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arXiv · 1710.07525

Boundary triplets, tensor products and point contacts to reservoirs

A. A. Boitsev·J. F. Brasche·M. M. Malamud·H. Neidhardt·I. Yu. Popov
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Abstract

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 $T$. 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.

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A. A. Boitsev, J. F. Brasche, M. M. Malamud, H. Neidhardt, I. Yu. Popov. 2017-10-20. Boundary triplets, tensor products and point contacts to reservoirs. https://doi.org/10.1007/s00023-018-0698-y

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