Mathematics – Spectral Theory
Scientific paper
2005-05-06
Math. Nachr. 279 (2006), 1041-1082
Mathematics
Spectral Theory
50 pages, more details in Sects. 2 and 3 are added
Scientific paper
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.
Gesztesy Fritz
Zinchenko Maxim
No associations
LandOfFree
On Spectral Theory for Schrödinger Operators with Strongly Singular Potentials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Spectral Theory for Schrödinger Operators with Strongly Singular Potentials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Spectral Theory for Schrödinger Operators with Strongly Singular Potentials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-489088