Self-Adjoint Extensions of Bipartite Hamiltonians

Abstract

We compute the deficiency spaces of operators of the form $H_A\hat{\otimes }I+I\hat{\otimes }{H}_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already by Ibort, Marmo and Pérez-Pardo, but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.