An example of unitary equivalence between self-adjoint extensions and their parameters

TL;DR

This paper explores the spectral properties of self-adjoint extensions of symmetric operators using boundary triplets, providing explicit constructions of unitary equivalences and applications to differential operators on metric graphs.

Contribution

It offers a constructive method to explicitly realize unitary equivalences between self-adjoint extensions and their parameters for a specific class of operators.

Findings

Explicit formulas for spectral projections as operator-valued integrals

Constructive proof of unitary equivalence in certain intervals

Applications to differential operators on metric graphs

Abstract

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.