The asymptotic nature of spectral functions in Sturm-Liouville problems with continuous spectrum
The author considers the spectral function \(\rho_\alpha (\mu)\) associated with the Sturm-Liouville equation \[ y''(x)+ \bigl[\lambda- q(x)\bigr] y(x)=0 \] and the boundary condition \(y(0)\cos \alpha+ y'(0)\sin \alpha=0\). The asymptotic behavior of \(\rho_\alpha (\mu)\) as \(\mu\to \infty\) is discussed and an explicit formula is given for the Dirichlet case, \(\alpha=0\), which has not been covered previously. Similar formulas can be obtained with the same methods for all values of \(\alpha\) and hence this paper generalizes results obtained by Atkinson, Harris, Levitan, and Marçenko.
scientific article; zbMATH DE number 1115566
scientific article