Self-Adjoint Extensions of Bipartite Hamiltonians

TL;DR

This paper analyzes the deficiency spaces of bipartite Hamiltonians composed of symmetric and self-adjoint operators, enabling the construction of their self-adjoint extensions using von Neumann's theory, extending previous results.

Contribution

It generalizes the structure of deficiency spaces for bipartite Hamiltonians beyond the case where one operator has a discrete, non-degenerate spectrum.

Findings

Computed deficiency spaces for a class of bipartite Hamiltonians.

Established conditions for the existence of self-adjoint extensions.

Extended previous results to more general operator spectra.

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 P\'erez-Pardo, but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.