Christoph Neuner: Generalized Tichmarsh-Weyl functions and super singular perturbations

When studying Sturm-Liouville differential expressions there are two fruitful approaches that can be followed to analyse these expressions, namely ODE theory and operator theory. For a certain class of strongly singular potentials, as introduced by Gesztesy and Zinchenko, the ODE side has in the past been better understood insofar as these potentials allow to find a so-called m-function that encapsulates spectral information of the underlying problem. Our aim now is to further understanding on the operator theory side.

On the one hand, we show how we can employ a model for supersingular perturbations to handle perturbed spherical Schrödinger operators, i.e., we deal with potentials of the form \(1/x^2 + q(x)\) where q is a perturbation term. In this model a so-called Q-function appears and it agrees, up to a polynomial term of low degree, with the above m-function. Hence, the approaches match up in this situation.

On the other hand, we also give an indication of the boundaries of the used perturbation model. For a potential of the form \(1/x^4\) the setup breaks down and thus further investigation is needed.