  We build up local, time translation covariant Boundary Quantum Field Theory
nets of von Neumann algebras A_V on the Minkowski half-plane M_+ starting with
a local conformal net A of von Neumann algebras on the real line and an element
V of a unitary semigroup E(A) associated with A. The case V=1 reduces to the
net A_+ considered by Rehren and one of the authors; if the vacuum character of
A is summable A_V is locally isomorphic to A_+. We discuss the structure of the
semigroup E(A). By using a one-particle version of Borchers theorem and
standard subspace analysis, we provide an abstract analog of the Beurling-Lax
theorem that allows us to describe, in particular, all unitaries on the
one-particle Hilbert space whose second quantization promotion belongs to of
E(A^(0)) with A^(0) the U(1)-current net. Each such unitary is attached to a
scattering function or, more generally, to a symmetric inner function. We then
obtain families of models via any Buchholz-Mach-Todorov extension of A^(0). A
further family of models comes from the Ising model.
