Stability criteria for the Weyl \(m\)-function
The paper presents a new approach to the spectral theory of the Schrödinger operator on the half-line, based on the analysis of Weyl-Titchmarsh \(m\)-function and its boundary values. The investigation is based on the Riccati equation: \({d\over dx}h(x,\lambda)= V(x)-\lambda- (h(x,\lambda))^2\), corresponding to the Schrödinger operator: \(-{d^2\over dx^2}+V(x)\). \(\lambda\) is the (real) spectral parameter. The authors show large \(x\) asymptotics of families of solutions to the Riccati equation, as \(\lambda\) is varied, imply explicit bounds of \(m_+(\lambda)\), the boundary value of the \(m\)-function (or of \(m_\alpha\) if Neumann or other boundary conditions are imposed). A family of solutions exhibiting a so-called \(\delta\)-clustering property is shown to imply precise estimates for the complex boundary values of the Weyl \(m\)-function and the spectral measure, and leads to an analysis of the absolutely continuous component of the spectral measure in terms of stability criteria for the Riccati equation.
scientific article; zbMATH DE number 1247684