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

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Abstract: 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 operatordomain. In the case \$H 0=-\Delta +V\$ we construct these Hamiltonians explicitlyshowing that they can be written in the form \$H=H 0+ B\$, where \$B\$ is asingular boundary potential and \$H\$ is self-adjoint on its maximal domain. Anapplication to the deformation quantization of one-dimensional systems withboundaries is also presented.

Author: Nuno Costa Dias, Andrea Posilicano, Joao Nuno Prata

Source: https://arxiv.org/