Advanced counting: we do it in spectral gaps!!!
What the zeros of solutions of Schrödinger's equation tell us about Krein's spectral shift function.

Helge Krüger
(University of Vienna)

Abstract: Remembering that we learnt in elementary school to count: one, two, three, four, and so on, we will add a new twist to counting and call it advanced counting for further references. Instead of starting with one, we will start with zero.

Then we will illustrate the concept of advanced counting by applications to the eigenvalue problem for Schrödinger operators. We will show what counting the number of zeros of a solution tells us about eigenvalues, and after it we will apply the same method to Wronskians, obtaining additional information about eigenvalues. Another big difference is that the method involving Wronskians is applicable in spectral gaps. For technical reasons, our results Krein's spectral shift function, on which a few words will be lost.