Boundary Triplets, Tensor Products and Point Contacts to Reservoirs

We consider symmetric operators of the form S: = A⊗ IT+ IH⊗ 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 Π S for S∗ preserving the tensor structure. The corresponding γ-field and Weyl function are expressed by means of the γ-field and Weyl function corresponding to the boundary triplet Π A for A∗ and the spectral measure of T. An application to 1-D Schrödinger 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. © 2018, Springer International Publishing AG, part of Springer Nature.