dot.