# References of Deficiency indices theorem (von Neumann)

I am looking for proof or some interpretation around why the domain of the new extension $$D(A_U)$$ in the Theorem below is given by its specific formula.

I have already searched in papers and here but I could find only applications of the Theorem. Does anybody know a reference with it or can give me an idea of the behavior of extension domain $$D(A_U)$$?

Theorem: Let $$A$$ be a Hermitian, densely defined, and closed operator, $$A^\dagger$$ its conjugate transpose, $$K_\pm = \{ \psi_\pm \in D(A^\dagger) : A^\dagger \psi_\pm = \pm i\eta\psi_\pm \}$$ and $$n_\pm = \dim (K_\pm)$$. Then:

1) if $$n_+ = n_- = 0$$, A is self-adjoint;

2) if $$n_+ \neq n_-$$, A is not self-adjoint and it has no self-adjoint extensions;

3) if $$n_+ = n_- > 0$$, A has infinitely many self-adjoint extensions $$A_U$$ parametrized by $$n \times n$$ unitary matrix maps $$U: K_+ \rightarrow K_-$$. For a choice of $$U$$, $$D(A) \subset D(A_U) \subset D(A^\dagger)$$, and $$D(A_U) = \{\psi = \phi + \psi_+ + U\psi_+ : \phi \in D(A), \psi_+ \in K_+ \}$$.

• Would Mathematics be a better home for this question? Oct 28, 2019 at 22:07
• This question arised from a Physics article, so my first thought was to post here, but I'll consider it, thanks! Oct 28, 2019 at 22:48
• I found this. Oct 29, 2019 at 1:50
• This is a question in pure mathematics, albeit with applications in QM. I remember this theorem, aside from the book by Tyutin and Gitman referenced by @KeithMcClary, also in the Hilbert Space Methods by Blank and Exner. Oct 29, 2019 at 6:13
• Thank you both! Nov 7, 2019 at 21:57