Fritz Gesztesy and Maxim Zinchenko
On Spectral Theory for Schr\"odinger Operators with Strongly Singular 
Potentials
(167K, LaTeX)

ABSTRACT.  We examine two kinds of spectral theoretic situations:  First, we
recall the case of  self-adjoint half-line Schr\"odinger operators
on [a,\infty), a\in\bbR, with a regular finite end point a and the
case of Schr\"odinger operators on the real line with locally integrable
potentials, which naturally lead to Herglotz functions and 2\times 2
matrix-valued Herglotz functions representing the associated
Weyl-Titchmarsh coefficients. Second, we contrast this with the case
of self-adjoint half-line Schr\"odinger operators on (a,\infty) with
a potential strongly singular at the end point a. We focus on situations
where the potential is so singular that the associated maximally
defined Schr\"odinger operator is self-adjoint (equivalently, the
associated minimally defined Schr\"odinger operator is essentially
self-adjoint) and hence no boundary condition is required at the finite
end point a. For this case we show that the Weyl-Titchmarsh coefficient
in this strongly singular context still determines the associated spectral
function, but ceases to posses the Herglotz property. However, as
will be shown, Herglotz function techniques continue to play a decisive
role in the spectral theory for strongly singular Schr\"odinger operators.