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 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.

Symmetric and antisymmetric tensor products for the function-theoretic operator theorist

“…In the simplest case, where the systems do not interact, the Hamiltonian of the many-body system corresponds to a symmetric tensor product of operators [20,Chapter 4,Section 9]. Recently, there has been an endeavor within the physics community to study self-adjoint extensions of symmetric tensor products of operators [18,19,22]. Furthermore, the symmetric part of a quantum geometric tensor can be exploited as a tool to detect quantum phase transitions in PT-symmetric quantum mechanics [34].…”