Boundary triplets, tensor products and point contacts to reservoirs

TL;DR

This paper develops a boundary triplet framework for symmetric tensor product operators, with applications to quantum transport models involving point contacts and quantum dots.

Contribution

It constructs boundary triplets for tensor product operators that preserve structure and expresses associated functions using component operators, with applications to quantum physics.

Findings

Boundary triplet construction for tensor product operators.

Explicit formulas for gamma-field and Weyl function.

Application to quantum dot electron transport models.

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.