Hermitian Symplectic Spaces, von Neumann’s Extension Theory, and Scattering on Quantum Graphs

We begin with the definition of a skew-Hermitian form and the corresponding Hermitian symplectic group. We motivate these definitions with a discussion of their relevance to self-adjoint extensions of Hamiltonian operators. In doing so, we introduce the basics of von Neumann’s extension theory. Next, we develop the necessary tools from Hermitian symplectic linear algebra to study self-adjoint extensions of Hamiltonian operators on simple one-dimensional regions. We apply these concepts to the scattering problem on non-compact quantum star graphs. Further, we suggest an experiment to determine the particular self-adjoint extension at play. Throughout the discussion, we make explicit note of the appearance of the unitary group U(n), as it parametrizes the set of self-adjoint extensions, the Lagrangian Grassmannian and the possible scattering matrices for a non-compact quantum star graph. ∗This material is based upon work supported by the National Science Foundation under agreement No. DMS-1055897. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.