The spectral function for Sturm-Liouville problems where the potential is of Wigner-von Neumann type or slowly decaying
The spectral functions \(\rho_0(\lambda)\) of Sturm-Liouville operators of the form \(-d^2/dx^2+ q\) in \(L^2[0,\infty)\) are investigated. Here, \(q\) is real-valued, continuously differentiable, \(q(x)\to 0\) as \(x\to\infty\) but \(q\not\in L^1[0,\infty)\). The domain of the Sturm-Liouville operator is specified via Dirichlet boundary conditions at \(x= 0\). One derives conditions which are sufficient for \(\rho''(A)\) to exist and to satisfy \(\rho''(\lambda)> 0\) for every \(\lambda> \Lambda_1\) for some \(\Lambda_1> 0\). In particular, there do not exist points of spectral concentration of the Sturm-Liouville operator on \((\Lambda_1,\infty)\). Moreover, one derives conditions which are sufficient for \(\rho(\lambda)\) to have an absolutely and uniformly convergent series representation for every \(\lambda> \Lambda_0\) for some \(\Lambda_0> 0\), so that \(\rho'(\lambda)\) exists for every \(\lambda> \Lambda_0\) and hence, the spectrum of the Sturm-Liouville operator is purely absolutely continuous on \((\Lambda_0,\infty)\). Both \(\Lambda_0\) and \(\Lambda_1\) can be computed explicitly. The computation is performed for a variety of potentials \(q\), for instance Wigner-von Neumann potentials.
scientific article; zbMATH DE number 2104529