An example of unitary equivalence between self-adjoint extensions and   their parameters
  The spectral problem for self-adjoint extensions is studied using the machinery of boundary triplets. For a class of symmetric operators having Weyl functions of a special type we calculate explicitly the spectral projections in the form of operator-valued integrals. This allows one to give a constructive proof of the fact that, in certain intervals, the resulting self-adjoint extensions are unitarily equivalent to a certain parameterizing operator acting in a smaller space, and one is able to provide an explicit form the associated unitary transform. Applications to differential operators on metric graphs and to direct sums are discussed. 
Konstantin Pankrashkin
math.FA math-ph math.MP math.SP