Abstract:
The representation of the resolvent as an integral operator,
the m function, and the associated spectral representation
are fundamental topics in the spectral theory of self-adjoint ordinary
differential operators. Versions of these are developed here
for canonical systems Ju'=-zHu of arbitrary order. A classical
result shows that canonical systems of order two can be used to
realize arbitrary spectral data, in the form of m functions from
the upper half plane to itself. In this paper, canonical systems on
graphs, not necessarily compact but with finitely many vertices,
are introduced and proved to be unitarily equivalent to certain
higher order canonical systems. It is shown that any Schr¨odinger
operator on a graph is unitarily equivalent to a canonical system
on the same graph. Consequently, for an arbitrary canonical system
or Schr¨odinger operator on a graph, a representation of the
resolvent as an integral operator and a spectral representation are
obtained.