Boundary Triplets, Tensor Products and Point Contacts to Reservoirs
2018
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.
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