Boundary triplets, tensor products and point contacts to reservoirs

Abstract

We consider symmetric operators of the form $S:=A\otimes I_{\mathfrak{T}}+I_{\mathfrak{H}}\otimes T$ where $A$ is symmetric and $T=T^{\ast }$ is (in general) unbounded. Such operators naturally arise in problems of simulating point contacts to reservoirs. We construct a boundary triplet ${\Pi }_S$ for $S^{\ast }$ 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 $A^{\ast }$ 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.

References in corpus (4)

• Spectra of self-adjoint extensions and applications to solvable Schroedinger operators
• Schrödinger operators with delta and delta'-potentials supported on hypersurfaces
• On the spectral theory of Gesztesy-Šeba realizations of 1-D Dirac operators with point interactions on a discrete set
• Electron transport through a quantum dot assisted by cavity photons

Cited by in corpus (1)

• Exactly solvable Schrödinger operators related to the confluent equation