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

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  • For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(R^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(R^d)$ that dynamically confine the system to an open set $\Omega \subset \RE^d$ while reproducing the action of $ H_0$ on an appropriate operator domain. In the case $H_0=-\Delta +V$ 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.
    Mathematics Subject Classification: Primary: 81Q10, 47B25; Secondary: 81S30.


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