We consider symmetric operators of the form S := A ⊗ I T + I H ⊗ 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 ΠS 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 ΠA for A * and the spectral measure of T . Applications to 1-D Schrödinger 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.Mathematics Subject Classification: 47A80, 47B25, 81Q05, 81Q37

We consider a model of point-like interaction between electrons and bosons in a cavity. The electrons are relativistic and are described by a Dirac operator on a bounded interval while the bosons are treated by second quantization. The model fits into the extension theory of symmetric operators. Our main technical tool to handle the model is the so-called boundary triplet approach to extensions of symmetric operators. The approach allows explicit computation of the Weyl function.

This review discusses three types of soft matter and liquid molecular materials, namely hydrogels, liquid crystals and gas bubbles in liquids, which are explored with an emergent machine learning approach....

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