Harald Woracek: A function space model for certain canonical systems with two singular endpoints

Abstract:

We consider \(2\)-dimensional canonical systems on the half-line with a positive semidefinite Hamiltonian, and assume that at the right endpoint Weyl's limit point case takes place (i.e., that the Hamiltonian is not integrable towards that endpoint). The selfadjoint operator A associated with the system is known to have spectral multiplicity at most \(2\).

(lcc): If the Hamiltonian is integrable towards the left endpoint (terminology: limit circle case), then A has simple spectrum and one scalar spectral measure can be found via the integral representation of the Weyl coefficient, which is a Nevanlinna function (i.e., maps the upper half-plane analytically into itself).

(lpc): If at the left endpoint limit point case takes place, A may have spectral multiplicity \(1\) or \(2\). The case that we still have simple spectrum can be characterised, but in general one doesn't have a construction of a scalar spectral measure in such a canonical way as in lcc. However, if the Hamiltonian (which now is not integrable towards the left endpoint) grows only very moderately, then an analogue of the Weyl coefficient can be constructed, and again we find a canonical way to obtain a scalar spectral measure.

A nice scale of examples which illustrates these phenomena is given by the Bessel equation and we will briefly discuss this, and related, examples.

The construction of the analogue of the Weyl coefficient rests on a more general theory about an operator model acting in a Pontryagin space instead of a Hilbert space. This operator model is rather complicated and not very explicit. However, in the above outlined situation, one can give a more concrete description of the model and we will discuss this model.