Essential self-adjointness of the SLH-network Hamiltonian

Establish the essential self-adjointness of the network Hamiltonian on the domain obtained by gathering the boundary conditions of the subcomponents in the Gough–James quantum feedback network construction.

Background

Gough and James define quantum feedback networks by summing the Chebotarev–Gregoratti Hamiltonians of the component systems, connecting inputs to outputs to create internal lines, and using the resulting boundary conditions to specify the domain of the network Hamiltonian. Essential self-adjointness on this domain is required to justify the corresponding Hamiltonian dynamics rigorously.

The paper identifies this property as an unresolved conjecture in the earlier Gough–James construction. Instead of resolving it, the paper avoids the issue by defining the network evolution directly through a unitary cocycle from the quantum stochastic differential equation, periodically interrupted by unitary braid maps representing the feedback connections.