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 $D(A_U) = \{\psi = \phi + \psi_+ + U\psi_+ : \phi \in D(A), \psi_+ \in K_+ \}$.

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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ Oct 28, 2019 at 22:07
  • $\begingroup$ This question arised from a Physics article, so my first thought was to post here, but I'll consider it, thanks! $\endgroup$ Oct 28, 2019 at 22:48
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    $\begingroup$ I found this. $\endgroup$ Oct 29, 2019 at 1:50
  • $\begingroup$ 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. $\endgroup$
    – DanielC
    Oct 29, 2019 at 6:13
  • $\begingroup$ Thank you both! $\endgroup$ Nov 7, 2019 at 21:57

1 Answer 1


Not to leave this nice question unanswered, a truly thorough analysis of the subject of self-adjoint extensions of symmetric operations is offered in the whole 3rd chapter of the book ”Self-adjoint Extensions in Quantum Mechanics. General Theory and Applications to Schrödinger and Dirac Equations with Singular Potentials” by D.M. Gitman, I.V. Tyutin, B.L. Voronov (Birkhäuser, 2010). A link to a Google books excerpt is offered by the comment of @Keith McClary above. If you read very carefully, you can find an answer to your questions.

A shorter presentation would be offered in section X.1 (2nd volume of Reed & Simon), just the first pages from 135 to 141. As they mention in the notes to chapter X, they just polished the presentation from Dunford & Schwarz, Volume 2.

A third option would be section 7.4 of Chapter 12 of Berezanskii & Sheftel's "Functional Analysis", Volume 2 (Birkhäuser, 1996).

  • $\begingroup$ Thank you for the effort, I'll look at it! $\endgroup$ Nov 7, 2019 at 21:57

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