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.