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Subordinate solutions and spectral measures of canonical systems. (English) Zbl 0967.34073

Summary: The theory of \(2\times 2\) trace-normed canonical systems of differential equations on \(\mathbb{R}^+\) can be put in the framework of abstract extension theory, cf. S. Hassi, H. S. V. de Snoo and H. Winkler [Boundary value problems for two-dimensional canonical systems, Integral Equations Oper. Theory (to appear)]. This includes the theory of strings as developed by I. S. Kac and M. G. Kreĭn.
In the present paper, the spectral properties of such canonical systems are characterized by means of subordinate solutions. The theory of subordinacy for Schrödinger operators on the halfline \(\mathbb{R}^+\), was originally developed by D. J. Gilbert and D. B. Pearson. Its extension to the framework of canonical systems makes it possible to describe the spectral measure of any Nevanlinna function in terms of subordinate solutions to the corresponding trace-normed canonical system, which is uniquely determined by a result of L. de Branges.

MSC:

34L05 General spectral theory of ordinary differential operators
34B20 Weyl theory and its generalizations for ordinary differential equations
47B25 Linear symmetric and selfadjoint operators (unbounded)
47E05 General theory of ordinary differential operators
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