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

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

2007-07-06

arxiv.org/abs/0707.0948v4

Communications on Pure and Applied Analysis, 10, 6 (2011) 1687-1706

Physics

Mathematical Physics

25 pages, published version

Scientific paper

For a general self-adjoint Hamiltonian operator $H_0$ on the Hilbert space $L^2(\RE^d)$, we determine the set of all self-adjoint Hamiltonians $H$ on $L^2(\RE^d)$ 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.

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