Self-adjoint extensions and deficiency indices: agreement pointer

Assume the Axiom of Choice (The Axiom of Choice). The extension theorem is proved on the companion A page of this pair: self-adjoint extensions of a closed symmetric operator $T$ correspond bijectively to the unitary operators $K_{+}\to K_{-}$ (Von Neumann parameterization of self-adjoint extensions), and such a unitary exists exactly when the deficiency dimensions agree (Existence of self-adjoint extensions is equality of deficiency indices, Deficiency subspaces and deficiency indices). The parameterization determines the extension's domain and action, not merely the number of extensions, and when a unitary is supplied no further choice is used to produce the extension. Equality of deficiency dimensions by itself does not exhibit a unitary: the Hilbert-basis input producing one is recorded on the A page. The minimal derivative operator of The minimal derivative has deficiency indices (1,1) and many self-adjoint extensions is the one-dimensional instance: the unitaries $K_{+}\to K_{-}$ are there the scalars $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }=\mathrm{\parallel }{e}^{-x}\mathrm{\parallel }\mathrm{/}\mathrm{\parallel }{e}^x\mathrm{\parallel }=1\mathrm{/}e$, in bijection with the boundary parameters $\mu$ of modulus one in the conditions $f(1)=\mu f(0)$.