Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems Ju'=-zHu and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also discuss the exponential types of the transfer matrices.

We study the minimum of the essential spectrum of canonical systems Ju ′ = −zHu. Our results can be described as a generalized and more quantitative version of the characterization of systems with purely discrete spectrum, which was recently obtained by Romanov and Woracek [6]. Our key tool is oscillation theory.

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 Schrödinger operator on a graph is unitarily equivalent to a canonical system on the same graph. Consequently, for an arbitrary canonical system or Schrödinger operator on a graph, a representation of the resolvent as an integral operator and a spectral representation are obtained.

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