Christoph Neuner: On supersingular perturbations

Abstract: This thesis consists of four papers and deals with supersingular rank one perturbations of self-adjoint operators and their models in Hilbert or Pontryagin spaces. Here, the term supersingular describes perturbation elements that are outside the underlying space but still obey a certain regularity conditions.

The first two papers study certain Sturm-Liouville differential expressions that can be realised as Schrödinger operators.
In Paper I we show that for the potential consisting of the inverse square plus a comparatively well-behaved term we can employ an existing model due to Kurasov to describe these operators in a Hilbert space. In particular, this approach is in good agreement with ODE techniques.

In Paper II we study the inverse fourth power potential.
While it is known that the ODE techniques still work, we show that the above model fails and thus that there are limits to the above operator theoretic approach.

In Paper III we concentrate on generalising Kurasov's model.
The original formulation assumes that the self-adjoint operator is semi-bounded, whereas we drop this requirement. We give two models with a Hilbert and Pontryagin space structure, respectively, and study the connections between the resulting constructions.

Finally, in Paper IV, we consider the concrete case of the operator of multiplication by the independent variable, a self-adjoint operator whose spectrum covers the real line, and study its perturbations. This illustrates some of the formalism that was developed in the previous paper, and a number of more explicit results are obtained, especially regarding the spectra of the appearing perturbed operators.