Extreme self‐adjoint extensions of a semibounded ‐difference operator

Abstract

For a certain q‐difference operator introduced and studied in a series of articles by the same authors, we investigate its extreme self‐adjoint extensions, i.e., the so‐called Friedrichs and Kreĭn extensions. We show that for the interval of parameters under consideration, the Friedrichs extension and the Kreĭn extension are distinct and give values of the parameter in the von Neumann formulas that correspond to those extensions and describe their resolvent operators. A crucial rôle in our investigation plays the fact that both the Friedrichs and the Kreĭn extensions are scale invariant.