J. Behrndt and M. Langer,
On the adjoint of a symmetric operator,
J. Lond. Math. Soc. (2) 82 (2010), 563–580

Abstract:
In general it is a non-trivial task to determine the adjoint S* of an unbounded symmetric operator S in a Hilbert or Krein space. We propose a method to specify S* explicitly which makes use of two boundary mappings that satisfy an abstract Green's identity and a surjectivity condition, and give rise to a self-adjoint extension of S. We show for various concrete examples how convenient and easily applicable this technique is.

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