Self-adjoint, globally defined Hamiltonian operators for systems with boundaries

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2({\text{RE}}^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2({\text{RE}}^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$whilereproducingtheactionof$ H_0$ on an appropriate operator domain. In the case $H_0=-\mathrm{\Delta }+V$ we construct these Hamiltonians explicitly showing that they can be written in the form $H=H_0+B$, where $B$ is a singular boundary potential and $H$ is self-adjoint on its maximal domain. An application to the deformation quantization of one-dimensional systems with boundaries is also presented.