Mateusz Piorkowski: Regularization of Schrödinger operators via Darboux transforms

Abstract:

 In this talk I will show how to associate to a Schrödinger differential expression on I = (a,b) with trace class resolvents a pair of natural numbers (or plus infinity), one for each endpoint a, b, which quantify how singular the operator is at a,b. It turns out that these "regularizations indices" are zero exactly in the so-called limit circle case, which plays a prominent role in the theory of self-adjoint extensions. Moreover, a Darboux transformation always changes the index by \pm 1, implying that a Schrödinger differential expression can be Darboux transformed to a regular expression if and only if both indices are finite.

This talk is based on recent work with Jonathan Stanfill found in arXiv:2407.04847