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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 in Ibort et al. [Boundary dynamics driven entanglement, J. Phys. A: Math. Theor.47(38) (2014) 385301], but only proven under the restriction of $H_B$
having discrete, non-degenerate spectrum.
